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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]

In MAXSPACE, given a set of ads A , one wants to schedule a subset A ′ ⊆ A into K slots B 1 , ⋯ , B K of size L. Each ad A i ∈ A has a size s i and a frequency w i . A schedule is feasible if the total size of ads in any slot is at most L, and each ad A i ∈ A ′ appears in exactly w i slots and at most once per slot. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We consider a generalization called MAXSPACE-R for which an ad A i also has a release date r i and may only appear in a slot B j if j ≥ r i . For this variant, we give a 1/9-approximation algorithm. Furthermore, we consider MAXSPACE-RDV for which an ad A i also has a deadline d i (and may only appear in a slot B j with r i ≤ j ≤ d i ), and a value v i that is the gain of each assigned copy of A i (which can be unrelated to s i ). We present a polynomial-time approximation scheme for this problem when K is bounded by a constant. This is the best factor one can expect since MAXSPACE is strongly NP-hard, even if K = 2 . [ABSTRACT FROM AUTHOR]

In offline scheduling models, jobs are given with their exact processing times. In their online counterparts, jobs arrive in sequence together with their processing times and the scheduler makes irrevocable decisions on how to execute each of them upon its arrival. We consider a semi-online variant which has equally rich application background, called scheduling with testing, where the exact processing time of a job is revealed only after a required testing operation is finished, or otherwise the job has to be executed for a given possibly over-estimated length of time. For multiprocessor scheduling with testing to minimize the total job completion time, we present several first approximation algorithms with constant competitive ratios for various settings, including a 2 φ -competitive algorithm for the non-preemptive general testing case and a (0.0382 + 2.7925 (1 - 1 2 m)) -competitive randomized algorithm, when the number of machines m ≥ 37 or otherwise 2.7925-competitive, where φ = (1 + 5) / 2 < 1.6181 is the golden ratio and m is the number of machines, a (3.5 - 3 2 m) -competitive algorithm allowing job preemption when m ≥ 3 or otherwise 3-competitive, and a (φ + φ + 1 2 (1 - 1 m)) -competitive algorithm for the non-preemptive uniform testing case when m ≥ 5 or otherwise (φ + 1) -competitive. Our results improve three previous best approximation algorithms for the single machine scheduling with testing problems, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of finding a regression in a version control system (VCS), such as git. The set of versions is modelled by a directed acyclic graph (DAG) where vertices represent versions of the software, and arcs are the changes between different versions. We assume that somewhere in the DAG, a bug was introduced, which persists in all of its subsequent versions. It is possible to query a vertex to check whether the corresponding version carries the bug. Given a DAG and a bugged vertex, the Regression Search Problem consists in finding the first vertex containing the bug in a minimum number of queries in the worst-case scenario. This problem is known to be NP-complete. We study the algorithm used in git to address this problem, known as git bisect. We prove that in a general setting, git bisect can use an exponentially larger number of queries than an optimal algorithm. We also consider the restriction where all vertices have indegree at most 2 (i.e. where merges are made between at most two branches at a time in the VCS), and prove that in this case, git bisect is a 1 log 2 (3 / 2) -approximation algorithm, and that this bound is tight. We also provide a better approximation algorithm for this case. Finally, we give an alternative proof of the NP-completeness of the Regression Search Problem, via a variation with bounded indegree. [ABSTRACT FROM AUTHOR]

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time r j , deadline d j , and processing time p j . They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their [ r j , d j ] window. This problem has been studied extensively (even for the case of m = 1 ). Several special cases of the problem remain open. Bar-Noy et al. (Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1–4, 1999, Atlanta, Georgia, USA, pp. 622–631. ACM, 1999, https://doi.org/10.1145/301250.301420) presented an algorithm with ratio 1 - 1 / (1 + 1 / m) m for m machines, which approaches 1 - 1 / e as m increases. For m = 1 , Chuzhoy et al. (42nd Annual Symposium on Foundations of Computer Science (FOCS) 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 348–356. IEEE Computer Society, 2001) presented an algorithm with approximation with ratio 1 - 1 e - ε (for any ε > 0 ). Recently Im et al. (SIAM J Discrete Math 34(3):1649–1669, 2020) presented an algorithm with ratio 1 - 1 / e + ε 0 for some absolute constant ε 0 > 0 for any fixed m. They also presented an algorithm with ratio 1 - O (log m / m) - ε for general m which approaches 1 as m grows. The approximability of the problem for m = O (1) remains a major open question. Even for the case of m = 1 and c = O (1) distinct processing times the problem is open (Sgall in: Algorithms - ESA 2012 - 20th Annual European Symposium, Ljubljana, Slovenia, September 10–12, 2012. Proceedings, pp 2–11, 2012). In this paper we study the case of m = O (1) and show that if there are c distinct processing times, i.e. p j 's come from a set of size c, then there is a randomized (1 - ε) -approximation that runs in time O (n m c 7 ε - 6 log T) , where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning. [ABSTRACT FROM AUTHOR]

