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Given a graph, the general problem to cover the maximum number of vertices by a collection of vertex-disjoint long paths seems to escape from the literature. A path containing at least k vertices is considered long. When k ≤ 3 , the problem is polynomial time solvable; when k is the total number of vertices, the problem reduces to the Hamiltonian path problem, which is NP-complete. For a fixed k ≥ 4 , the problem is NP-hard and the best known approximation algorithm for the weighted set packing problem implies a k-approximation algorithm. To the best of our knowledge, there is no approximation algorithm directly designed for the general problem; when k = 4 , the problem admits a 4-approximation algorithm which was presented recently. We propose the first (0.4394 k + O (1)) -approximation algorithm for the general problem and an improved 2-approximation algorithm when k = 4 . Both algorithms are based on local improvement, and their theoretical performance analyses are done via amortization and their practical performance is examined through simulation studies. [ABSTRACT FROM AUTHOR]

Parameterization above (or below) a guarantee is a successful concept in parameterized algorithms. The idea is that many computational problems admit "natural" guarantees bringing to algorithmic questions whether a better solution (above the guarantee) could be obtained efficiently. For example, for every boolean CNF formula on m clauses, there is an assignment that satisfies at least m/2 clauses. How difficult is it to decide whether there is an assignment satisfying more than m / 2 + k clauses? Or, if an n-vertex graph has a perfect matching, then its vertex cover is at least n/2. Is there a vertex cover of size at least n / 2 + k for some k ≥ 1 and how difficult is it to find such a vertex cover? The above guarantee paradigm has led to several exciting discoveries in the areas of parameterized algorithms and kernelization. We argue that this paradigm could bring forth fresh perspectives on well-studied problems in approximation algorithms. Our example is the longest cycle problem. One of the oldest results in extremal combinatorics is the celebrated Dirac's theorem from 1952. Dirac's theorem provides the following guarantee on the length of the longest cycle: for every 2-connected n-vertex graph G with minimum degree δ (G) ≤ n / 2 , the length of a longest cycle L is at least 2 δ (G) . Thus the "essential" part in finding the longest cycle is in approximating the "offset" k = L - 2 δ (G) . The main result of this paper is the above-guarantee approximation theorem for k. Informally, the theorem says that approximating the offset k is not harder than approximating the total length L of a cycle. In other words, for any (reasonably well-behaved) function f, a polynomial time algorithm constructing a cycle of length f(L) in an undirected graph with a cycle of length L, yields a polynomial time algorithm constructing a cycle of length 2 δ (G) + Ω (f (k)) . [ABSTRACT FROM AUTHOR]

k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.82, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm by Agarwal and Procopiuc [Algorithmica 2002] yields an O (n log k) + (1 / ϵ) O (2 d k 1 - 1 / d log k) -time (1 + ϵ) -approximation for Euclidean k-center, where d is the dimension. We show for a closely related problem, k-supplier, the double-exponential dependence on dimension is unavoidable if one hopes to have a sub-linear dependence on k in the exponent. We also derive similar algorithmic results to the ones by Agarwal and Procopiuc for both k-center and k-supplier. We use a relatively new tool, called Voronoi separator, which makes our algorithms and analyses substantially simpler. Furthermore we consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2 O (k log k) n 2 time 3-approximation for NUkC in general metrics, and a 2 O ((k log k) / ϵ) d n time (1 + ϵ) -approximation for Euclidean NUkC. The latter time bound matches the bound for k-center. [ABSTRACT FROM AUTHOR]

We study rectangle stabbing problems in which we are given n axis-aligned rectangles in the plane that we want to stab, that is, we want to select line segments such that for each given rectangle there is a line segment that intersects two opposite edges of it. In the horizontal rectangle stabbing problem (Stabbing), the goal is to find a set of horizontal line segments of minimum total length such that all rectangles are stabbed. In the horizontal–vertical stabbing problem (HV-Stabbing), the goal is to find a set of rectilinear (that is, either vertical or horizontal) line segments of minimum total length such that all rectangles are stabbed. Both variants are NP-hard. Chan et al. (ISAAC, 2018) initiated the study of these problems by providing constant approximation algorithms. Recently, Eisenbrand et al. (A QPTAS for stabbing rectangles, 2021) have presented a QPTAS and a polynomial-time 8-approximation algorithm for Stabbing, but it was open whether the problem admits a PTAS. In this paper, we obtain a PTAS for Stabbing, settling this question. For HV-Stabbing, we obtain a (2 + ε) -approximation. We also obtain PTASs for special cases of HV-Stabbing: (i) when all rectangles are squares, (ii) when each rectangle's width is at most its height, and (iii) when all rectangles are δ -large, that is, have at least one edge whose length is at least δ , while all edge lengths are at most 1. Our result also implies improved approximations for other problems such as generalized minimum Manhattan network. [ABSTRACT FROM AUTHOR]

In the non-uniform sparsest cut problem, we are given a supply graph G and a demand graph D, both with the same set of nodes V. The goal is to find a cut of V that minimizes the ratio of the total capacity on the edges of G crossing the cut over the total demand of the crossing edges of D. In this work, we study the non-uniform sparsest cut problem for supply graphs with bounded treewidth k. For this case, Gupta et al. (ACM STOC, 2013) obtained a 2-approximation with polynomial running time for fixed k, and it remained open the question of whether there exists a c-approximation algorithm for a constant c independent of k, that runs in FPT time. We answer this question in the affirmative. We design a 2-approximation algorithm for the non-uniform sparsest cut with bounded treewidth supply graphs that runs in FPT time, when parameterized by the treewidth. Our algorithm is based on rounding the optimal solution of a linear programming relaxation inspired by the Sherali-Adams hierarchy. In contrast to the classic Sherali-Adams approach, we construct a relaxation driven by a tree decomposition of the supply graph by including a carefully chosen set of lifting variables and constraints to encode information of subsets of nodes with super-constant size, and at the same time we have a sufficiently small linear program that can be solved in FPT time. [ABSTRACT FROM AUTHOR]

