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We consider the numerical taxonomy problem of fitting a positive distance function \({\mathcal {D}:{S\choose 2}\rightarrow \mathbb {R}_{\gt 0}}\) by a tree metric. We want a tree T with positive edge weights and including S among the vertices so that their distances in T match those in \(\mathcal {D}\). A nice application is in evolutionary biology where the tree T aims to approximate thebranching process leading to the observed distances in \(\mathcal {D}\) [Cavalli-Sforza and Edwards 1967]. We consider the total error, that is, the sum of distance errors over all pairs of points. We present a deterministic polynomial time algorithm minimizing the total error within a constant factor. We can do this both for general trees and for the special case of ultrametrics with a root having the same distance to all vertices in S. The problems are APX-hard, so a constant factor is the best we can hope for in polynomial time. The best previous approximation factor was O((log n)(log log n)) by Ailon and Charikar [2005], who wrote "determining whether an O(1) approximation can be obtained is a fascinating question." [ABSTRACT FROM AUTHOR]

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]

This paper is concerned with designing algorithms for and analyzing the computational complexity of the partial vertex cover problem in trees. Graphs (and trees) are frequently used to model risk management in various systems. In particular, Caskurlu et al. in [4] have considered a system which essentially represents a tripartite graph. The goal in this model is to reduce the risk in the system below a predefined risk threshold level. It can be shown that the main goal in this risk management system can be formulated as a Partial Vertex Cover problem on bipartite graphs. In this paper, we focus on a special case of the partial vertex cover problem, when the input graph is a tree. We consider four possible versions of this setting, depending on whether or not, the vertices and edges are weighted. Two of these versions, where edges are assumed to be unweighted, are known to be polynomial-time solvable. However, the computational complexity of this problem with weighted edges, and possibly with weighted vertices, remained open. The main contribution of this paper is to resolve these questions by fully characterizing which variants of partial vertex cover remain NP-hard in trees, and which can be solved in polynomial time. In the paper, we propose two pseudo-polynomial DP-based algorithms for the most general case in which weights are present on both the edges and the vertices of the tree. One of these algorithms leads to a polynomial-time procedure, when weights are confined to the edges of the tree. The insights used in this algorithm are combined with additional scaling ideas to derive an FPTAS for the general case. A secondary contribution of this work is to propose a novel way of using centroid decompositions in trees, which could be useful in other settings as well. [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]

The Power of Traffic Tolls The paper "In Congestion Games, Taxes Achieve Optimal Approximation" by Paccagnan and Gairing investigates the power and limitations of taxes as interventions in congestion games. Perhaps surprisingly, they find that efficiently computed taxes can achieve the same performance as the best polynomial time algorithm, even when the algorithm has full control over the agents' actions. The authors establish three main results to support this claim. First, they prove that minimizing congestion is NP-hard to approximate within a given factor based on the latency functions. Second, they demonstrate how convex optimization tools can be used to design optimal taxes. Third, they provide a polynomial time algorithm with an approximation factor matching the hardness barrier. The upshot of their contribution can be summarized as follows: Judiciously designed taxes achieve optimal approximation, and no other tractable intervention can improve upon this result. In this work, we address the problem of minimizing social cost in atomic congestion games. For this problem, we present lower bounds on the approximation ratio achievable in polynomial time and demonstrate that efficiently computable taxes result in polynomial time algorithms matching such bounds. Perhaps surprisingly, these results show that indirect interventions, in the form of efficiently computed taxation mechanisms, yield the same performance achievable by the best polynomial time algorithm, even when the latter has full control over the agents' actions. It follows that no other tractable approach geared at incentivizing desirable system behavior can improve upon this result, regardless of whether it is based on taxations, coordination mechanisms, information provision, or any other principle. In short: Judiciously chosen taxes achieve optimal approximation. Three technical contributions underpin this conclusion. First, we show that computing the minimum social cost is NP -hard to approximate within a given factor depending solely on the admissible cost functions. Second, we design a polynomially computable taxation mechanism whose efficiency (price of anarchy) matches this hardness factor, and thus is optimal among all tractable mechanisms. As these results extend to coarse correlated equilibria, any no-regret algorithm inherits the same performances, allowing us to devise polynomial time algorithms with optimal approximation. Supplemental Material: The e-companion is available at https://doi.org/10.1287/opre.2021.0526. [ABSTRACT FROM AUTHOR]