In this work, we introduce a multi-vehicle (or multi-postman) extension of the classical Mixed Rural Postman Problem, which we call the Min–Max Mixed Rural Postmen Cover Problem (MRPCP). The MRPCP is defined on a mixed graph G = (V , E , A) , where V is the vertex set, E denotes the (undirected) edge set and A represents the (directed) arc set. Let F ⊆ E ( H ⊆ A ) be the set of required edges (required arcs). There is a nonnegative weight associated with each edge and arc. The objective is to determine no more than k closed walks to cover all the required edges in F and all the required arcs in H such that the weight of the maximum weight closed walk is minimized. By replacing closed walks with (open) walks in the MRPCP, we obtain the Min–Max Mixed Rural Postmen Walk Cover Problem (MRPWCP). The Min–Max Mixed Chinese Postmen Cover Problem (MCPCP) is a special case of the MRPCP where F = E and H = A . The Min–Max Stacker Crane Cover Problem (SCCP) is another special case of the MRPCP where F = ∅ and H = A For the MRPCP with the input graph satisfying the weakly symmetric condition, i.e., for each arc there exists a parallel edge whose weight is not greater than this arc, we devise a 27 4 -approximation algorithm. This algorithm achieves an approximation ratio of 33 5 for the SCCP with the weakly symmetric condition. Moreover, we obtain the first 5-approximation algorithm (4-approximation algorithm) for the MRPWCP (MCPCP) with the weakly symmetric condition. [ABSTRACT FROM AUTHOR]

We consider single-machine scheduling with a non-renewable resource. In this setting, we are given a set of jobs, each characterized by a processing time, a weight, and a resource requirement. At fixed points in time, certain amounts of the resource are made available to be consumed by the jobs. The goal is to assign the jobs non-preemptively to time slots on the machine, so that each job has enough resource available at the start of its processing. The objective that we consider is the minimization of the sum of weighted completion times. The main contribution of the paper is a PTAS for the case of 0 processing times ( 1 | r m = 1 , p j = 0 | ∑ w j C j ). In addition, we show strong NP-hardness of the case of unit resource requirements and weights ( 1 | r m = 1 , a j = 1 | ∑ C j ), thus answering an open question of Györgyi and Kis. We also prove that the schedule corresponding to the Shortest Processing Time First ordering provides a 3/2-approximation for the latter problem. Finally, we investigate a variant of the problem where processing times are 0 and the resource arrival times are unknown. We present a (4 + ϵ) -approximation algorithm, together with a (4 - ε) -inapproximability result, for any ε > 0 . [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 considers several two-machine open shop problems with two agents. Each agent has an independent set of nonpreemptive jobs, and the objective is to find either a schedule minimizing a linear combination of the makespans of both agents, a schedule minimizing the makespan of one agent with the makespan of the other agent not exceeding a threshold, or all Pareto-optimal schedules with respect to the makespans of both agents. We present a number of results for the problems above, including a polynomial algorithm and a pseudo-polynomial algorithm for special cases, non-approximability, two approximation algorithms, and a fully polynomial-time approximation scheme. Finally, we run numerical experiments to show the effectiveness of the pseudo-polynomial algorithm and the approximation algorithms. [ABSTRACT FROM AUTHOR]

We consider bicriteria flowshop scheduling problems with two machines to simultaneously minimize the makespan and the total completion time without any prioritization between the two objectives. An approximation relative to the optimal value of the problem with regard to each objective is adopted, even though a schedule with both objectives reaching their minimum at the same time may not exist. We consider several problems with ordered aspects on processing times such as job ordered and machine ordered. For each problem, we propose a fast (ρ 1 , ρ 2) -approximation algorithm, where ρ 1 and ρ 2 indicate the approximation ratios with regard to the makespan and the total completion time, respectively, and we explore the problem's inapproximability region that cannot be reached by any algorithm. [ABSTRACT FROM AUTHOR]

This study addresses the energy efficiency challenge in cloud data centers by examining the Virtual Machine Placement (VMP) problem. VMP involves mapping virtual machines (VMs) to physical machines (PMs) under capacity constraints. The paper focuses on the bin packing with linear usage cost (BPLUC) variant of bin packing, which includes fixed and variable costs in the calculation of the cost of a used bin. We prove that every approximation algorithm for the bin and vector bin packing can be used for BPLUC and VBPLUC, respectively. We propose a more power-efficient approach to VMP by applying a vector bin packing algorithm to minimize power consumption in data centers. We test the proposed algorithm on various synthetic and real workloads, and the experimental results demonstrate that it is more power-efficient than existing algorithms for VMP. The findings suggest that the proposed algorithm has significant implications for energy-efficient strategies in cloud data centers. Generally, this study makes contributes to the development of energy-efficient approaches to VMP that can help reduce power consumption and improve the sustainability of cloud data centers. [ABSTRACT FROM AUTHOR]