We consider the weighted completion time minimization problem for capacitated parallel machines, which is a fundamental problem in modern cloud computing environments. In our setting, the processed jobs may be of varying duration, require different resources, and be of unequal importance (weight). Each server (machine) can process multiple concurrent jobs up to its capacity. We study heuristic approaches with provable approximation guarantees and offer an algorithm that prioritizes the jobs with the smallest volume-by-weight ratio. We bound the algorithm's approximation ratio using a decreasing function of the ratio between the highest resource demand of any job and the server's capacity. Thereafter, we create a hybrid, constant approximation algorithm for two or more machines. We also develop a constant approximation algorithm for the case of a single machine. Via a numerical study and a mixed-integer linear program of the problem, we demonstrate the performance of the suggested algorithm with respect to the optimal solutions and alternative scheduling methods. We show that the suggested scheduling method can be applied to both offline and online problems that may arise in real-world settings. This research is the first, to the best of our knowledge, to propose a polynomial-time algorithm with a constant approximation ratio for minimizing the weighted sum of job completion times for capacitated parallel machines. [ABSTRACT FROM AUTHOR]

We present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant Δ , and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and 1 + Δ . Our goal is to maximize the number of near exact covered bins. If Δ = 0 or Δ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that P ≠ N P ). However, for the case where Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution. [ABSTRACT FROM AUTHOR]

In the Determinant Maximization problem, given an n × n positive semi-definite matrix A in Q n × n and an integer k, we are required to find a k × k principal submatrix of A having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a f (k) n O (1) -time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the W[1]-hardness with respect to the rankr of an input matrix. Our result is stronger than Koutis' result in the sense that any k × k principal submatrix is singular whenever k > r . We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of 2 - c k for some universal constant c > 0 . Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an ε -additive approximation algorithm that runs in ε - r 2 · r O (r 3) · n O (1) time for the rank r of an input matrix, provided that the diagonal entries are bounded. [ABSTRACT FROM AUTHOR]

We study two scheduling problems in a proportionate flow shop environment, where job processing times are machine independent. In contrast to classical proportionate flow shop models, we assume (in both problems) that processing times are step-deteriorating. Accordingly, each job J j has a normal processing time, a j , if it starts to be processed in the shop no later than its deteriorating date, δ j . Otherwise, the job's processing time increases by b j (the job's deterioration penalty). Our aim is to find a job schedule that minimizes either the makespan or the total load. These two problems are known to be N P -hard for the special case of a single machine, even when all jobs have the same deteriorating date. In this paper, we derive several positive results in relation to the two problems. We first show that the two problems can be represented in a unified way. We then prove that the unified problem is only ordinary N P -hard by providing a pseudo-polynomial time algorithm for its solution. We also show that the pseudo-polynomial time algorithm can be converted into a fully polynomial time approximation scheme (FPTAS). Finally, we analyze the parameterized complexity of the problem with respect to the number of different deteriorating dates in the instance, v δ . We show that although the problem is N P -hard when v δ = 1 , it is fixed parameterized tractable (FPT) for the combined parameters (i) (ν δ , ν a) and (ii) (ν δ , ν b) , where ν a is the number of different normal processing times in the instance, and ν b is the number of different deterioration penalties in the instance. [ABSTRACT FROM AUTHOR]

In 2015, Haviv introduced the Remote set problem (RSP) and studied the complexity of the covering radius problem (CRP), which is a classical problem in lattices. The RSP aims to identify a set containing a point that is sufficiently distant from a given lattice L . It introduced a new method for analyzing the complexity of CRP. An open question in RSP is whether we can obtain the approximation factor γ = 1 / 2 . This paper investigates this question and proposes a probabilistic polynomial-time algorithm for RSP with an approximation factor of 1 / 2 - 1 / (c λ n (p)) , where c ∈ Z + and λ n (p) is the n-th successive minima in lattice under l p -norm. For a given lattice L with rank n and positive integer d, our algorithm outputs a set S of size d in polynomial time. This set S includes a point at least (1 2 - 1 c λ n (p) ) ρ (p) (L) from lattice L with a probability greater than 1 - 1 / 2 d . Here, c is a positive integer and ρ (p) (L) denotes the covering radius of L in l p -norm( 1 ≤ p ≤ ∞ ). Based on this, we obtain that GAPCRP 2 + 1 / 2 O (n) belongs to the complexity class coRP, and we provide new reductions from GAPCRP to GAPCVP. [ABSTRACT FROM AUTHOR]

This paper constructs an interval partition linearization algorithm for solving minimax linear fractional programming (MLFP) problem. In this algorithm, MLFP is converted and decomposed into a series of linear programs by dividing the outer 1-dimensional space of the equivalent problem (EP) into polynomially countable intervals. To improve the computational efficiency of the algorithm, two new acceleration techniques are introduced, in which the regions in outer space where the optimal solution of EP does not exist are largely deleted. In addition, the global convergence of the proposed algorithm is summarized, and its computational complexity is illustrated to reveal that it is a fully polynomial time approximation scheme. Finally, the numerical results demonstrate that the proposed algorithm is feasible and effective. [ABSTRACT FROM AUTHOR]

This paper introduces the d-distanceb-matching problem, in which we are given a bipartite graph G = (S , T ; E) with S = { s 1 , ⋯ , s n } , a weight function on the edges, an integer d ∈ Z + and a degree bound function b : S ∪ T → Z + . The goal is to find a maximum-weight subset M ⊆ E of the edges satisfying the following two conditions: (1) the degree of each node v ∈ S ∪ T is at most b(v) in M, (2) if s i t , s j t ∈ M , then | i - j | ≥ d . In the cyclic version of the problem, the nodes in S are considered to be in cyclic order. We get back the (cyclic)d-distance matching problem when b (s) = 1 for s ∈ S and b (t) = ∞ for t ∈ T . We prove that the d-distance matching problem is APX-hard, even in the unweighted case. We show that 2 - 1 d is a tight upper bound on the integrality gap of the natural integer programming model for the cyclic d-distance b-matching problem provided that (2 d - 1) divides the size of S. For the non-cyclic case, the integrality gap is shown to be at most (2 - 2 d) . The proofs give approximation algorithms with guarantees matching these bounds, and also improve the best known algorithms for the (cyclic) d-distance matching problem. In a related problem, our goal is to find a permutation of S maximizing the weight of the optimal d-distance b-matching. This problem can be solved in polynomial time for the (cyclic) d-distance matching problem — even though the (cyclic) d-distance matching problem itself is NP-hard and also hard to approximate arbitrarily. For (cyclic) d-distance b-matchings, however, we prove that finding the best permutation is NP-hard, even if b ≡ 2 or d = 2 , and we give e-approximation algorithms. [ABSTRACT FROM AUTHOR]