We study a shipment consolidation problem commonly faced by companies that outsource logistics operations and operate in a commit-to-delivery mode. It involves delivering a given set of orders to their destinations by their committed due times using multiple shipping methods at the minimum total shipping and inventory cost. The shipping cost is generally nonlinear in shipping quantity and can be represented by a subadditive piecewise linear function. We investigate two shipping scenarios, one involving long-haul shipping only and the other involving joint long-haul and short-haul shipping. We develop analytical results and solution algorithms for the shipment consolidation problem under each shipping scenario. The problem under the first shipping scenario is shown to be strongly NP -hard. We find that a simple policy, called the First-Due-First-Delivered (FDFD) policy, which assigns orders with earlier delivery due times to shipping methods with earlier destination arrival times, is very effective. This policy enables us to develop a polynomial time algorithm, which not only solves the problem under the concave shipping cost structure optimally but also achieves a performance guarantee of 2 for the problem under the general subadditive shipping cost structure. For the problem under the second shipping scenario, we extend the FDFD policy for long-haul shipping and derive another policy, called the No-Wait policy, for short-haul shipping. We use these policies to develop a polynomial time algorithm and analyze its performance guarantee. Our computational experiments show that the algorithm significantly outperforms a commercial optimization solver, and its performance is robust across different parameter settings that reflect various practical situations. This paper was accepted by Jeannette Song, operations management. Supplemental Material: The data and e-companion are available at https://doi.org/10.1287/mnsc.2023.4835. [ABSTRACT FROM AUTHOR]

The use of drones for collecting information and detecting bugs in orchards covered by nets is a challenging problem. The nets help in reducing pest damage, but they also constrain the drone's flight path, making it longer and more complex. To address this issue, we model the orchard as an aisle-graph, a regular data structure that represents consecutive aisles where trees are arranged in straight lines. The drone flies close to the trees and takes pictures at specific positions for monitoring the presence of bugs, but its energy is limited, so it can only visit a subset of positions. To tackle this challenge, we introduce the Single-drone Orienteering Aisle-graph Problem (SOAP), a variant of the orienteering problem, where likely infested locations are prioritized by assigning them a larger profit. Additionally, the drone's movements have a cost in terms of energy, and the objective is to plan a drone's route in the most profitable locations under a given drone's battery. We show that SOAP can be optimally solved in polynomial time, but for larger orchards/instances, we propose faster approximation and heuristic algorithms. Finally, we evaluate the algorithms on synthetic and real datasets to demonstrate their effectiveness and efficiency. [ABSTRACT FROM AUTHOR]

We study linear Fisher markets with satiation. In these markets, sellers have earning limits, and buyers have utility limits. Beyond applications in economics, they arise in the context of maximizing Nash social welfare when allocating indivisible items to agents. In contrast to markets with either earning or utility limits, markets with both limits have not been studied before. They turn out to have fundamentally different properties. In general, the existence of competitive equilibria is not guaranteed. We identify a natural property of markets (termed money clearing) that implies existence. We show that the set of equilibria is not always convex, answering a question posed in the literature. We design an FPTAS to compute an approximate equilibrium and prove that the problem of computing an exact equilibrium lies in the complexity class continuous local search (CLS ; i.e., the intersection of polynomial local search (PLS) and polynomial parity arguments on directed graphs (PPAD)). For a constant number of buyers or goods, we give a polynomial-time algorithm to compute an exact equilibrium. We show how (approximate) equilibria can be rounded and provide the first constant-factor approximation algorithm (with a factor of 2.404) for maximizing Nash social welfare when agents have capped linear (also known as budget-additive) valuations. Finally, we significantly improve the approximation hardness for additive valuations to 8/7>1.069. Funding: J. Garg was supported by the National Science Foundation [Grant CCF-1942321 (CAREER)]. M. Hoefer was supported by Deutsche Forschungsgemeinschaft [Grants Ho 3831/5-1, Ho 3831/6-1, and Ho 3831/7-1]. [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]

The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). On the other hand, these problems are much easier for “uninitialised” variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided by polynomial time algorithms (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialised and uninitialised variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighbourhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. Interestingly, this is the first Diophantine hardness result on a variant of the Skolem problem. On the other hand, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable in their full generality, with PSPACE complexity. Our analysis is based on the set of initial configurations such that positivity holds, which leads to new insights into these difficult problems, and interesting geometrical interpretations. [ABSTRACT FROM AUTHOR]

Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fréchet distance. A shortcoming of existing map matching algorithms under the Fréchet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this article, we investigate map matching queries under the Fréchet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in O((pq)1-δ) query time for any δ > 0, where p and q are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this article. We show that for c-packed graphs, one can construct a data structure of \(\tilde{O}(cp)\) size that can answer (1+ε)-approximate map matching queries in \(\tilde{O}(c^4 q \log ^4 p)\) time, where \(\tilde{O}(\cdot)\) hides lower-order factors and dependence on ε. [ABSTRACT FROM AUTHOR]