We consider the robust version of the classic k-center clustering problem, where we wish to remove up to z points (outliers), so as to be able to cluster the remaining points in k clusters with minimum maximum radius. We study such a problem under the fully dynamic adversarial model, where points can be inserted or deleted arbitrarily. In this setting, the main goal is to design algorithms that maintain a high quality solution at any point in time, while requiring a "small" amortized cost, i.e. a "small" number of operations per insertion or deletion, on average. In our work, we provide the first constant bi-criteria approximation algorithm for such a problem with its amortized cost being independent of both z and the size of the current input. We also complement our positive result with a lower bound showing that any constant (non bi-criteria) approximation algorithm has amortized cost at least linear in z. Finally, we conduct an in-depth experimental analysis of our algorithm on Twitter, Flickr, and Air-Quality datasets showing the effectiveness of our approach. [ABSTRACT FROM AUTHOR]

In this paper, the first fixed-ratio approximation algorithms are proposed for a series of asymmetric settings of well-known combinatorial routing problems. Among them are the Steiner cycle problem, the prize-collecting traveling salesman problem, the minimum cost cycle cover problem by a bounded number of cycles, etc. Almost all of the proposed algorithms rely on original reductions of the considered problems to auxiliary instances of the asymmetric traveling salesman problem and employ the breakthrough approximation results for this problem obtained recently by O. Svensson, J. Tarnawski, L. Végh, V. Traub, and J. Vygen. On the other hand, approximation of the cycle cover problem was proved by applying a deeper extension of their approach. [ABSTRACT FROM AUTHOR]

We study the following scheduling problem with transportation. Given a set of n jobs that need to be processed on a single machine, we need to deliver the finished jobs to one of the two destinations using a single transporter. The goal is to minimize the time that all the jobs are delivered to their destination, such that the transporter returns. We propose a 11 / 6 + ε -approximation algorithm which improves upon the best-known approximation ratio of 2. [ABSTRACT FROM AUTHOR]

We investigate the optimal piecewise linear interpolation of the bivariate product xy over rectangular domains. More precisely, our aim is to minimize the number of simplices in the triangulation underlying the interpolation, while respecting a prescribed approximation error. First, we show how to construct optimal triangulations consisting of up to five simplices. Using these as building blocks, we construct a triangulation scheme called crossing swords that requires at most - times the number of simplices in any optimal triangulation. In other words, we derive an approximation algorithm for the optimal triangulation problem. We also show that crossing swords yields optimal triangulations in the case that each simplex has at least one axis-parallel edge. Furthermore, we present approximation guarantees for other well-known triangulation schemes, namely for the red refinement and longest-edge bisection strategies as well as for a generalized version of K1-triangulations. Thereby, we are able to show that our novel approach dominates previous triangulation schemes from the literature, which is underlined by illustrative numerical examples. [ABSTRACT FROM AUTHOR]

For a graph G = (V , E) , a collection P of vertex-disjoint (simple) paths is called a path cover of G if every vertex v ∈ V is contained in exactly one path of P . The Path Cover problem (PC for short) is to find a minimum cardinality path cover of G. In this paper, we introduce generalizations of PC, where each path is associated with a weight (cost or profit). Our problem, Minimum (Maximum) Weighted Path Cover [MinPC (MaxPC)], is defined as follows: Let U = { 0 , 1 , ⋯ , n - 1 } . Given a graph G = (V , E) and a weight function f : U → R ∪ { + ∞ , - ∞ } that defines a weight for each path based on its length, the objective of MinPC (MaxPC) is to find a path cover P of G such that the total weight of the paths in P is minimized (maximized). Let L be a subset of U, and P L be the set of paths such that each path is of length ℓ ∈ L . We consider Min P L PC with binary cost, i.e., the cost function is f (ℓ) = 1 if ℓ ∈ L ; otherwise, f (ℓ) = 0 . We also consider Max P L PC with f (ℓ) = ℓ + 1 , if ℓ ∈ L ; otherwise, f (ℓ) = 0 . Many well-known graph theoretic problems such as the Hamiltonian Path and the Maximum Matching problems can be modeled using Min P L PC and Max P L PC. In this paper, we first show that deciding whether Min P { 0 , 1 , 2 } PC has a 0-weight solution is NP-complete for planar bipartite graphs of maximum degree three, and consequently, (i) for any constant σ ≥ 1 , there is no polynomial-time approximation algorithm with approximation ratio σ for Min P { 0 , 1 , 2 } PC unless P = NP, and (ii) Max P { 3 , ⋯ , n - 1 } PC is NP-hard for the same graph class. Next, we present a polynomial-time algorithm for Min P { 0 , 1 , ⋯ , k } PC on graphs with bounded treewidth for a fixed k. Lastly, we present a 4-approximation algorithm for Max P { 3 , ⋯ , n - 1 } PC, which becomes a 2.5-approximation algorithm for subcubic graphs. [ABSTRACT FROM AUTHOR]

We consider the max hypergraph 3-cut problem with limited unbalance (MH3C-LU). The objective is to divide the vertex set of an edge-weighted hypergraph H = (V , E , w) into three disjoint subsets V 1 , V 2 , and V 3 such that the sum of edge weights cross different parts is maximized subject to | | V i | - | V l | | ≤ B ( ∀ i ≠ l ∈ { 1 , 2 , 3 } ) for a given parameter B. This problem is NP-hard because it includes some well-known problems like the max 3-section problem and the max 3-cut problem as special cases. We formulate the MH3C-LU as a ternary quadratic program and present a randomized approximation algorithm based on the complex semidefinite programming relaxation technique. [ABSTRACT FROM AUTHOR]