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by n , m , w max , and w min the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T 0 ≥ w max and multiplicative cooling schedule with factor 1 - 1 / ℓ , where ℓ = ω (m n ln (m)) , with probability at least 1 - 1 / m computes in time O (ℓ (ln ln (ℓ) + ln (T 0 / w min))) a spanning tree with weight at most 1 + κ times the optimum weight, where 1 + κ = (1 + o (1)) ln (ℓ m) ln (ℓ) - ln (m n ln (m)) . Consequently, for any ϵ > 0 , we can choose ℓ in such a way that a (1 + ϵ) -approximation is found in time O ((m n ln (n)) 1 + 1 / ϵ + o (1) (ln ln n + ln (T 0 / w min))) with probability at least 1 - 1 / m . In the special case of so-called (1 + ϵ) -separated weights, this algorithm computes an optimal solution (again in time O ((m n ln (n)) 1 + 1 / ϵ + o (1) (ln ln n + ln (T 0 / w min))) ), which is a significant speed-up over Wegener's runtime guarantee of O (m 8 + 8 / ϵ) . Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the (1 + 1) EA can lead to stronger runtime guarantees than either algorithm alone. [ABSTRACT FROM AUTHOR]

In this paper, we study restricted variations of the following obnoxious facility location problem in the plane: locate k new obnoxious facilities amidst n ordered demand points (existing facility sites) in the plane so that none of the existing facility sites are affected. We study this problem in two restricted settings. The obnoxious facilities are constrained to be centered on (i) a predetermined horizontal line segment pq ¯ , and (ii) the boundary arc of a predetermined disk C . For the problem in (i), we first give an (1 - ϵ) -approximation algorithm that runs in O (n + k) log | | p q | | 2 (k - 1) ϵ time, where ϵ > 0 and ||pq|| is the length of pq ¯ . We then propose two exact polynomial time algorithms, one based on doing binary search on all candidates and the other with the parametric search technique of Megiddo (J ACM 30:852–865, 1983); that will take O (n k) 2 time and O (n + k) 2 time respectively. Continuing further, using the improved parametric technique, we give an O (n log n) -time algorithm for k = 2 . We also show that the (1 - ϵ) -approximation algorithm of (i) can be easily adapted to solve the problem (ii) with an extra multiplicative factor of n in the running time. Finally, we give a O (n 3 k) -time dynamic programming solution to the min-sum obnoxious facility location problem restricted to a line segment where the demand points are associated with weights, and the goal is to minimize the sum of the weights of the points covered. [ABSTRACT FROM AUTHOR]

We study several two-machine shop scheduling problems, namely flow shop, job shop and open shop scheduling problems under linear constraints. In these problems, the processing times of two stages of jobs are also decision variables and satisfy a system of linear constraints. The goal of each problem is to determine the processing time of each job, and to schedule the jobs to the shop machine such that the makespan, i.e., the completion time of all jobs, is minimized. These problems can find application in various areas, such as industrial production, advertising and automotive maintenance. We study the computational complexity and propose polynomial-time optimal or approximation algorithms for them. In particular, we show that although a two-machine flow shop scheduling problem and a two-machine job shop scheduling problem without recirculation can be solved in polynomial time, the problems where processing times satisfy linear constraints are generally NP-hard in the strong sense. Then, we design algorithms for various settings of these problems. We design polynomial-time algorithms for them when there are a fixed number of constraints. For the general case, we first propose a simple 2-approximation algorithm, and then design a polynomial-time approximation schemes. In contrast to the previous two problems, we show that the two-machine open shop scheduling problem under linear constraints can be solved in polynomial time. [ABSTRACT FROM AUTHOR]

A graph is called a sum graph if its vertices can be labelled by distinct positive integers such that there is an edge between two vertices if and only if the sum of their labels is the label of another vertex of the graph. Most papers on sum graphs consider combinatorial questions like the minimum number of isolated vertices that need to be added to a given graph to make it a sum graph. In this paper, we initiate the study of sum graphs from the viewpoint of computational complexity. Notice that every n-vertex sum graph can be represented by a sorted list of n positive integers where edge queries can be answered in O (log n) time. Therefore, upper-bounding the numbers used as vertex labels also upper-bounds the space complexity of storing the graph in the database. We show that every n-vertex, m-edge, d-degenerate graph can be made a sum graph by adding at most m isolated vertices to it, such that the largest numbers used as vertex labels grows as O (n 2 d) . This enables us to store the graph using O (m log n) bits of memory. For sparse graphs (graphs with O (n) edges), this matches the trivial lower bound of Ω (n log n) . As planar graphs and forests have constant degeneracy, our result implies an upper bound of O (n 2) on their label numbers. The previously best known upper bound on the numbers needed for labelling general graphs with the minimum number of isolated vertices was O (4 n) , due to Kratochvíl, Miller & Nguyen (2001). Furthermore, their proof was existential, whereas our labelling can be constructed in polynomial time. [ABSTRACT FROM AUTHOR]

In the Strip Packing problem, we are given a vertical half-strip [ 0 , W ] × [ 0 , + ∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. It is NP-hard to approximate this problem within a factor (3 / 2 - ε) for any constant ε > 0 by a simple reduction from the Partition problem, while the current best approximation factor for it is (5 / 3 + ε) . It seems plausible that Strip Packing admits a (3 / 2 + ε) -approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ > 0 , we call large the rectangles with width at least δ W and height at least δ O P T , and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, the skewed case is still NP-hard to approximate within a factor (3 / 2 - ε) , and we provide an (almost) tight (3 / 2 + ε) -approximation algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we give an algorithm that finds an ϵ -approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in polynomial time if the rank of the quadratic function and the number of integer variables are fixed. The running time of the algorithm is expected unless P = NP. In order to design this algorithm we introduce the novel concepts of spherical form MIQP and of aligned vectors, and we provide a number of results of independent interest. In particular, we give a strongly polynomial algorithm to find a symmetric decomposition of a matrix, and show a related result on simultaneous diagonalization of matrices. [ABSTRACT FROM AUTHOR]

We study algorithmic Bayesian persuasion problems in which the principal (a.k.a. the sender) has to persuade multiple agents (a.k.a. receivers) by using public communication channels. Specifically, our model follows the multi-receiver model with no inter-agent externalities introduced by Arieli and Babichenko (J Econ Theory 182:185–217, 2019). It is known that the problem of computing a sender-optimal public persuasive signaling scheme is not approximable even in simple settings. Therefore, prior works usually focus on determining restricted classes of the problem for which efficient approximation is possible. Typically, positive results in this space amounts to finding bi-criteria approximation algorithms yielding an almost optimal and almost persuasive solution in polynomial time. In this paper, we take a different perspective and study the persuasion problem in the general setting where the space of the states of nature, the action space of the receivers, and the utility function of the sender can be arbitrary. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling scheme in such settings. In particular, we show that, assuming the Exponential Time Hypothesis, solving this problem requires at least a quasi-polynomial number of steps even in instances with simple utility functions and binary action spaces such as an election with the k-voting rule. In doing so, we prove that a relaxed version of the Maximum Feasible Subsystem of Linear Inequalities problem requires at least quasi-polynomial time to be solved. Finally, we close the gap by providing a quasi-polynomial time bi-criteria approximation algorithm for arbitrary public persuasion problems that, under mild assumptions, yields a QPTAS. [ABSTRACT FROM AUTHOR]