Theoretical studies on evolutionary algorithms have developed vigorously in recent years. Many such algorithms have theoretical guarantees in both running time and approximation ratio. Some approximation mechanism seems to be inherently embedded in many evolutionary algorithms. In this paper, we identify such a relation by proposing a unified analysis framework for a global simple multiobjective evolutionary algorithm (GSEMO) and apply it on a minimum weight general cover problem, which is general enough to subsume many important problems including the minimum submodular cover problem in which the submodular function is real-valued, and the minimum connected dominating set problem for which the potential function is nonsubmodular. We show that GSEMO yields theoretically guaranteed approximation ratios matching those achievable by a greedy algorithm in expected polynomial time when the potential function g is polynomial in the input size and the minimum gap between different g-values is a constant. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was supported by National Natural Science Foundation of China [11771013, U20A2068]; Zhejiang Provincial Natural Science Foundation of China [LD19A010001]. [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]

We consider a single machine rescheduling problem where a set of original jobs have been scheduled to minimize the total weighted completion time. However, before formal processing begins, a new set of jobs arrives and creates a disruption. The decision maker can reject a subset of the new jobs by paying certain rejection penalties, and reschedule the original and the remaining new jobs without excessively disrupting the original schedule, which is measured by the maximum completion time deviation for any original jobs between the original and adjusted schedules. The objective is to minimize the sum of total weighted completion time of the original jobs and the accepted new jobs in the adjusted schedule, the weighted maximum completion time deviation and the total rejection cost. We first provide a dynamic programming-based exact algorithm running in pseudo-polynomial time, and then propose a fully polynomial time approximation scheme. Given the NP-hardness for the studied problem, our result is the best possible from the approximation algorithm perspective. In addition, we conduct extensive computational experiments to evaluate the performance of our proposed methods, and provide some managerial insights on their applicability under different situations. [ABSTRACT FROM AUTHOR]

A dominator coloring of a graph G is a proper vertex coloring with a domination property that each vertex dominates a color class. The minimum number of colors used in a dominator coloring of G is called dominator chromatic number of G and is denoted as χ d (G). Graphs with χ d (G) ≤ 3 have characterizations and such graphs can be recognized in polynomial time. However, for k ≥ 4 , deciding whether χ d (G) ≤ k is NP-hard. In this paper, we investigate the computational complexity of minimum q -dominator partization problem (Min- q -D-Partz). In Min- q -D-Partz, given a graph G , the objective is to find a vertex set S of minimum size such that χ d (G [ V ∖ S ]) ≤ q. We prove that Min- 2 -D-Partz is APX-complete and has a two-factor approximation algorithm. We also prove that Min- 3 -D-Partz cannot be approximated within any constant factor and can be approximated within a factor of O (log n). [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]

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around rounding the solution to a Semidefinite Program. However, it is likely that important combinatorial or algebraic insights are needed in order to break the no(1) threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs that arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form H = (V,E) with V = V (K3 G) and E = E(K3 G) s E, where E E(K3 G) is any edge set such that no vertex has more than an -fraction of its edges in E. We show that one can construct H = K3 \times G with V (H) = V (H) that is close to H. For arbitrary G, H satisfies | E(H)Δ E H)| O(\epsilon | E(H)|). Additionally, when G is a mild expander, we provide a 3-coloring for H in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on G, we show that it is NP-hard to 3-color H. [ABSTRACT FROM AUTHOR]

We consider a selection problem with stochastic probing. There is a set of items whose values are drawn from independent distributions. The distributions are known in advance. Each item can be tested repeatedly. Each test reduces the uncertainty about the realization of its value. We study a testing model, where the first test reveals whether the realized value is smaller or larger than the c-quantile of the underlying distribution of some constant c \in (0, 1). Subsequent tests allow us to further narrow down the interval in which the realization is located. There is a limited number of possible tests, and our goal is to design near-optimal testing strategies that allow us to maximize the expected value of the chosen item. We study both identical and nonidentical distributions and develop polynomial-time algorithms with constant approximation factors in both scenarios. [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]

For several decades, much effort has been put into identifying classes of CNF formulas whose satisfiability can be decided in polynomial time. Classic results are the linear-time tractability of Horn formulas (Aspvall, Plass, and Tarjan, 1979) and Krom (i.e., 2CNF) formulas (Dowling and Gallier, 1984). Backdoors, introduced by Williams, Gomes and Selman (2003), gradually extend such a tractable class to all formulas of bounded distance to the class. Backdoor size provides a natural but rather crude distance measure between a formula and a tractable class. Backdoor depth, introduced by Mählmann, Siebertz, and Vigny (2021), is a more refined distance measure, which admits the utilization of different backdoor variables in parallel. We propose FPT approximation algorithms to compute backdoor depth into the classes Horn and Krom. This leads to a linear-time algorithm for deciding the satisfiability of formulas of bounded backdoor depth into these classes. [ABSTRACT FROM AUTHOR]