In this paper, we propose and investigate the individually fair k-center with outliers (IFkCO). In the IFkCO, we are given an n-sized vertex set in a metric space, as well as integers k and q. At most k vertices can be selected as the centers and at most q vertices can be selected as the outliers. The centers are selected to serve all the not-an-outlier (i.e., served) vertices. The so-called individual fairness constraint restricts that every served vertex must have a selected center not too far way. More precisely, it is supposed that there exists at least one center among its ⌈ (n - q) / k ⌉ closest neighbors for every served vertex. Because every center serves (n - q) / k vertices on the average. The objective is to select centers and outliers, assign every served vertex to some center, such that the maximum fairness ratio over all served vertices is minimized, where the fairness ratio of a vertex is defined as the ratio between its distance with the assigned center and its distance with a ⌈ (n - q) / k ⌉ th closest neighbor. As our main contribution, a 4-approximation algorithm is presented, based on which we develop an improved algorithm from a practical perspective. Extensive experiment results on both synthetic datasets and real-world datasets are presented to illustrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

Matrix decomposition is a widely used tool in machine learning with many applications such as dimension reduction or visualization. In this paper we consider decomposing X, a matrix of size n × m , to a product WS where we require that S, a matrix of size n × k , needs to have consecutive ones property. More specifically, we require that each row of S needs to be in the form of 0 , ... , 0 , 1 , ... , 1 , 0 , ... , 0 . Such decompositions are particularly meaningful if X is a matrix where each row represents a time series; in such a case the ones in each row in S represent a time segment. We show that the optimization problem is inapproximable. To solve the problem we propose 5 different algorithms. The first two algorithms are based on solving iteratively S while keeping W fixed and then solving W while keeping S fixed. The next two algorithms are based on greedily optimizing a single row in S and the corresponding column in W. The last algorithm first finds the optimal decomposition of with 2 k - 1 non-overlapping rows, and then greedily combines the rows until k rows remain. We compare the algorithms experimentally, focusing on the quality of the decomposition as well as the computational time. We show experimentally that our algorithms yield interpretable results in practical time. [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]

In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the matrix variables in the lifted space, we derive some enhanced semidefinite relaxations of the complex quadratic programming problems. Then, we compare the proposed semidefinite relaxations with existing ones, and show that the newly proposed semidefinite relaxations could be strictly tighter than the previous ones. Moreover, the proposed semidefinite relaxations can be applied to more general cases of complex quadratic programming problems, whereas the previous ones are only designed for special cases. Numerical results indicate that the proposed semidefinite relaxations not only provide tighter relaxation bounds, but also improve some existing approximation algorithms by finding better sub-optimal solutions. [ABSTRACT FROM AUTHOR]

In wireless sensor networks (WSNs), the virtual backbone (VB) consists of a subset of nodes, which are responsible for routing tasks. Fault-tolerant VBs are desirable for overcoming the effects of node or link failure in WSNs. Usually, a homogeneous WSN (VB) is abstracted as a unit disk graph (UDG) (connected dominating set(CDS)). This paper introduces the concept of a fault-tolerant CDS in a UDG called a ((2, 2), m)-CDS, which is different from a traditional fault-tolerant CDS ((k, m)-CDS). A ((2, 2), m)-CDS can still function even if one edge or one node fails, which implies that it possesses fault-tolerant properties for both nodes and edges, in contrast to traditional (k, m)-CDSs, which possess fault tolerance only for nodes. Then, we propose a 5 α -approximation algorithm for computing a ((2, 2), m)-CDS, where α is the performance ratio for computing a (2, m) -CDS, and analyze its time complexity. In simulations, we compare our algorithm with an existing algorithm for fault-tolerant CDS construction based on CDS size. From the simulations, we find that our algorithm outperforms its competitor in constructing a quality VB. [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]

An efficient partial ellipsoid approximation scheme is presented to find a 1 ⌈ m 2 ⌉ -approximation solution to the nonconvex homogeneous quadratic optimization with m convex quadratic constraints, where ⌈ x ⌉ is the smallest integer larger than or equal to x. If there is an additional nonconvex quadratic constraint beyond the m convex constraints, we can use the new scheme to find a 1 m -approximation solution. [ABSTRACT FROM AUTHOR]