We consider scheduling monotonic moldable tasks on multiple platforms, where each platform contains a set of processors. A moldable task can be split into several pieces of equal size and processed simultaneously on multiple processors. Tasks are not allowed to be processed spanning over platforms. This scheduling model has many applications, ranging from parallel computing to the berth and quay crane allocation and the workforce assignment problem. We develop several approximation algorithms aiming at minimizing the makespan. More precisely, we provide a 2-approximation algorithm for identical platforms, a Fully Polynomial Time Approximation Scheme (FPATS) under the assumption of large processor counts and a 2-approximation algorithm for a fixed number of heterogeneous platforms. Most of the proposed algorithms combine a dual approximation scheme with a novel approach to improve the dual approximation algorithm. All results can be extended to the contiguous case, i.e., a task can only be executed by contiguously numbered processors. [ABSTRACT FROM AUTHOR]

We give a 2-approximation algorithm for the Maximum Agreement Forest problem on two rooted binary trees. This NP-hard problem has been studied extensively in the past two decades, since it can be used to compute the rooted Subtree Prune-and-Regraft (rSPR) distance between two phylogenetic trees. Our algorithm is combinatorial and its running time is quadratic in the input size. To prove the approximation guarantee, we construct a feasible dual solution for a novel exponential-size linear programming formulation. In addition, we show this linear program has a smaller integrality gap than previously known formulations, and we give an equivalent compact formulation, showing that it can be solved in polynomial time. [ABSTRACT FROM AUTHOR]

The Unsplittable Flow Cover problem (UFP-cover) models the well-studied general caching problem and various natural resource allocation settings. We are given a path with a demand on each edge and a set of tasks, each task being defined by a subpath and a size. The goal is to select a subset of the tasks of minimum cardinality such that on each edge e the total size of the selected tasks using e is at least the demand of e. There is a polynomial time 4-approximation for the problem (Bar-Noy et al. STOC 2001) and also a QPTAS (Höhn et al. ICALP 2018). In this paper we study fixed-parameter algorithms for the problem. We show that it is W[1]-hard but it becomes FPT if we can slighly violate the edge demands (resource augmentation) and also if there are at most k different task sizes. Then we present a parameterized approximation scheme (PAS), i.e., an algorithm with a running time of f (k) ⋅ n O 휖 (1) that outputs a solution with at most (1 + 휖)k tasks or asserts that there is no solution with at most k tasks. In this algorithm we use a new trick that intuitively allows us to pretend that we can select tasks from OPT multiple times. We show that the other two algorithms extend also to the weighted case of the problem, at the expense of losing a factor of 1 + 휖 in the cost of the selected tasks. [ABSTRACT FROM AUTHOR]

We revisit the deadline version of the discrete time-cost tradeoff problem for the special case of bounded depth. Such instances occur for example in VLSI design. The depth of an instance is the number of jobs in a longest chain and is denoted by d. We prove new upper and lower bounds on the approximability. First we observe that the problem can be regarded as a special case of finding a minimum-weight vertex cover in a d-partite hypergraph. Next, we study the natural LP relaxation, which can be solved in polynomial time for fixed d and—for time-cost tradeoff instances—up to an arbitrarily small error in general. Improving on prior work of Lovász and of Aharoni, Holzman and Krivelevich, we describe a deterministic algorithm with approximation ratio slightly less than d 2 for minimum-weight vertex cover in d-partite hypergraphs for fixed d and given d-partition. This is tight and yields also a d 2 -approximation algorithm for general time-cost tradeoff instances, even if d is not fixed. We also study the inapproximability and show that no better approximation ratio than d + 2 4 is possible, assuming the Unique Games Conjecture and P ≠ NP . This strengthens a result of Svensson [21], who showed that under the same assumptions no constant-factor approximation algorithm exists for general time-cost tradeoff instances (of unbounded depth). Previously, only APX-hardness was known for bounded depth. [ABSTRACT FROM AUTHOR]

The main contribution of this paper is the first constant factor approximation algorithm for red-blue set cover problem with unit disks. To achieve this, we first give a polynomial time algorithm for line-separable red-blue set cover problem with unit disks. We next obtain a factor 2 approximation algorithm for strip-separable red-blue set cover problem with unit disks. Finally, we obtain a constant factor approximation algorithm for red-blue set cover problem with unit disks by combining our algorithm for the strip-separable problem with the results of Ambühl et al. [1]. Our methods involve a novel decomposition of the optimal solution to line-separable problem into blocks with special structure and extensions of the sweep-line technique of Erlebach and van Leeuwen [9]. [ABSTRACT FROM AUTHOR]

We give better approximation ratios for two Steiner Tree variants by combining known algorithms: the optimum 3 -decomposition and iterative randomized rounding. The first problem is Steiner Tree with minimum number of Steiner points and bounded edge length problem ( S M T - M S P ). The input consists of a set of terminals R in the Euclidean space R 2 . A feasible solution is a Steiner tree τ spanning R with Steiner points S such that every edge in τ has length at most 1 . The objective is to minimize S . Previously, the best approximation ratio for S M T - M S P was 1 + ln (4) + ϵ ≈ 2.386 . We present a polynomial time algorithm with ratio 2.277 . The second problem is Steiner Tree in quasi-bipartite graphs. It is a Steiner Tree problem on graph G = (V , E , c) with terminal set R when the edge set E does not include any edge between two vertices in V \ R . The best-known approximation ratio for this problem is 73 60 , We improve this ratio to 298 245 . [ABSTRACT FROM AUTHOR]

Automatic engraving machines have several industrial applications, such as nameplates, identification codes, and bar code engraving. It can also be used for engraving messages or pictures on decorative items in homes. Additionally, the proposed algorithm can be extended for obtaining the paths for 3D printers. The engraving process happens when the designs, messages, or pictures are carved into any item. The cost of engraving can be optimized by minimizing the movements of the automatic engraving machine. The movements include engraving motions and air motions. This work proposes a fast algorithm for obtaining the engraving path that guarantees the air motions are at most 150% of the optimal. Optimum engraving paths are commonly obtained using exponential time algorithms. The proposed algorithm is a 3/2 approximation algorithm that obtains the engraving path in polynomial time. The experimental results indicate that the proposed algorithm computes an efficient engraving path in a short time. [ABSTRACT FROM AUTHOR]

