LP-Based Algorithms for Capacitated Facility Location

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding. © 2014 IEEE.

LP-Based Algorithms For Capacitated Facility Location

Hyung Chan An, Ola Nils Anders Svensson

Linear programming (LP) has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and the primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no LP relaxation was known to efficiently approximate the problem. In this paper, we present an LP relaxation with a constant integrality gap for the capacitated facility location. We demonstrate that the fundamental theories of multicommodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.

Society for Industrial and Applied Mathematics2017

Applications of Strong Convex Relaxations to Allocation Problems

Christos Kalaitzis

Approximation algorithms are a commonly used tool for designing efficient algorithmic solutions for intractable problems, at the expense of the quality of the output solution. A prominent technique for designing such algorithms is the use of Linear Programming (LP) relaxations. An optimal solution to such a relaxation provides a bound on the objective value of the optimal integral solution, to which we compare the integral solution we return. In this context, when studying a specific problem, two natural questions often arise: What is a strong LP relaxation for this problem, and how can we exploit it? Over the course of the past few decades, a significant amount of effort has been expended by the research community in order to answer these questions for a variety of interesting intractable problems. Although there exist multiple problems for which we have designed LP relaxations that achieve best-possible guarantees, there still exist numerous problems for which we either have no strong LP relaxations, or do not know how to use them. The main focus of this thesis is extending our understanding of such strong relaxations. We focus on designing good approximation algorithms for certain allocation problems, by employing a class of strong LP relaxations, called configuration-LPs. For many such allocation problems, the best-known results are derived by using simple and natural LP relaxations, whereas configuration-LPs have been used successfully on several occasions in order to break pre-existing barriers set by weaker relaxations. However, our understanding of configuration-LPs is far from complete for many problems. Therefore, understanding and using these relaxations to the farthest extent possible is a quite intriguing question. Answering this question could result in improved approximation algorithms for a wide variety of allocation problems. The first problem we address in this thesis is the restricted max-min fair allocation problem. Prior to our work, the best known result provided an

\Omega(1)

-approximation that ran in polynomial time. Also, it was known how to estimate the value of an optimal solution to the problem within a factor of

1/(4+c)

, for any

c>0

, by solving the corresponding configuration-LP. Our first contribution in this thesis is the design of a

1/13

-approximation algorithm for the problem, using the configuration-LP. Specifically, although our algorithm is fully combinatorial, it consists of a local-search procedure that is guaranteed to succeed only when the configuration-LP is feasible. In order to establish the correctness and running time of the algorithm, it is crucial to use the configuration-LP in our analysis. The second problem we study is the scheduling of jobs on unrelated machines in order to minimize the sum of weighted completion times. For this problem, the best known approximation algorithm achieves a ratio of

3/2-r

, for some small

r>0

. Our second contribution in this thesis is the improvement of this ratio to

(1+\sqrt{2})/2+c

, for any

c>0

, for the special case of the problem where the jobs have uniform Smith ratios. To achieve this ratio, we design a randomized rounding algorithm that rounds solutions to the corresponding configuration-LP. Through a careful examination of the distribution this randomized algorithm outputs, we identify the one that maximizes the approximation ratio, and we then upper bound the ratio this worst-case distribution exhibits by

(1+\sqrt{2})/2+c

EPFL2017

No Small Linear Program Approximates Vertex Cover within a Factor 2 - e

Abbas Bazzi, Ola Nils Anders Svensson

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regevproved that the problem is NP-hard to approximate within a factor2 - ∈, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best in approximability result for the problem is due to Dinur and Safra: vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP)relaxations of the problem: every LP relaxation that approximates vertex cover within a factor of 2-∈ has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomially many inequalities. © 2015 IEEE.

IEEE Computer Society2015

Applications of Strong Convex Relaxations to Allocation Problems

Christos Kalaitzis

\Omega(1)

-approximation that ran in polynomial time. Also, it was known how to estimate the value of an optimal solution to the problem within a factor of

1/(4+c)

, for any

c>0

, by solving the corresponding configuration-LP. Our first contribution in this thesis is the design of a

1/13

3/2-r

, for some small

r>0

. Our second contribution in this thesis is the improvement of this ratio to

(1+\sqrt{2})/2+c

, for any

c>0

(1+\sqrt{2})/2+c

EPFL2017