We study geometric variations of the discriminating code problem. In the discrete version of the problem, a finite set of points P and a finite set of objects S are given in R d . The objective is to choose a subset S ∗ ⊆ S of minimum cardinality such that for each point p i ∈ P , the subset S i ∗ ⊆ S ∗ covering p i satisfies S i ∗ ≠ ∅ , and each pair p i , p j ∈ P , i ≠ j , we have S i ∗ ≠ S j ∗ . In the continuous version of the problem, the solution set S ∗ can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case ( d = 1 ), the points in P are placed on a horizontal line L, and the objects in S are finite-length line segments aligned with L (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. This is also in contrast with most geometric covering problems, which are usually polynomial-time solvable in one dimension. Still for the 1-dimensional discrete version, we design a polynomial-time 2-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case ( d = 2 ) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce (16 · O P T + 1) -approximate and (64 · O P T + 1) -approximate solutions respectively, using rounding of suitably defined integer linear programming problems. Finally, we apply our techniques to a related variant of the discrete problem, where instead of points and geometric objects we just have a set S of objects. The goal is to select a small subset S ∗ of objects so that all objects of S are discriminated by their intersection with the objects of S ∗ . This problem can be viewed as a graph problem by stating it in terms of the vertices of the geometric intersection graph of S. Under this graph-theoretical form, it is known as the identifying code problem. We show that the identifying code problem for axis-parallel unit square intersection graphs (in d = 2 ) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects described above, and all our positive approximation results still hold in this setting. [ABSTRACT FROM AUTHOR]

We consider the intractable problem of finding several edge-disjoint spanning trees of the minimum total weight with a given diameter in complete undirected graph in current paper. The weights of edges of a graph are random variables from several continuous distributions: uniform, biased truncated exponential, biased truncated normal. The approximation algorithm with time complexity (n2), where n is number of vertices in graph, is proposed for solving this problem. The asymptotic optimality conditions for constructed algorithm is presented for each considered probabilistic distribution. [ABSTRACT FROM AUTHOR]

We consider the problem of interdicting a directed graph by deleting nodes with the goal of minimizing the local edge connectivity of the remaining graph from a given source to a sink. We introduce and study a general downgrading variant of the interdiction problem where the capacity of an arc is a function of the subset of its endpoints that are downgraded, and the goal is to minimize the downgraded capacity of a minimum source-sink cut subject to a node downgrading budget. This models the case when both ends of an arc must be downgraded to remove it, for example. For this generalization, we provide a bicriteria (4, 2)-approximation that downgrades nodes with total weight at most 4 times the budget and provides a solution where the downgraded connectivity from the source to the sink is at most 2 times that in an optimal solution. We accomplish this with an LP relaxation and rounding using a ball-growing algorithm based on the LP values. Furthermore, we show that other bicriteria approximations exist where one can worsen the approximation factor for one of the costs in order to improve the other. We further generalize the downgrading problem to one where each vertex can be downgraded to one of k levels, and the arc capacities are functions of the pairs of levels to which its ends are downgraded. We generalize our LP rounding to get a (4k, 4k)-approximation for this case. Trade-offs between the two approximation ratios similar to the two-level case also exist for the generalized problem. By transferring node values to edge values, we also derive new bicriteria approximation results for the vertex interdiction versions of the multiway cut problem in digraphs and multicut problems in undirected graphs. [ABSTRACT FROM AUTHOR]

We give the first 2-approximation algorithm for the cluster vertex deletion problem. This approximation factor is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algorithm combines previous approaches, based on the local ratio technique and the management of twins, with a novel construction of a "good" cost function on the vertices at distance at most 2 from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem. [ABSTRACT FROM AUTHOR]

In this paper, we devise improved approximation algorithms for the Min–Max Rural Postmen Cover Problem (RuralPostCover) and the Min–Max Chinese Postmen Cover Problem (ChinesePostCover), which are natural extensions of the classical Rural Postman Problem and the Chinese Postman Problem where multiple postmen are available. These results are based on some key observations, a new approach to derive closed walks from (open) walks and an efficient postmen allocation procedure in the literature. As an application of the algorithm for RuralPostCover, we give the first constant-factor approximation algorithms for the Min–Max Subtree Cover Problem (SubtreeCover) and its generalization, called the Min–Max Steiner Tree Cover Problem with Vertex Weights (SteinerTreeCover), using simple approximation preserving reductions. Moreover, we devise specialized algorithms for SteinerTreeCover (SubtreeCover) with better approximation ratios. [ABSTRACT FROM AUTHOR]

Exploiting drones as flying base stations to assist the terrestrial cellular networks or replace them is promising in 5G and beyond. One of the challenging problems is optimally deploying multiple drones to achieve coverage for the ground users. Usually, the goal is to find the minimum number of drones and their placement when all users are served. In this work, we consider a more realistic scenario. We focus on the situation where the number of drones is given in advance, and this number is significantly smaller than the number required to cover all ground users. This assumption is reasonable in emergency cases or battlefields where the number of ground users (for example, soldiers or firefighters) is much larger than the number of drones. Additionally, we consider the case when the ground users have a rank, interpreted as weight, and we aim to deploy drones' swarm such that the sum of the weights of the ground users covered by the swarm is maximized while the drones in the swarm are connected (without involving a third party entity that provides connectivity in the swarm). Our solution significantly improves currently best known approximation ratio for the problem from 1/144 to 1/28. [ABSTRACT FROM AUTHOR]