Tverberg's theorem states that for any k ≥ 2 and any set P ⊂ R d of at least (d + 1) (k - 1) + 1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class CLS = PPAD ∩ PLS , but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set P ⊂ R d and any k ∈ { 2 , ⋯ , n } in O (n d ⌈ log k ⌉) time a k-partition of P such that there is a ball of radius O ((k / n) diam (P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria's method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem. [ABSTRACT FROM AUTHOR]

In this paper, we address the line-capacitated minimum Steiner tree problem (the Lc-MStT problem, for short), which is a variant of the (Euclidean) capacitated minimum Steiner tree problem and defined as follows. Given a set X = { r 1 , r 2 , ... , r n } of n terminals in R 2 , a demand function d : X → N and a positive integer C, we are asked to determine the location of a line l and a Steiner tree T l to interconnect these n terminals in X and at least one point located on this line l such that the total demand of terminals in each maximal subtree (of T l ) connected to the line l, where the terminals in such maximal subtree are all located at the same side of this line l, does not exceed the bound C. The objective is to minimize total weight ∑ e ∈ T l w (e) of such a Steiner tree T l among all line-capacitated Steiner trees mentioned-above, where weight w (e) = 0 if two endpoints of that edge e ∈ T l are located on the line l and otherwise weight w(e) is the Euclidean distance between two endpoints of that edge e ∈ T l . In addition, when this line l is as an input in R 2 and ∑ r ∈ X d (r) ≤ C holds, we refer to this version as the 1-line-fixed minimum Steiner tree problem (the 1Lf-MStT problem, for short). We obtain three main results. (1) Given a ρ st -approximation algorithm to solve the Euclidean minimum Steiner tree problem and a ρ 1 L f -approximation algorithm to solve the 1Lf-MStT problem, respectively, we design a (ρ st ρ 1 L f + 2) -approximation algorithm to solve the Lc-MStT problem. (2) Whenever demand of each terminal r ∈ X is less than C 2 , we provide a (ρ 1 L f + 2) -approximation algorithm to resolve the Lc-MStT problem. (3) Whenever demand of each terminal r ∈ X is at least C 2 , using the Edmonds' algorithm to solve the minimum weight perfect matching as a subroutine, we present an exact algorithm in polynomial time to resolve the Lc-MStT problem. [ABSTRACT FROM AUTHOR]

Consider an arc-capacitated network N through which an integer-valued flow must be sent from several source nodes to a sink node. Each feasible flow defines a corresponding multigraph with the same vertices as N and an edge for each arc of N , where the edge multiplicity is the flow in the respective arc. The maximum flow degree of a feasible flow is the maximum sum of the flow entering and leaving a node of N , i.e. the maximum degree of the corresponding multigraph. The minimum maximum flow degree problem (MMFDP) consists in determining on N a feasible flow such that its maximum flow degree is minimum. We present a polynomial time algorithm for this problem. We use its optimum value to derive an improved upper bound for the flow coloring problem (FCP), which consists in finding a feasible flow whose corresponding multigraph has the minimum chromatic index. Based on this procedure, we design an approximation algorithm for the FCP that improves the best known approximation factor. [ABSTRACT FROM AUTHOR]

We study the nonlinear newsvendor problem concerning goods of a non-discrete nature, and a class of stochastic dynamic programs with several application areas such as supply chain management and economics. The class is characterized by continuous state and action spaces, either convex or monotone cost functions that are accessed via value oracles, and affine transition functions. We establish that these problems cannot be approximated to any degree of either relative or additive error, regardless of the scheme used. To circumvent these hardness results, we generalize the concept of fully polynomial-time approximation scheme allowing arbitrarily small additive and multiplicative error at the same time, while requiring a polynomial running time in the input size and the error parameters. We develop approximation schemes of this type for the classes of problems mentioned above. In light of our hardness results, such approximation schemes are "best possible". A computational evaluation shows the promise of this approach. [ABSTRACT FROM AUTHOR]

It is known that problems like Vertex Cover, Feedback Vertex Set and Odd Cycle Transversal are polynomial time solvable in the class of chordal graphs. We consider these problems in a graph that has at most k vertices whose deletion results in a chordal graph when parameterized by k. While this investigation fits naturally into the recent trend of what is called 'structural parameterizations', here we assume that the deletion set is not given. One method to solve them is to compute a k-sized or an approximate (f(k) sized, for a function f) chordal vertex deletion set and then use the structural properties of the graph to design an algorithm. This method leads to at least k O (k) n O (1) running time when we use the known parameterized or approximation algorithms for finding a k-sized chordal deletion set on an n vertex graph. In this work, we design 2 O (k) n O (1) time algorithms for these problems. Our algorithms do not compute a chordal vertex deletion set (or even an approximate solution). Instead, we construct a tree decomposition of the given graph in 2 O (k) n O (1) time where each bag is a union of four cliques and O (k) vertices. We then apply standard dynamic programming algorithms over this special tree decomposition. This special tree decomposition can be of independent interest. Our algorithms are, what are sometimes called permissive in the sense that given an integer k, they detect whether the graph has no chordal vertex deletion set of size at most k or output the special tree decomposition and solve the problem. We also show lower bounds for the problems we deal with under the strong exponential time hypothesis. [ABSTRACT FROM AUTHOR]

Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA's performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified. [ABSTRACT FROM AUTHOR]

Given a set of players P, a set of indivisible resources R, and a set of non-negative values { v pr } p ∈ P , r ∈ R , an allocation is a partition of R into disjoint subsets { C p } p ∈ P so that each player p is assigned the resources in C p . The max-min fair allocation problem is to determine the allocation that maximizes min p ∑ r ∈ C p v pr . In the restricted case of this problem, each resource r has an intrinsic value v r , and v pr = v r for every player p who desires r and v pr = 0 for every player p who does not. We study the restricted max-min fair allocation problem in this paper. For this problem, the configuration LP has played an important role in estimating and approximating the optimal solution. Our first result is an upper bound of 3 21 26 on the integrality gap, which is currently the best. It is obtained by a tighter analysis of the local search of Asadpour et al. [TALG'12]. It remains unknown whether this local search runs in polynomial time or not. Our second result is a polynomial-time algorithm that achieves an approximation ratio of 4 + δ for any constant δ ∈ (0 , 1) . Our algorithm can be seen as a generalization of the aforementioned local search. [ABSTRACT FROM AUTHOR]

