Michel X. Goemans

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least.87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of 1/2 for MAX CUT and 3/4 or MAX 2SAT. Slight extensions of our analysis lead to a.79607-approximation algorithm for the maximum directed cut problem (MAX DICUT) and a.758-approximation algorithm for MAX SAT, where the best previously known approximation algorithms had performance guarantees of 1/4 and 3/4, respectively. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

A General Approximation Technique for Constrained Forest Problems

We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our technique produces approximation algorithms that run in <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic>) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of <italic>O</italic>(<italic>n</italic><supscrpt>2</supscrpt> log <italic>n</italic>) time compares favorably with the best strongly polynomial exact algorithms running in <italic>O</italic>(<italic>n</italic><supscrpt>3</supscrpt>) time for dense graphs. A similar result is obtained for the 2-matching problem and its variants.We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [4].

Approximating the value of two power proof systems, with applications to MAX 2SAT and MAX DICUT

It is well known that two prover proof systems are a convenient tool for establishing hardness of approximation results. In this paper, we show that two prover proof systems are also convenient starting points for establishing easiness of approximation results. Our approach combines the Feige-Lovasz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semidefinite programs, as introduced by Goemans and Williamson (STOC94). As a consequence of our approach, we present improved approximation algorithms for MAX 2SAT and MAX DICUT. The algorithms are guaranteed to deliver solutions within a factor of 0.931 of the optimum for MAX 2SAT and within a factor of 0.859 for MAX DICUT, improving upon the guarantees of 0.878 and 0.796 of Goemans and Williamson (1994).<<ETX>>

.879-approximation algorithms for MAX CUT and MAX 2SAT

We present randomized approximation algorithms for the MAX CUT and MAX 2SAT problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. We then show how to derandomize the algorithm to obtain approximation algorithms with the same performance guarantee of .87856. The previous best-known approximation algorithms for these problems had performance guarantees of ~ for MAX CUT and ~ for MAX 2SAT. A slight extension of our analysis leads to a .79607-approximation algorithm for the maximum directed cut problem, where a & approximation algorithm was the previous best-known algorithm. Our algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and, to the best of our knowledge, represents the first use of semidefinite programming in the design of approximation algorithms. *Address: Dept. of Mathematics, Room 2-372, M. I. T., Cambridge, MA 02139. Email: goemans@nath. mit. edu. Research supported in part by NSF contract 9302476-CCR, Air Force contract F49620-92-J-0125 and DARPA contract NOOO14-92-J1799. tAddress: School of Operations Research and Industrial En~ineering, 237 ETC 13uildin~, Cornell University, Ithaca, NY 14853. Email: dpw@cs. cornell. edu. Research supported by an NSF Postdoctoral Fellowship. This research was conducted while the author was visiting MIT. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association of Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. David P. Williamson Cornell University

Semidefinite programming in combinatorial optimization

We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovász theta function and its applications to stable sets, perfect graphs, and coding theory, (ii) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relationship to the sparsest cut problem.

Tight approximation algorithms for maximum general assignment problems

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, f ij , for assigning item j to bin i; and a separate packing constraint for each bin - i.e. for bin i, a family L i of subsets of items that fit in bin i. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1) and a distributed caching problem (DCP) described in this paper.Given a β-approximation algorithm for finding the highest value packing of a single bin, we give1. A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm.2. A simple polynomial-time local search (β/β+1 - e) - approximation algorithm, for any e > 0.Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 - 1/e - e)-approximation and a local search algorithm with (1/2-e)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 - 1/e. The best previously known approximation algorithm for GAP is a 1/2-approximation by Shmoys and Tardos; and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap.To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 -1/e unless NP⊆ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem.We extend the (1 - 1/e)-approximation algorithm to a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a 1/2-e approximation algorithm for the k-median problem with hard capacities. Finally, we study naturally defined game-theoretic versions of these problems, and show that they have price of anarchy of 2. We also prove the existence of cycles of best response moves, and exponentially long best-response paths to (pure or sink) equilibria.

Sink equilibria and convergence

We introduce the concept of a sink equilibrium. A sink equilibrium is a strongly connected component with no outgoing arcs in the strategy profile graph associated with a game. The strategy profile graph has a vertex set induced by the set of pure strategy profiles; its arc set corresponds to transitions between strategy profiles that occur with nonzero probability. (Here our focus will just be on the special case in which the strategy profile graph is actually a best response graph; that is, its arc set corresponds exactly to best response moves that result from myopic or greedy behaviour). We argue that there is a natural convergence process to sink equilibria in games where agents use pure strategies. This leads to an alternative measure of the social cost of a lack of coordination, the price of sinking, which measures the worst case ratio between the value of a sink equilibrium and the value of the socially optimal solution. We define the value of a sink equilibrium to be the expected social value of the steady state distribution induced by a random walk on that sink. We illustrate the value of this measure in three ways. Firstly, we show that it may more accurately reflects the inefficiency of uncoordinated solutions in competitive games when the use of pure strategies is the norm. In particular, we give an example (a valid-utility game) in which the game converges to solutions which are a factor n worse than socially optimal. The price of sinking is indeed n, but the price of anarchy is close to 1. Secondly, sink equilibria always exist. Thus, even in games in which pure strategy Nash equilibria (PSNE) do not exist, we can still calculate the price of sinking. Thirdly, we show that bounding the price of sinking can have important implications for the speed of convergence to socially good solutions in games where the agents make best response moves in a random order. We present two examples to illustrate our ideas. (i) Unsplittable selfish routing (and weighted congestion games):we prove that the price of sinking for the weighted unsplittable flow version of the selfish routing problem (for bounded-degree polynomial latency functions) is at most O(2/sup 2d/ d/sup 2d + 3/). In comparison, we give instances of these games without any PSNE. Moreover, our proof technique implies fast convergence to socially good (approximate) solutions. This is in contrast to the negative result of Fabrikant, Papadimitriou, and Talwar (2004) showing the existence of exponentially long best-response paths. (ii) Valid-utility games: we show that for valid-utility games the price of sinking is at most n+1; thus the worst case price of sinking in a valid-utility game is between it and n+1. We use our proof to show fast convergence to constant factor approximate solutions in basic-utility games. In addition, we present a hardness result which shows that, in general, there might be states that are exponentially far from any sink equilibrium in valid-utility games. We prove this by showing that the problem of finding a sink equilibrium (or a PSNE) in valid-utility games is PLS-complete.

New

Yannakakis recently presented the first

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-Approximation Algorithms for the Maximum Satisfiability Problem

-approximation algorithm for the Maximum Satisfiability Problem (MAX SAT). His algorithm makes nontrivial use of solutions to maximum flow problems. New, simple