We consider a knapsack-constrained maximization problem of a nonnegative monotone DR-submodular function f over a bounded integer lattice [ B ] in R + n , max { f (x) : x ∈ [ B ] and ∑ i = 1 n x (i) c (i) ≤ 1 } , where n is the cardinality of a ground set N and c (·) is a cost function defined on N. Soma and Yoshida [Math. Program., 172 (2018), pp. 539-563] present a (1 - e - 1 - O (ϵ)) -approximation algorithm for this problem by combining threshold greedy algorithm with partial element enumeration technique. Although the approximation ratio is almost tight, their algorithm runs in O (n 3 ϵ 3 log 3 τ [ log 3 B ∞ + n ϵ log B ∞ log 1 ϵ c min ]) time, where c min = min i c (i) and τ is the ratio of the maximum value of f to the minimum nonzero increase in the value of f. Besides, Ene and Nguy e ˇ ~ n [arXiv:1606.08362, 2016] indirectly give a (1 - e - 1 - O (ϵ)) -approximation algorithm with O ((1 ϵ) O (1 / ϵ 4) n log ‖ B ‖ ∞ log 2 (n log ‖ B ‖ ∞)) time. But their algorithm is random. In this paper, we make full use of the DR-submodularity over a bounded integer lattice, carry forward the greedy idea in the continuous process and provide a simple deterministic rounding method so as to obtain a feasible solution of the original problem without loss of objective value. We present a deterministic algorithm and theoretically reduce its running time to a new record, O ( (1 ϵ) O (1 / ϵ 5) · n log 1 c min log ‖ B ‖ ∞) , with the same approximate ratio. [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]

We consider the k -Prize-Collecting Steiner Tree Problem. An instance is composed of an integer k and a graph G with costs on edges and penalties on vertices. The objective is to find a tree spanning at least k vertices which minimizes the cost of the edges in the tree plus the penalties of vertices not in the tree. This is one of the most fundamental network design problems and is a common generalization of the Prize-Collecting Steiner Tree Problem and the k -Minimum Spanning Tree Problem. Our main result is a 2-approximation algorithm, which improves on the currently best known approximation factor of 3.96 and has a faster running time. The algorithm builds on a modification of the primal-dual framework of Goemans and Williamson, and reveals interesting properties that can be applied to other similar problems. [ABSTRACT FROM AUTHOR]

In healthcare management, one important problem is how to efficiently allocate resources with limited capacities. From the point of view of a central decision maker with a long planning horizon, many healthcare resources such as appointment slots, operation rooms, and hospital beds can be regarded as reusable resources. Motivated by this observation, we study a new version of the assignment problem of reusable resources to sequentially arriving users. Different from previous works on reusable resources, we assume that each user will tell us the intended usage duration of each resource upfront before a decision needs to be made. In the setting where users arrive randomly according to a certain distribution, we propose an algorithm based on linear approximations of the optimal value functions, and show that it performs better than common methods developed for the case where usage times are known upon return in different simulations, as well as in a case study with real data. When users arrive in an adversarial order, we show by example that it is hard to design an algorithm with a lower-bounded competitive ratio. As users are much less likely to arrive in an adversarial order in reality, our proposed algorithm can be used to effectively schedule resources such as appointment slots and operation rooms in hospitals. [ABSTRACT FROM AUTHOR]

We study variants of the Strip Cover problem (SC) in which sensors with limited battery power are deployed on a line barrier, and the goal is to cover the barrier as long as possible. The energy consumption of a sensor depends on its sensing radius: Energy is drained in proportion to the sensor radius raised to a constant exponent α ≥ 1 . In the Set Once Strip Cover problem (OnceSC), the radius of each sensor can be set once, and the sensor can be activated at any time. SC k and O N C E SC k are variants of SC and OnceSC, respectively, in which each sensor is associated with a set of at most k predetermined radii. It was previously known that OnceSC is NP-hard when α = 1 , and the complexity of the case where α > 1 remained open. We extend the above-mentioned NP-hardness result in two ways: We show that OnceSC is NP-hard for every α > 1 and that OnceSC is strongly NP-hard for α = 1 . In addition, we show that O N C E SC k , for k ≥ 2 , is NP-hard, for any α ≥ 1 , even for uniform radii sets. On the positive side, we present (i) a 5 γ α -approximation algorithm for O N C E SC k , for k ≥ 1 , where γ is the maximum ratio between two radii associated with the same sensor; (ii) a 5-approximation algorithm for SC k , for every k ≥ 1 ; and (iii) a (5 + ε) -approximation algorithm for Strip Cover, for any constant ε > 0 . Finally, we present an O (n log n) -time algorithm for a variant of O N C E SC k in which all sensors must be activated at the same time. [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]

This work studies solution methods for approximating a given tensor by a sum of R rank-1 tensors with one or more of the latent factors being orthonormal. Such a problem arises from applications such as image processing, joint singular value decomposition, and independent component analysis. Most existing algorithms are of the iterative type, while algorithms of the approximation type are limited. By exploring the multilinearity and orthogonality of the problem, we introduce an approximation algorithm in this work. Depending on the computation of several key subproblems, the proposed approximation algorithm can be either deterministic or randomized. The approximation lower bound is established, both in the deterministic and the expected senses. The approximation ratio depends on the size of the tensor, the number of rank-1 terms, and is independent of the problem data. When reduced to the rank-1 approximation case, the approximation bound coincides with those in the literature. Moreover, the presented results fill a gap left in Yang (SIAM J Matrix Anal Appl 41:1797–1825, 2020), where the approximation bound of that approximation algorithm was established when there is only one orthonormal factor. Numerical studies show the usefulness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min–avg minimizes the average tour cost and min–max minimizes the maximum tour cost over all days. For min–max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min–avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open. [ABSTRACT FROM AUTHOR]