Metric data plays an important role in various settings, for example, in metric-based indexing, clustering, classification, and approximation algorithms in general. Due to measurement error, noise, or an inability to completely gather all the data, a collection of distances may not satisfy the basic metric requirements, most notably the triangle inequality. In this paper we initiate the study of the metric violation distance problem: given a set of pairwise distances, modify the minimum number of distances such that the resulting set forms a metric. Three variants of the problem are considered, based on whether distances are allowed to only decrease, only increase, or the general case which allows both decreases and increases. We show that while the decrease only variant is polynomial time solvable, the increase only and general variants are NP-Complete, and moreover cannot in polynomial time be approximated to any ratio better than the minimum vertex cover problem. We then provide approximation algorithms for the increase only and general variants of the problem, by proving interesting necessary and sufficient conditions on the optimal solution, which are used to approximately reduce to a purely combinatorial problem for which we provide matching asymptotic upper and lower bounds. [ABSTRACT FROM AUTHOR]

In recent years, several compressed indexes based on variants of the Burrows–Wheeler transform have been introduced. Some of these are used to index structures far more complex than a single string, as was originally done with the FM-index (Ferragina and Manzini in J. ACM 52(4):552–581, https://doi.org/10.1145/1082036.1082039, 2005). As such, there has been an increasing effort to better understand under which conditions such an indexing scheme is possible. This has led to the introduction of Wheeler graphs (Gagie et al. in Theor Comput Sci 698:67–78, https://doi.org/10.1016/j.tcs.2017.06.016, 2017). Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs, and that Wheeler graphs can be indexed in a space-efficient way. Hence, being able to recognize whether a given graph is a Wheeler graph, or being able to approximate a given graph by a Wheeler graph, could have numerous applications in indexing. Here we resolve the open question of whether there exists an efficient algorithm for recognizing if a given graph is a Wheeler graph. We show: The problem of recognizing whether a given graph G = (V , E) is a Wheeler graph is NP-complete for any edge label alphabet of size σ ≥ 2 , even when G is a DAG. This holds even on a restricted subset of graphs called d-NFAs for d ≥ 5 . This is in contrast to recent results demonstrating the problem can be solved in polynomial time for d-NFAs where d ≤ 2 . We also show that the recognition problem can be solved in linear time for σ = 1 on graphs without self-loops; There exists an 2 e log σ + O (n + e) time exact algorithm where n = | V | and e = | E | . This algorithm relies on graph isomorphism being computable in strictly sub-exponential time; We define an optimization variant of the problem called Wheeler Graph Violation, abbreviated WGV, where the aim is to identify the smallest set of edges that have to be removed from a graph to obtain a Wheeler graph. We show WGV is APX-hard, even when G is a DAG, implying there exists a constant C > 1 for which there is no C-approximation algorithm (unless P = NP). Also, conditioned on the Unique Games Conjecture, for all C > 1 , it is NP-hard to find a C-approximation, implying WGV is not in APX; We define the Wheeler Subgraph problem, abbreviated WS, where the aim is to find the largest subgraph which is a Wheeler Graph (the dual of WGV). In contrast to WGV, we give an O (σ) -approximation algorithm for the WS problem, implying it is in APX for σ = O (1) . The above findings suggest that most problems under this theme are computationally difficult. However, we identify a class of graphs for which the recognition problem is polynomial-time solvable, raising the question of which properties determine this problem's difficulty. [ABSTRACT FROM AUTHOR]

The Non-Uniform k-center (NUkC) problem has recently been formulated by Chakrabarty et al. [ICALP, 2016; ACM Trans Algorithms 16(4):46:1–46:19, 2020] as a generalization of the classical k-center clustering problem. In NUkC, given a set of n points P in a metric space and non-negative numbers r 1 , r 2 , ... , r k , the goal is to find the minimum dilation α and to choose k balls centered at the points of P with radius α · r i for 1 ≤ i ≤ k , such that all points of P are contained in the union of the chosen balls. They showed that the problem is NP -hard to approximate within any factor even in tree metrics. On the other hand, they designed a "bi-criteria" constant approximation algorithm that uses a constant times k balls. Surprisingly, no true approximation is known even in the special case when the r i 's belong to a fixed set of size 3. In this paper, we study the NUkC problem under perturbation resilience, which was introduced by Bilu and Linial (Comb Probab Comput 21(5):643–660, 2012). We show that the problem under 2-perturbation resilience is polynomial time solvable when the r i 's belong to a constant-sized set. However, we show that perturbation resilience does not help in the general case. In particular, our findings imply that even with perturbation resilience one cannot hope to find any "good" approximation for the problem. [ABSTRACT FROM AUTHOR]

We study the generalized minimum Manhattan network (GMMN) problem: given a set P of pairs of points in the Euclidean plane R 2 , we are required to find a minimum-length geometric network which consists of axis-aligned segments and contains a shortest path in the L 1 metric (a so-called Manhattan path) for each pair in P . This problem commonly generalizes several NP-hard network design problems that admit constant-factor approximation algorithms, such as the rectilinear Steiner arborescence (RSA) problem, and it is open whether so does the GMMN problem. As a bottom-up exploration, Schnizler (2015) focused on the intersection graphs of the rectangles defined by the pairs in P , and gave a polynomial-time dynamic programming algorithm for the GMMN problem whose input is restricted so that both the treewidth and the maximum degree of its intersection graph are bounded by constants. In this paper, as the first attempt to remove the degree bound, we provide a polynomial-time algorithm for the star case, and extend it to the general tree case based on an improved dynamic programming approach. [ABSTRACT FROM AUTHOR]

We study a security game over a network played between a defender and kattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of λ nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for λ constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is NP -hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any λ ; this is the worst equilibrium defense ratio over all graphs. [ABSTRACT FROM AUTHOR]