The fair k -center problem has been paid lots of attention recently. In the fair k -center problem, we are given a set X of points in a metric space and a parameter k ∈ Z + , where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k -center problem objective is minimized. In this paper, we consider the fair k -center problem with outliers and capacity constraints, denoted as the fair k -center with outliers (F k CO) problem and the capacitated fair k -center (CF k C) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1 + ϵ) -approximations with FPT time for the F k CO and CF k C problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the F k CO problem with some reasonable assumptions. [ABSTRACT FROM AUTHOR]

The coordination among drones and ground vehicles for last-mile delivery has gained significant interest in recent years. In this paper, we study multiple drone delivery scheduling problem (MDSP) [1] for last-mile delivery, where we have a set of drones with an identical battery budget and a set of delivery locations; along with profit for delivery, cost, and delivery time intervals. The objective of the MDSP is to find conflict-free schedules for each drone such that the total profit gained is maximum subject to the battery constraint of the drones. This paper proposes an optimal pseudo-polynomial algorithm and a fully polynomial time approximation scheme (FPTAS) for the single drone delivery scheduling problem (SDSP). Also, we propose two approximation algorithms of factor 1 3 for the MDSP with some constraints on the number of drones. We formulate the problem for the heterogeneous case, where the drones have non-identical battery capacities, and propose a 1 6 ρ -approximation algorithm, where ρ is the ratio between the minimum and maximum battery capacity of the drones. • An optimal pseudo-polynomial algorithm and FPTAS for the single drone delivery scheduling problem is proposed. • Two approximation algorithms for the multiple drone delivery scheduling problem are proposed. • An approximation algorithm for the multiple heterogeneous drone delivery scheduling problem is proposed. • We analyzed the approximation factors of all the aforementioned proposed algorithms and time complexities. [ABSTRACT FROM AUTHOR]

We initiate the study of a generalization of the class cover problem [Cannon and Cowen [1] , Bereg et al. [2] ] the generalized class cover problem , where we are allowed to misclassify some points provided we pay an associated positive penalty for every misclassified point. Two versions: single coverage and multiple coverage , of the generalized class cover problem are investigated. We study five different variants of both versions of the generalized class cover problem with axis-parallel strips and axis-parallel half-strips extending to different directions in the plane, thus extending similar work by Bereg et al. (2012) [2] on the class cover problem. We prove that the multiple coverage version of the generalize class cover problem with axis-parallel strips are in P , whereas the single coverage version is NP -hard. A factor 2 approximation algorithm is provided for the later problem. The APX -hardness result is also shown for the single coverage version. For half-strips extending to exactly one direction, both the single and multiple coverage versions can be solved in polynomial time using dynamic programming. In the case of half-strips extending to two orthogonal directions, we prove the class cover problem is NP -hard followed by APX -hard. This gives improve hardness results compare to Bereg et al. (2012) [2] , where they proved the class cover problem with half-strips oriented in four different directions is NP -hard. These NP - and APX -hardness results can directly apply to both single and multiple versions. Finally, constant factor approximation algorithms are provided for half-strips extending to more than one direction. [ABSTRACT FROM AUTHOR]

In large-scale disasters, evacuation using public transport, such as buses, is always essential, especially with limitations on the time window and resources available. Unfortunately, a Bus Evacuation Problem (BEP) is NP-hard in nature. This means it is impossible to find an optimal evacuation plan within a reasonable time window. Therefore, it is efficacious and requisite to use an approximation algorithm such that a sub-optimal plan can be derived quickly in large-scale disasters. In the literature, an approximation algorithm for BEP has been presented. However, there is not only a lack of rigorous proof of the approximation ratio, but also the computation time of the approximation algorithm is influenced by the number of evacuees. In this work, firstly the approximation ratio is derived as 2 n b + 7 , where n b is the number of buses, which is fundamentally more precise than the existing work. Further, the approximation ratio can be reduced to n b + 6 through a new proof provided in this work. More importantly, a new approximation algorithm is proposed in this work, with an aggregated network flow model that can be quickly solved as a linear programming problem in polynomial time. The approximation ratio of the new algorithm is proven to be n b + 1 when there is a pile of buses, and the computation time is insensitive to the number of evacuees. Finally, 10 sets of random cases and a real-life disaster, the flooding in Xingguo, China in 2019 are studied to illustrate the efficiency and practical applicability of the proposed approximation algorithm. [ABSTRACT FROM AUTHOR]