Resource Minimization Fire Containment (RMFC) is a natural model for optimal inhibition of harmful spreading phenomena on a graph. In the RMFC problem on trees, we are given an undirected tree G, and a vertex r where the fire starts at, called root. At each time step, the firefighters can protect up to B vertices of the graph while the fire spreads from burning vertices to all their neighbors that have not been protected so far. The task is to find the smallest B that allows for saving all the leaves of the tree. The problem is hard to approximate up to any factor better than 2 even on trees unless P = NP (King and MacGillivray in Discret Math 310(3):614–621, 2010). Chalermsook and Chuzhoy (In: Proceedings of the 21st annual ACM-SIAM symposium on discrete algorithms, SODA 2010, Austin, Texas, USA, 17–19 Jan 2010, SIAM, pp 1334–1349, 2010) presented a Linear Programming (LP) based O (log ∗ n) approximation for RMFC on trees that matches the integrality gap of the natural Linear Programming relaxation. This was recently improved by Adjiashvili et al. (ACM Trans Algorithms 15(2):20:1–20:33, 2019) to a 12-approximation through a combination of LP rounding along with several new techniques. In this paper we present an asymptotic QPTAS for RMFC on trees. More specifically, let ϵ > 0 , and I be an instance of RMFC where the optimum number of firefighters to save all the leaves is O P T (I) . We present an algorithm which uses at most ⌈ (1 + ϵ) O P T (I) ⌉ many firefighters at each time step and runs in time n O (log log n / ϵ) . This suggests that the existence of an asymptotic PTAS is plausible especially since the exponent is O (log log n) , not O (log n) . Our result combines a more refined height reduction lemma than the one in Adjiashvili et al. (2019) with LP rounding and dynamic programming to find the solution. We also apply our height reduction lemma to the algorithm provided in Adjiashvili et al. (2019) plus a more careful analysis to improve their 12-approximation and provide a polynomial time ( 5 + ϵ )-approximation. [ABSTRACT FROM AUTHOR]

In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of value at least half the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing at most half the value thus providing a 1/4-approximation algorithm and integrality gap for maximum integer multiflows in the plane. [ABSTRACT FROM AUTHOR]

Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense tensor BR1Approx, and is a higher-order extension of the sparse matrix BR1Approx, is one of the most important problems in sparse tensor decomposition and related problems arising from statistics and machine learning. By exploiting the multilinearity as well as the sparsity structure of the problem, four polynomial-time approximation algorithms are proposed, which are easily implemented, of low computational complexity, and can serve as initial procedures for iterative algorithms. In addition, theoretically guaranteed approximation lower bounds are derived for all the algorithms. We provide numerical experiments on synthetic and real data to illustrate the efficiency and effectiveness of the proposed algorithms; in particular, serving as initialization procedures, the approximation algorithms can help in improving the solution quality of iterative algorithms while reducing the computational time. [ABSTRACT FROM AUTHOR]

In this paper, we consider the group prize-collecting Steiner tree problem with submodular penalties (GPCST-SP problem). In this problem, we are given an undirected connected graph G = (V , E) with a pre-specified root r and a partition V = { V 0 , V 1 , ... , V k } of V with r ∈ V 0 . Assume c : E → R + is an edge cost function and p : 2 V → R + is a submodular penalty function, where R + is the set of nonnegative real numbers. For a group V i ∈ V , we say it is spanned by a tree if the tree contains at least one vertex of that group. The goal of the GPCST-SP problem is to find an r-rooted tree that minimizes the costs of the edges in the tree plus the penalty cost of the subcollection S containing these groups not spanned by the tree. Our main result is a 2I-approximation algorithm for the problem, where I = max { | V i | ∣ i = 0 , 1 , 2 , ... , k } . [ABSTRACT FROM AUTHOR]

We propose an algorithm to generate inner and outer polyhedral approximations to the upper image of a bounded convex vector optimization problem. It is an outer approximation algorithm and is based on solving norm-minimizing scalarizations. Unlike Pascoletti–Serafini scalarization used in the literature for similar purposes, it does not involve a direction parameter. Therefore, the algorithm is free of direction-biasedness. We also propose a modification of the algorithm by introducing a suitable compact subset of the upper image, which helps in proving for the first time the finiteness of an algorithm for convex vector optimization. The computational performance of the algorithms is illustrated using some of the benchmark test problems, which shows promising results in comparison to a similar algorithm that is based on Pascoletti–Serafini scalarization. [ABSTRACT FROM AUTHOR]

