We study the classic weighted maximum throughput problem on unrelated machines. We give a (1-1/e-epsilon)-approximation algorithm for the preemptive case with no migrations. To our knowledge this is the first ever approximation result for this problem. It is an immediate consequence of a polynomial-time reduction we design, that uses any rho-approximation algorithm for the single-machine problem to obtain an approximation factor of (1-1/e)rho-epsilon for the corresponding unrelated-machines problem, for any epsilon>0. The reduction works both for the preemptive and the non-preemptive setting. On a single machine we present a PTAS for the non-preemptive version of the problem for the special case of a constant number of distinct due dates or distinct release dates. By our reduction this yields an approximation factor of (1-1/e)-epsilon for the non-preemptive problem on unrelated machines when there is a constant number of distinct due dates or release dates on each machine.

Motivated by time-sharing systems with deadlines, we introduce the study of the following problem. We are given m machines and n jobs, as well as a set of tolerance capacities u(ij) >= 0 for every job j and machine i. Can we assign the jobs so that, if job j ends up on machine i, the total size of jobs that are processed on i is at most u(ij)? We define two natural optimization versions: (i) Maximize the total weight of jobs that can be assigned without violating the tolerance capacities. (ii) Minimize the amount rho >= 1 by which capacities have to be scaled so that all jobs can be assigned. For (i), we provide constant-factor approximations even in the presence of additional side-constraints. For (ii), we provide a strong inapproximability result and integrality gap lower bounds for two key relaxations. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The irrelevant vertex technique provides a powerful tool for the design of parameterized algorithms for a wide variety of problems on graphs. A common characteristic of these problems, permitting the application of this technique on surface-embedded graphs, is the fact that every graph of large enough treewidth contains a vertex that is irrelevant, in the sense that its removal yields an equivalent instance of the problem. The straightforward application of this technique yields algorithms with running time that is quadratic in the size of the input graph. This running time is due to the fact that it takes linear time to detect one irrelevant vertex and the total number of irrelevant vertices to be detected is linear as well. Using advanced techniques, sub-quadratic algorithms have been designed for particular problems, even in general graphs. However, designing a general framework for linear-time algorithms has been open, even for the bounded-genus case. In this paper we introduce a general framework that enables finding in linear time an entire set of irrelevant vertices whose removal yields a bounded-treewidth graph, provided that the input graph has bounded genus. Our technique consists of decomposing any surface-embedded graph into a tree-structured collection of bounded-treewidth subgraphs where detecting globally irrelevant vertices can be done locally and independently. Our method is applicable to a wide variety of known graph containment or graph modification problems where the irrelevant vertex technique applies. Examples include the (Induced) Minor Folio problem, the (Induced) Disjoint Paths problem, and the ℱ-Minor-Deletion problem.

We study the approximability of covering problems when the set of items chosen to satisfy the covering constraints must form an ideal of a given partial order. We examine the general case with multiplicity constraints, where item i can be chosen up to d_i times. For the basic precedence-constrained knapsack problem (PCKP) we answer an open question of McCormick et al. (Algorithmica 783:771–787, 2017) and show the existence of approximation algorithms with strongly-polynomial bounds. PCKP is a special case, with a single covering constraint, of a precedence-constrained covering integer program (PCCP). For a general PCCP where the number of covering constraints is m ≥ 1, we show that an algorithm of Pritchard and Chakrabarty (Algorithmica 611:75–93, 2011) for covering integer programs can be extended to yield an f -approximation, where f is the maximum number of variables with nonzero coefficients in a covering constraint. This is nearly-optimal under standard complexity-theoretic assumptions and rather surprisingly matches the bound achieved for the problem without precedence constraints.

Motivated by time-sharing systems with deadlines, such as 2-way synchronization of Digital Twins, we introduce the study of a very natural problem which can be abstracted as follows. We are given m machines and n jobs, as well as a set of tolerance capacities u_ij≥ 0 for every job j and machine i. Can we assign the jobs so that, if job j ends up on machine i, at most u_ij jobs in total are processed on i? We define two natural optimization versions: (i) Maximize the total weight of jobs that can be assigned without violating the tolerance capacities u_ij , and (ii) minimize the amount ρ≥ 1, by which capacities have to be scaled so that all jobs can be assigned. For the first problem and its generalizations we provide an (1-1/e) -approximation algorithm. For the second problem we show that it is n^1/2-ε -inapproximable and provide linear integrality gap lower bounds for two key relaxations.

