David P. Williamson

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].

.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

Adversarial queuing theory

We consider packet routing when packets are injected continuously into a network. We develop an adversarial theory of queuing aimed at addressing some of the restrictions inherent in probabilistic analysis and queuing theory based on time-invariant stochastic generation. We examine the stability of queuing networks and policies when the arrival process is adversarial, and provide some preliminary results in this direction. Our approach sheds light on various queuing policies in simple networks, and paves the way for a systematic study of queuing with few or no probabilistic assumptions.

Adversarial queueing theory

We introduce a new approach to the study of dynamic (or continuous) packet routing, where packets are being continuously injected into a network. Our objective is to study what happens to packet routing under continuous injection as a function of network load, for various queueing policies. Our approach is based on the adversarial generation of packets, so that the results are more robust in that they do not hinge upon particular probabilistic assumptions. In suggesting a new approach to studying a classical phenomenon, it is important to give careful consideration to all the relevant previous work in packet routing, queueing theory and probabilistic analysis. We give a more detailed account of previous work in Appendix A, to permit comparison with our work. Here we summarize the salient features of prior work in order to motivate our model. Most prior work on packet routing has been in the static model in which there is a fixed initial set of packet routing requests; when these packets are delivered, the problem is considered solved and the analysis stops there. Static packet routing is a basic problem in the context of parallel computation models, but for the setting of communications networks it is essential to study the case of continuous injection of packets. While it is possible to try modelling such continuous problems statically, by delaying the entry of packets using synchronization barriers, a much more natural approach is to analyze standard (local–control) routing algorithms in this fully dynamic setting. Nearly all previous work in this regard has used probabilistic models for the generation (and sometimes, delivery) of packets. Such work can broadly be classified into: ∗Department of Computer Science, University of Toronto, Toronto, Canada M5S 1A4. Part of this work was performed while visiting the IBM T.J. Watson Research Center. †Laboratory for Computer Science, MIT, Cambridge, MA 02139. Supported by an ONR Graduate Fellowship. Part of this work was performed while visiting the IBM T.J. Watson Research Center. ‡IBM Almaden Research Center, 650 Harry Road, San Jose, CA 95120. §IBM T.J. Watson Research Center, Box 218, Yorktown Heights, NY 10598. 1. Queueing-theoretic approaches, where packets are generated by a Poisson injection process; frequently, each packet is assumed to have a random destination. A common assumption in queueing theory is that the time for a packet to pass through a server (i.e., an edge) is exponentially distributed whereas for packet routing this time is a constant. This apparently slight difference poses a world of subtle difficulties in adapting queueing theory to continuous packet routing.

Short Shop Schedules

We consider the open shop, job shop, and flow shop scheduling problems with integral processing times. We give polynomial-time algorithms to determine if an instance has a schedule of length at most 3, and show that deciding if there is a schedule of length at most 4 is 𝒩𝒫-complete. The latter result implies that, unless 𝒫 = 𝒩𝒫, there does not exist a polynomial-time approximation algorithm for any of these problems that constructs a schedule with length guaranteed to be strictly less than 5/4 times the optimal length. This work constitutes the first nontrivial theoretical evidence that shop scheduling problems are hard to solve even approximately.

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

Scheduling Parallel Machines On-line

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