For a family of graphs F , Weighted F -Deletion is the problem for which the input is a vertex weighted graph G = (V , E) and the goal is to delete S ⊆ V with minimum weight such that G \ S ∈ F . Designing a constant-factor approximation algorithm for large subclasses of perfect graphs has been an interesting research direction. Block graphs, 3-leaf power graphs, and interval graphs are known to admit constant-factor approximation algorithms, but the question is open for chordal graphs and distance-hereditary graphs. In this paper, we add one more class to this list by presenting a constant-factor approximation algorithm when F is the intersection of chordal graphs and distance-hereditary graphs. They are known as ptolemaic graphs and form a superset of both block graphs and 3-leaf power graphs above. Our proof presents new properties and algorithmic results on inter-clique digraphs as well as an approximation algorithm for a variant of Feedback Vertex Set that exploits this relationship (named Feedback Vertex Set with Precedence Constraints), each of which may be of independent interest. [ABSTRACT FROM AUTHOR]

We are given a set of parallel jobs that have to be executed on a set of speed-scalable processors varying their speeds dynamically. Running a job at a slower speed is more energy-efficient, however, it takes a longer time and affects the performance. Every job is characterized by the processing volume and the number or the set of the required processors. Our objective is to minimize the maximum completion time so that the energy consumption is not greater than a given energy budget. For various particular cases, we propose polynomial-time approximation algorithms, consisting of two stages. At the first stage, we give an auxiliary convex program. By solving this problem, we get processing times of jobs and a lower bound on the makespan. Then, at the second stage, we transform our problem into the corresponding scheduling problem with the constant speed of processors and construct a feasible schedule. We also obtain an "almost exact" solution for the preemptive settings based on a configuration linear program. [ABSTRACT FROM AUTHOR]

Energy harvesting technologies allow wireless devices to be recharged by the surrounding environment, providing wireless sensor networks (WSNs) with higher performance and longer lifetime. However, directly building a wireless sensor network with energy harvesting nodes is very costly. A compromise is upgrading existing networks with energy harvesting technologies. In this paper, we focus on prolonging the lifetime of WSNs with the help of energy harvesting relays (EHRs). EHRs are responsible for forwarding data for sensor nodes, allowing them to become terminals and thus extending their lifetime. We aim to deploy a minimum number of relays covering the whole network. As EHRs have several special properties such as the energy harvesting and depletion rate, it brings great research challenges to seek an optimal deployment strategy. To this end, we propose an approximation algorithm named Effective Relay Deployment Algorithm, which can be divided into two phases: disk covering and connector insertion using the partitioning technique and the Steinerization technique, respectively. Based on probabilistic analysis, we further optimize the performance ratio of our algorithm to (5 + 6/K) where K is an integer denoting the side length of a cell after partitioning. Our extensive simulation results show that our algorithm can reduce the number of EHRs to be deployed by up to 45% compared with previous work and thus validate the efficiency and effectiveness of our solution. [ABSTRACT FROM AUTHOR]

We study the problem of finding maximum weakly stable matchings when preference lists are incomplete and contain one-sided ties of bounded length. We show that if the tie length is at most L, then it is possible to achieve an approximation ratio of 1 + (1 − 1 L) L . We also show that the same ratio is an upper bound on the integrality gap, which matches the known lower bound. In the case where the tie length is at most 2, our result implies an approximation ratio and integrality gap of 5 4 , which matches the known UG-hardness result. [ABSTRACT FROM AUTHOR]

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin. [ABSTRACT FROM AUTHOR]

In the classic maximum coverage problem, we are given subsets T 1 , ... , T m of a universe [n] along with an integer k and the objective is to find a subset S ⊆ [ m ] of size k that maximizes C (S) : = | ∪ i ∈ S T i | . It is well-known that the greedy algorithm for this problem achieves an approximation ratio of (1 - e - 1) and there is a matching inapproximability result. We note that in the maximum coverage problem if an element e ∈ [ n ] is covered by several sets, it is still counted only once. By contrast, if we change the problem and count each element e as many times as it is covered, then we obtain a linear objective function, C (∞) (S) = ∑ i ∈ S | T i | , which can be easily maximized under a cardinality constraint. We study the maximum ℓ -multi-coverage problem which naturally interpolates between these two extremes. In this problem, an element can be counted up to ℓ times but no more; hence, we consider maximizing the function C (ℓ) (S) = ∑ e ∈ [ n ] min { ℓ , | { i ∈ S : e ∈ T i } | } , subject to the constraint | S | ≤ k . Note that the case of ℓ = 1 corresponds to the standard maximum coverage setting and ℓ = ∞ gives us a linear objective. We develop an efficient approximation algorithm that achieves an approximation ratio of 1 - ℓ ℓ e - ℓ ℓ ! for the ℓ -multi-coverage problem. In particular, when ℓ = 2 , this factor is 1 - 2 e - 2 ≈ 0.73 and as ℓ grows the approximation ratio behaves as 1 - 1 2 π ℓ . We also prove that this approximation ratio is tight, i.e., establish a matching hardness-of-approximation result, under the Unique Games Conjecture. This problem is motivated by the question of finding a code that optimizes the list-decoding success probability for a given noisy channel. We show how the multi-coverage problem can be relevant in other contexts, such as combinatorial auctions. [ABSTRACT FROM AUTHOR]