In this paper, an efficient O ( n 2 ) algorithm is proposed to solve a special case of single-item lot-sizing problems (SILSP) in which both the production and holding costs are piecewise linear, there is an all-unit discount with one breakpoint for the production cost, and the inventory is bounded. The algorithm is based on a key structural property that may be of more general interest, that of a just-in-time ordering policy. Finally, we show that when the problem is extended to two items, it is NP-complete.

The Disjoint Paths problem asks whether a fixed number of pairs of terminals in a graph $G$ can be linked by pairwise disjoint paths. In the context of this problem, Robertson and Seymour introduced the celebrated irrelevant vertex technique that has since become standard in graph algorithms. The technique consists of detecting a vertex that is irrelevant in the sense that its removal creates an equivalent instance of the problem. That way, one may solve the problem in $O(n^2)$ steps, as the detection of an irrelevant vertex takes $O(n)$ time and at most $n$ vertices may need to be removed. In this paper we study the Planar Disjoint Paths problem where the input graph is planar. We introduce an extension of the irrelevant vertex technique where all the irrelevant vertices are removed simultaneously so that an instance of the Planar Disjoint Paths problem can be transformed in a linear number of steps to an equivalent one that has bounded treewidth. As a consequence, the Planar Disjoint Paths problem can be solved in linear time for every fixed number of terminals.

We introduce Planar Disjoint Paths Completion, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph G, k pairs of terminals, and a face F of G, find a minimum-size set of edges, if one exists, to be added inside F so that the embedding remains planar and the pairs become connected by k disjoint paths in the augmented network. Our results are twofold: first, we give an upper bound on the number of necessary additional edges when a solution exists. This bound is a function of k, independent of the size of G. Second, we show that the problem is fixed-parameter tractable, in particular, it can be solved in time f(k) · n.

The DISJOINT PATHS PROBLEM asks, given a graph a and a set of pairs of terminals (s(1),t(1)), ..., (s(k), t(k)), whether there is a collection of k pairwise vertex-disjoint paths linking s(i) and for i = 1,..., k. In their f(k).n(3) algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose - very technical - proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we give a new and self-contained proof of this result that strongly exploits the combinatorial properties of planar graphs and achieves g(k) = O(k(3/2).2(k)). Our bound is radically better than the bounds known for general graphs. (C) 2016 Elsevier Inc. All rights reserved.

Metric uncapacitated facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. Capacitated facility location (Cfl) is a generalization for which there are local-search-based constant-factor approximations, while there is no known compact relaxation with constant integrality gap. This paper produces, through a host of impossibility results, the first comprehensive investigation of the effectiveness of mathematical programming for metric capacitated facility location, with emphasis on lift-and-project methods. We show that the relaxations obtained from the natural LP at \(\varOmega (n)\) levels of the semidefinite Lovász–Schrijver hierarchy for mixed programs, and at \(\varOmega (n)\) levels of the Sherali–Adams hierarchy, have an integrality gap of \(\varOmega (n)\), where \(n\) is the number of facilities, partially answering an open question of An et al. (Centrality of trees for capacitated k-center, 2013), Li and Svensson (Proceedings of 45th ACM Symposium on Theory of Computing, STOC ’13. ACM, pp 901–910, 2013). For the families of instances we consider, both hierarchies yield at the \(n\)th level an exact formulation for Cfl. Thus our bounds are asymptotically tight. Building on our methodology for the Sherali–Adams result, we prove that the standard Cfl relaxation enriched with the submodular inequalities of Aardal et al. (Math Oper Res 20:562–582, 1995), a generalization of the flow-cover valid inequalities, has also an \(\varOmega (n)\) gap and thus not bounded by any constant. This disproves a long-standing conjecture of Levi et al (Math Program 131(1–2):365–379, 2012). We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for Cfl through a sharp threshold phenomenon.

We introduce Planar Disjoint Paths Completion, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a plane graph G, k pairs of terminals, and a face F of G, find a minimum-size set of edges, if one exists, to be added inside F so that the embedding remains planar and the pairs become connected by k disjoint paths in the augmented network. Our results are twofold: first, we give an explicit bound on the number of necessary additional edges if a solution exists. This bound is a function of k, independent of the size of G. Second, we show that the problem is fixed-parameter tractable, in particular, it can be solved in time f(k)·n 2.