This paper introduces the d-distance matching problem, in which we are given a bipartite graph G = (S , T ; E) with S = { s 1 , ⋯ , s n } , a weight function on the edges and an integer d ∈ Z + . The goal is to find a maximum-weight subset M ⊆ E of the edges satisfying the following two conditions: (i) the degree of every node of S is at most one in M, (ii) if s i t , s j t ∈ M , then | j - i | ≥ d . This question arises naturally, for example, in various scheduling problems. We show that the problem is NP-complete in general and admits a simple 3-approximation. We give an FPT algorithm parameterized by d and also show that the case when the size of T is constant can be solved in polynomial time. From an approximability point of view, we show that the integrality gap of the natural integer programming model is at most 2 - 1 2 d - 1 , and give an LP-based approximation algorithm for the weighted case with the same guarantee. A combinatorial (2 - 1 d) -approximation algorithm is also presented. Several greedy approaches are considered, and a local search algorithm is described that achieves an approximation ratio of 3 / 2 + ϵ for any constant ϵ > 0 in the unweighted case. The novel approaches used in the analysis of the integrality gap and the approximation ratio of locally optimal solutions might be of independent combinatorial interest. [ABSTRACT FROM AUTHOR]

We consider a natural generalization of classical scheduling problems to a setting in which using a time unit for processing a job causes some time-dependent cost, the time-of-use tariff, which must be paid in addition to the standard scheduling cost. We focus on preemptive single-machine scheduling and two classical scheduling cost functions, the sum of (weighted) completion times and the maximum completion time, that is, the makespan. While these problems are easy to solve in the classical scheduling setting, they are considerably more complex when time-of-use tariffs must be considered. We contribute optimal polynomial-time algorithms and best possible approximation algorithms. For the problem of minimizing the total (weighted) completion time on a single machine, we present a polynomial-time algorithm that computes for any given sequence of jobs an optimal schedule, i.e., the optimal set of time slots to be used for preemptively scheduling jobs according to the given sequence. This result is based on dynamic programming using a subtle analysis of the structure of optimal solutions and a potential function argument. With this algorithm, we solve the unweighted problem optimally in polynomial time. For the more general problem, in which jobs may have individual weights, we develop a polynomial-time approximation scheme (PTAS) based on a dual scheduling approach introduced for scheduling on a machine of varying speed. As the weighted problem is strongly NP-hard, our PTAS is the best possible approximation we can hope for. For preemptive scheduling to minimize the makespan, we show that there is a comparably simple optimal algorithm with polynomial running time. This is true even in a certain generalized model with unrelated machines. [ABSTRACT FROM AUTHOR]

We develop new approximation algorithms for classical graph and set problems in the RAM model under space constraints. As one of our main results, we devise an algorithm for d - H I T T I N G S E T that runs in time n O (d 2 + (d / ϵ)) , uses O ((d 2 + (d / ϵ)) log n) bits of space, and achieves an approximation ratio of O ((d / ϵ) n ϵ) for any positive ϵ ≤ 1 and any d ∈ N . In particular, this yields a factor- O (log n) approximation algorithm which runs in time n O (log n) and uses O (log 2 n) bits of space (for constant d). As a corollary, we obtain similar bounds for V E R T E X C O V E R and several graph deletion problems. For bounded-multiplicity problem instances, one can do better. We devise a factor-2 approximation algorithm for V E R T E X C O V E R on graphs with maximum degree Δ , and an algorithm for computing maximal independent sets, both of which run in time n O (Δ) and use O (Δ log n) bits of space. For the more general d - H I T T I N G S E T problem, we devise a factor-d approximation algorithm which runs in time n O (d δ 2) and uses O (d δ 2 log n) bits of space on set families where each element appears in at most δ sets. For I N D E P E N D E N T S E T restricted to graphs with average degree d, we give a factor-(2d) approximation algorithm which runs in polynomial time and uses O (log n) bits of space. We also devise a factor- O (d 2) approximation algorithm for D O M I N A T I N G S E T on d-degenerate graphs which runs in time n O (log n) and uses O (log 2 n) bits of space. For d-regular graphs, we show how a known randomized factor- O (log d) approximation algorithm can be derandomized to run in time n O (1) and use O (log n) bits of space. Our results use a combination of ideas from the theory of kernelization, distributed algorithms and randomized algorithms. [ABSTRACT FROM AUTHOR]

Given a set of n segments and a query shape Q, the windowing length query asks for finding the sum of the lengths of the parts of the segments that lie inside Q. The popular places problem of a set of curves asks for the subset of the plane where each query shape centered at a point of that region intersects with at least f distinct curves. For square queries, an optimal O (n 2) time algorithm and a matching lower bound exist. We solve the length query problem for convex polygons and disks as query shapes, with O (log n + k) query time and polynomial preprocessing time that depends on the complexity of the query shape. We define a new version of the problem of finding popular places in a set of trajectories where the center of a query is a popular place if the length of the curves inside that query is at least f and use our data structure to solve the original problem as well as this new version. Other than length queries, we solve reporting queries that return the set of intersected segments. For disk queries, we design a point-location data structure for congruent disks with O (log n) query time and O (n 3 log n) preprocessing. We also give algorithms for computing the length query for c-packed curves, which are a class of curves for which the length of the curve inside a disk of radius r is upper-bounded by cr, where c is a constant. Also, we use length queries for polygons to approximate the minimum value c for which a curve is c-packed, if such a c exists. Our results extend to MRC and MPC models for MapReduce, where we address these problems on a set of x-monotone curves. The round complexities of our MapReduce algorithms are constant. In addition, we also implemented our popular places algorithms on trajectories on inputs as big as 15K points to evaluate the efficiency of our algorithms in practice. [ABSTRACT FROM AUTHOR]

A major challenge in wireless underground sensor networks is the signal attenuation originated from multi-environment transmission between underground sensor nodes and the above-ground base station. To overcome this issue, an efficient approach is deploying a set of relay nodes aboveground, thereby reducing transmission loss by shortening transmitting distance. However, this introduces several new challenges, including load balancing and transmission loss minimization. This paper tackles the problem of deploying relay nodes to reduce transmission loss under a load balancing constraint by proposing two approximation algorithms. The first algorithm is inspired by Beam Search, combined with a new selection scheme based on Boltzmann distribution. The second algorithm aims to further improve the solutions obtained by the former by reducing the transmission loss. We observe that we can find an optimal assignment between sensor nodes and a set of the chosen relay in polynomial time by reformulating the part of the problem as a bipartite matching problem with minimum cost. Experimental results indicate that the proposed methods perform better than the other existing ones in most of our test instances while reducing the execution time. [ABSTRACT FROM AUTHOR]

