Semidefinite and Linear Programming Integrality Gaps for Scheduling Identical Machines

Kurpisz, Adam; Mastrolilli, Monaldo; Mathieu, Claire; Mömke, Tobias; Verdugo, Victor; Wiese, Andreas

doi:10.1007/978-3-319-33461-5_13

Cited by 3 publications

(5 citation statements)

References 22 publications

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“…Proof. The example is almost similar to the example in [24] which was used to show integrality gap examples for strong LP relaxations for identical machines makespan minimization problem. We just sketch a proof here.…”

Section: Max-min Allocation Problems and Supply Polyhedramentioning

confidence: 89%

The Heterogeneous Capacitated k-Center Problem

Chakrabarty

Krishnaswamy

Kumar

2017

Integer Programming and Combinatorial Optimization

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In this paper we initiate the study of the heterogeneous capacitated k-center problem: given a metric space X = (F ∪ C, d), and a collection of capacities. The goal is to open each capacity at a unique facility location in F , and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities ci's are identical, the problem becomes the well-studied uniform capacitated k-center problem for which constant-factor approximations are known [8,23]. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem, as well the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical santa-claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results for Heterogeneous Cap-k-Center.• A quasi-polynomial time O(log n/ε)-approximation where every capacity is violated by 1 + ε.• A polynomial time O(1)-approximation where every capacity is violated by an O(log n) factor. We get improved results for the soft-capacities version where we can place multiple facilities in the same location.

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Section: Max-min Allocation Problems and Supply Polyhedramentioning

confidence: 89%

The Heterogeneous Capacitated k-Center Problem

Chakrabarty

Krishnaswamy

Kumar

2017

Integer Programming and Combinatorial Optimization

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“…Hard instances. We briefly describe the construction of a family of hard instances {I k } k∈N for the configuration linear program in [40]. Let T = 1023, and for each odd k ∈ N we have n = 15k jobs and 3k machines.…”

Section: Configuration Linear Programmentioning

confidence: 99%

“…This shows a natural limitation to the power of hierarchies as one-fit-all techniques. Quite remarkably, the instances used for obtaining lower bounds often have a very symmetric structure [43,21,56,40,58], which suggests a strong connection between the tightness of the relaxation given by these hierarchies and symmetries. The main purpose of this article is to study this connection for a specific relevant problem, namely, minimum makespan scheduling on identical machines.…”

Section: Introductionmentioning

confidence: 99%

“…The stronger configuration linear program, uses an exponential number of variables y iC which indicate whether the set of jobs assigned to i has C as a multiset of processing times. Kurpisz et al [40] showed that the configuration linear program has an integrality gap of at least 1024/1023 ≈ 1.0009 even after a linear number of rounds of the LS + or SA hierarchies. Hence, the same lower bound holds when the ground formulation is the assignment linear program.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, the same lower bound holds when the ground formulation is the assignment linear program. On the other hand, Kurpisz et al [40] leave open whether the SoS hierarchy applied to the configuration linear program has a (1 + ε) integrality gap after O ε (1) many rounds. Our first main contribution is a negative answer to this question.…”

Section: Introductionmentioning

confidence: 99%

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Breaking Symmetries to Rescue Sum of Squares: The Case of Makespan Scheduling

Verdugo

Verschae

2019

Integer Programming and Combinatorial Optimization

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The Sum of Squares (SoS) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of the hierarchy give integrality gaps matching the best known approximation algorithm. In many other, however, ad-hoc techniques give significantly better approximation ratios. Notably, the lower bounds instances, in many cases, are invariant under the action of a large permutation group. The main purpose of this paper is to study how the presence of symmetries on a formulation degrades the performance of the relaxation obtained by the SoS hierarchy. We do so for the special case of the minimum makespan problem on identical machines. Our first result is to show that a linear number of rounds of SoS applied over the configuration linear program yields an integrality gap of at least 1.0009. This improves on the recent work by Kurpisz et al. [40] that shows an analogous result for the weaker LS + and SA hierarchies. Then, we consider the weaker assignment linear program and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying the SoS hierarchy for O ε (1) rounds to this linear program reduces the integrality gap to (1 + ε). Our results suggest that for this classical problem the symmetries of the natural assignment linear program were the main barrier preventing the SoS hierarchy to give relaxations with integrality gap (1 + ε) after a constant number of rounds. We leave as an open question whether this phenomenon occurs for different problems where the SoS hierarchy yields weak relaxations.

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Integrality gaps for colorful matchings

Kelk

Stamoulis

2019

Discrete Optimization

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We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem. We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances. We complement these results by showing that if we exclude certain simple sub-structures from our input graphs, then the integrality gap of the natural linear formulation strictly improves. To prove this, we adapt for our purposes the results of Füredi [Combinatorica, 1(2):155-162, 1981]. We further leverage this to show upper bounds on the performance of the Sherali-Adams hierarchy when applied to the natural LP relaxation of the BCM problem. say that edge e "has" color C j if e ∈ E j . Let C = ∪ i=1,...,k C i be the collection of all color classes. Each color class C j is associated with a positive number w j ≥ 1. Our goal is to find a maximum (weighted) matching M that contains at most w j edges of color C j i.e., a matching M such that |M ∩ E j | ≤ w j , ∀C j ∈ C.In [46] an LP-based approximation algorithm with approximation ratio 1/2 was given for the BCM problem, which matches the integrality gap of the natural LP relaxation for this problem. The algorithm is based on the elegant technique by Parekh [40] which gives an inductive process to write any basic feasible solution of the relaxed LP as an approximate sparse convex combination of integral solutions. The result holds for any bounds w j ≥ 1, integral or otherwise, since the analysis does not make use of the fact that w i ∈ Z + , ∀i, only the fact that w i ≥ 1 (otherwise the integrality gap could be unbounded). It has been further generalized by Parekh and Pritchard [41] to uniform hypergraphs.A very natural question occurs: a negative result based on a bad integrality gap instance rules out the possibility of a good relaxation-based approximation algorithm. But this holds only for the particular relaxation that we use. What about other, more complicated and sophisticated relaxations? As an illustrative example, if we take the normal (degree-constrained) relaxation for the classical matching problem, which has integrality gap of 3 /2, and enhance it with the blossom inequalities, we get an exact formulation of the convex hull of all integer points for the matching problem [16].Given the apparent difficulty of identifying stronger/tighter linear relaxations for combinatorial optimization problems, a large body of work has been dedicated in recent years to identifying systematic techniques to enhance the quality of a given linear (or semi-definite) program with valid inequalities (inequalities that are satisfied by all integral points). The hope is that the part of the polyhedron responsible for the bad integrality gap example will be eliminated. Many such "lift and project" methods have been proposed so far, in particular by Sherali and Adams (SA) [45], by Lovász and Schrijver (LS) [29], by Balas, Ceria and Cornuéjols (BCC) [5], by Lasserre [25] and by B...

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Semidefinite and Linear Programming Integrality Gaps for Scheduling Identical Machines

Cited by 3 publications

References 22 publications

The Heterogeneous Capacitated k-Center Problem

The Heterogeneous Capacitated k-Center Problem

Breaking Symmetries to Rescue Sum of Squares: The Case of Makespan Scheduling

Integrality gaps for colorful matchings

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