Exploring the power of linear programming for combinatorial optimization problems has been recently receiving renewed attention after a series of breakthrough impossibility results. From an algorithmic perspective, the related questions concern whether there are compact formulations even for problems that are known to admit polynomial-time algorithms. We propose a framework for proving lower bounds on the size of extended formulations. We do so by introducing a specific type of extended relaxations that we call product relaxations and is motivated by the study of the Sherali-Adams (SA) hierarchy. Then we show that for every approximate relaxation of a polytope P, there is a product relaxation that has the same size and is at least as strong. We provide a methodology for proving lower bounds on the size of approximate product relaxations by lower bounding the chromatic number of an underlying hypergraph, whose vertices correspond to gap-inducing vectors. We extend the definition of product relaxations and our methodology to mixed integer sets. However in this case we are able to show that mixed product relaxations are at least as powerful as a special family of extended formulations. As an application of our method we show an exponential lower bound on the size of approximate mixed product formulations for the metric capacitated facility location problem, a problem which seems to be intractable for linear programming as far as constant-gap compact formulations are concerned.

Metric facility location is a well-studied problem for which linear programming methods have been used with great success in deriving approximation algorithms. The capacity-constrained generalizations, such as capacitated facility location (CFL) and lower-bounded facility location (LBFL), have proved notorious as far as LP-based approximation is concerned: while there are local-search-based constant-factor approximations, there is no known linear relaxation with constant integrality gap. According to Williamson and Shmoys devising a relaxation-based approximation for CFL is among the top 10 open problems in approximation algorithms. This paper advances significantly the state-of-the-art on the effectiveness of linear programming for capacity-constrained facility location through a host of impossibility results for both CFL and LBFL. We show that the relaxations obtained from the natural LP at Omega(n) levels of the Sherali-Adams hierarchy have an unbounded gap, partially answering an open question from the literature. Here, n denotes the number of facilities in the instance. Building on the ideas for this result, we prove that the standard CFL relaxation enriched with the generalized flow-cover valid inequalities has also an unbounded gap. This disproves a long-standing conjecture of Levi et al. We finally introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation and captures general configuration-style LPs. We characterize the behavior of proper relaxations for CFL and LBFL through a sharp threshold phenomenon.

The metric capacitated facility location is a well-studied problem for which, while constant factor approximations are known, no efficient relaxation with constant integrality gap is known. The question whether there is such a relaxation is among the most important open problems of approximation algorithms \cite{ShmoysWbook}. In this paper we show that, if one is restricted to linear programs that use the natural encoding for facility location, at least an exponential number of constraints is needed to achieve a constant gap. Our proof does not assume any special property of the relaxation such as locality or symmetry.

The metric uncapacitated facility location problem (UFL) enjoys a special stature in approximation algorithms as a testbed for various techniques. Two generalizations of UFL are capacitated facility location (CFL) and lower-bounded facility location (LBFL). In the former, every facility has a capacity which is the maximum demand that can be assigned to it, while in the latter, every open facility is required to serve a given minimum amount of demand. Both CFL and LBFL are approximable within a constant factor but their respective natural LP relaxations have an unbounded integrality gap. According to Shmoys and Williamson, the existence of a relaxation-based algorithm for CFL is one of the top 10 open problems in approximation algorithms. In this paper we give the first results on this problem. We provide substantial evidence against the existence of a good LP relaxation for CFL by showing unbounded integrality gaps for two families of strengthened formulations. The first family we consider is the hierarchy of LPs resulting from repeated applications of the lift-and-project Lovász-Schrijver procedure starting from the standard relaxation. We show that the LP relaxation for CFL resulting after $\Omega(n)$ rounds, where $n$ is the number of facilities in the instance, has unbounded integrality gap. Note that the Lovász-Schrijver procedure is known to yield an exact formulation for CFL in at most $n$ rounds. We also introduce the family of proper relaxations which generalizes to its logical extreme the classic star relaxation, an equivalent form of the natural LP. We characterize the integrality gap of proper relaxations for both LBFL and CFL and show a threshold phenomenon under which it decreases from unbounded to 1.