We consider map labeling for the case that a map undergoes a sequence of operations such as rotation, zoom and translation over a specified time span. We unify and generalize several previous models for dynamic map labeling into one versatile and flexible model. In contrast to previous research, we completely abstract from the particular operations and express the labeling problem as a set of time intervals representing the labels' presences, activities and conflicts. One of the model's strength is manifested in its simplicity and broad range of applications. In particular, it supports label selection both for map features with fixed position as well as for moving entities (e.g., for tracking vehicles in logistics or air traffic control). We study the active range maximization problem in this model. We prove that the problem is NP-complete and W[1]-hard, and present constant-factor approximation algorithms. In the restricted, yet practically relevant case that no more than k labels can be active at any time, we give polynomial-time algorithms as well as constant-factor approximation algorithms. [ABSTRACT FROM AUTHOR]

We study the capacitated assortment and price optimization problem, where a retailer sells categories of substitutable products subject to a capacity constraint. The goal of the retailer is to determine the subset of products as well as their selling prices so as to maximize the expected revenue. We model the customer purchase behavior using the nested logit model and formulate this problem as a non-linear binary integer program. For this NP-complete problem, we show that there exists a pseudo polynomial time approximation scheme that finds its ϵ -approximate solution. We first convert the original problem into an equivalent fixed point problem. We then show that finding an ϵ -approximate solution to the fixed point problem can be achieved by binary search, where a non-linear auxiliary problem is repeatedly approximated by a dynamic programing based algorithm involving an approximation to a series of multiple-choice parametric knapsack problems. For the special case when the capacity constraints are cardinal and nest-specific, we develop an algorithm that finds the optimal solution in strongly polynomial time. Moreover, our algorithm can be directly applied to find an ϵ -approximate solution to the capacitated assortment optimization problem under the nested logit model, which is the first approximate algorithm that is polynomial with respect to the number of nests in the literature. [ABSTRACT FROM AUTHOR]

In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results. [ABSTRACT FROM AUTHOR]

Alamdari and Shmoys introduced the following variant of the k-median problem. In this variant, we are given an instance of the k-median problem and a threshold value. Then this variant is the same as the k-median problem except that if the distance between a client i and a facility j is more than the threshold value, then i is not allowed to be connected to j. In this paper, we consider a matroid generalization of this variant of the k-median problem. First, we introduce a generalization of this variant in which the constraint on the number of opened facilities is replaced by a matroid constraint. Then we propose a polynomial-time bicriteria approximation algorithm for this problem by combining the algorithm of Alamdari and Shmoys and the algorithm of Krishnaswamy, Kumar, Nagarajan, Sabharwal, and Saha for a matroid generalization of the k-median problem. [ABSTRACT FROM AUTHOR]

In recent years, the popularity of graph databases has grown rapidly. This paper focuses on single-graph as an effective model to represent information and its related graph mining techniques. In frequent pattern mining in a single-graph setting, there are two main problems: support measure and search scheme. In this paper, we propose a novel framework for designing support measures that brings together existing minimum-image-based and overlap-graph-based support measures. Our framework is built on the concept of occurrence/instance hypergraphs. Based on such, we are able to design a series of new support measures: minimum instance (MI) measure, and minimum vertex cover (MVC) measure, that combine the advantages of existing measures. More importantly, we show that the existing minimum-image-based support measure is an upper bound of the MI measure, which is also linear-time computable and results in counts that are close to number of instances of a pattern. We show that not only most major existing support measures and new measures proposed in this paper can be mapped into the new framework, but also they occupy different locations of the frequency spectrum. By taking advantage of the new framework, we discover that MVC can be approximated to a constant factor (in terms of number of pattern nodes) in polynomial time. In contrast to common belief, we demonstrate that the state-of-the-art overlap-graph-based maximum independent set (MIS) measure also has constant approximation algorithms. We further show that using standard linear programming and semidefinite programming techniques, polynomial-time relaxations for both MVC and MIS measures can be developed and their counts stand between MVC and MIS. In addition, we point out that MVC, MIS, and their relaxations are bounded within constant factor. In summary, all major support measures are unified in the new hypergraph-based framework which helps reveal their bounding relations and hardness properties. [ABSTRACT FROM AUTHOR]

This article introduces and solves a spatial keyword cover problem (SK-Cover for short), which aims to identify the group of spatio-textual objects covering all the keywords in a query and minimizing a distance cost function that leads to fewer objects in the answer set. In a broad sense, SK-Cover has been actively studied in the literature of spatial keyword search, such as the m-closest keywords query and the collective spatial keyword query. However, these existing works focus on minimizing only the largest pairwise distance even though the actual spatial cost is highly influenced by the number of objects in the answer group. Motivated by this, the present article further generalizes the problem definition in such a way that the total cost takes the cardinality of the group as well as the spatial distance. We prove that SK-Cover is not only NP-hard but also does not allow an approximation better than O (log | T |) in polynomial time, where T is the set of query keywords. We first establish an O (log | T |) -approximation algorithm, which is asymptotically optimal in terms of the approximability of SK-Cover, together with effective accessing strategies and pruning rules to improve the overall efficiency and scalability. Despite the NP-hardness of SK-Cover, this article also develops exact solutions that find the optimal group of objects in a reasonably fast manner in practice, especially when it is required to cover a relatively small number of query keywords. In addition to our algorithmic results, we empirically show that our approximation algorithm always achieves the best accuracy and the efficiency comparable to that of a state-of-the-art algorithm intended for m CK , a problem similar to yet theoretically easier than SK-Cover, and also demonstrate that our exact algorithm using the proposed approximation scheme runs much faster than the baseline algorithm adapted from the existing solution for m CK . [ABSTRACT FROM AUTHOR]

Social networks often provide only a binary perspective on social ties: two individuals are either connected or not. While sometimes external information can be used to infer the strength of social ties, access to such information may be restricted or impractical to obtain. Sintos and Tsaparas (KDD 2014) first suggested to infer the strength of social ties from the topology of the network alone, by leveraging the Strong Triadic Closure (STC) property. The STC property states that if person A has strong social ties with persons B and C, B and C must be connected to each other as well (whether with a weak or strong tie). They exploited this property to formulate the inference of the strength of social ties as a NP-hard maximization problem, and proposed two approximation algorithms. We refine and improve this line of work, by developing a sequence of linear relaxations of the problem, which can be solved exactly in polynomial time. Usefully, these relaxations infer more fine-grained levels of tie strength (beyond strong and weak), which also allows one to avoid making arbitrary strong/weak strength assignments when the network topology provides inconclusive evidence. Moreover, these relaxations allow us to easily change the objective function to more sensible alternatives, instead of simply maximizing the number of strong edges. An extensive theoretical analysis leads to two efficient algorithmic approaches. Finally, our experimental results elucidate the strengths of the proposed approach, while at the same time questioning the validity of leveraging the STC property for edge strength inference in practice. [ABSTRACT FROM AUTHOR]