In the Capacitated facility location (Cfl) problem we are given a set F of facilities and a set C of clients in a common metric space. Every facility has a hard capacity ui. Opening a facility i incurs a nonnegative cost fi, while assigning a client j to facility i incurs a nonnegative connection cost cij . The goal is to open a subset F ′ ⊆ F of the facilities and assign each client to an open facility so that at most ui clients are assigned to facility i and the total cost is minimized. Cfl is approximable within a constant factor via local search but the natural LP relaxation has an unbounded integrality gap. It is an open question whether there is a relaxation-based approximation algorithm for Cfl or even if an LP with constant integrality gap exists. We present the first study of a comprehensive family of strengthened linear programs for this problem. We consider the hierarchy of polytopes resulting from repeated applications of the Lovász-Schrijver lift-and-project operator on the standard fractional polytope. We show that the LP relaxation for Cfl resulting in Ω(n) rounds, where n is the number of facilities in the instance, has an unbounded integrality gap. The Lovász-Schrijver procedure is known to yield an exact formulation for Cfl in at most n rounds. ∗This research has been co-financed by the European Union (European Social Fund – ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) Research Funding Program: “Thalis. Investing in knowledge society through the European Social Fund”. †Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Panepistimiopolis Ilissia, Athens 157 84, Greece; (www.di.uoa.gr/ ̃sgk). Part of this work conducted while visiting the IEOR Department, Columbia University, New York, NY 10027. ‡Department of Informatics and Telecommunications, National and Kapodistrian University of Athens, Panepistimiopolis Ilissia, Athens 157 84, Greece; (gmoys@di.uoa.gr). Partially supported by an NKUAELKE graduate fellowship.

We consider the following special case of minimizing makespan. A set of jobs J and a set of machines M are given. Each job j∈J can be scheduled on a machine from a subset Mj of M. The processing time of j is the same on all machines in Mj. The jobs are of two sizes, namely b (big) and s (small). We present a polynomial-time algorithm that approximates the value of the optimal makespan within a factor of 1.883 and some further improvements when every job can be scheduled on at most two machines.

The Disjoint-Paths Problem asks, given a graph G and a set of pairs of terminals (s1, t1),..., (sk, tk), whether there is a collection of k pairwise vertex-disjoint paths linking si and ti, for i = 1,..., k. In their f(k) ċ n3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose - very technical - proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we prove this result for planar graphs achieving g(k) = 2O(k). Our bound is radically better than the bounds known for general graphs. Moreover, our proof is new and self-contained, and it strongly exploits the combinatorial properties of planar graphs. We also prove that our result is optimal, in the sense that the function g(k) cannot become better than exponential. Our results suggest that any algorithm for the DISJOINT-PATHS PROBLEM that runs in time better than 22o(k) ċnO(1) will probably require drastically different ideas from those in the irrelevant vertex technique.

We consider network congestion games in which a finite number of non-cooperative users select paths. The aim is to mitigate the inefficiency caused by the selfish users by introducing taxes on the network edges. A tax vector is strongly (weakly)-optimal if all (at least one of) the equilibria in the resulting game minimize(s) the total latency. The issue of designing optimal tax vectors for selfish routing games has been studied extensively in the literature. We study for the first time taxation for networks with atomic users which have unsplittable traffic demands and are heterogeneous, i.e., have different sensitivities to taxes. On the positive side, we show the existence of weakly-optimal taxes for single-source network games. On the negative side, we show that the cases of homogeneous and heterogeneous users differ sharply as far as the existence of strongly-optimal taxes is concerned: there are parallel-link games with linear latencies and heterogeneous users that do not admit strongly-optimal taxes.

A natural generalization of the selfish routing setting arises when some of the users obey a central coordinating authority, while the rest act selfishly. Such behavior can be modeled by dividing the users into an α fraction that are routed according to the central coordinator’s routing strategy ( Stackelberg strategy ), and the remaining 1− α that determine their strategy selfishly, given the routing of the coordinated users. One seeks to quantify the resulting price of anarchy, i.e., the ratio of the cost of the worst traffic equilibrium to the system optimum, as a function of α . It is well-known that for α =0 and linear latency functions the price of anarchy is at most 4/3 (J. ACM 49, 236–259, 2002). If α tends to 1, the price of anarchy should also tend to 1 for any reasonable algorithm used by the coordinator. We analyze two such algorithms for Stackelberg routing, LLF and SCALE. For general topology networks, multicommodity users, and linear latency functions, we show a price of anarchy bound for SCALE which decreases from 4/3 to 1 as α increases from 0 to 1, and depends only on α . Up to this work, such a tradeoff was known only for the case of two nodes connected with parallel links (SIAM J. Comput. 33, 332–350, 2004), while for general networks it was not clear whether such a result could be achieved, even in the single-commodity case. We show a weaker bound for LLF and also some extensions to general latency functions. The existence of a central coordinator is a rather strong requirement for a network. We show that we can do away with such a coordinator, as long as we are allowed to impose taxes (tolls) on the edges in order to steer the selfish users towards an improved system cost. As long as there is at least a fraction α of users that pay their taxes, we show the existence of taxes that lead to the simulation of SCALE by the tax-payers. The extension of the results mentioned above quantifies the improvement on the system cost as the number of tax-evaders decreases.