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 perfc~rmance guarantees of~for MAX CUT and~for 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 approxim ation algorithms had performance guarantees of~and~, 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 preliminary version has appeared in

Abstract. 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-cost spanning tree, minimum-weight perfect matching, traveling salesman, and Steiner tree problems.Our technique produces approximation algorithms that run in O(n log n) 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 minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n3) 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 nonfixed 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 [Math. Programming, 59 (1993), pp. 413-420].

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We consider a stochastic variant of the NP-hard 0/1 knapsack problem, in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution “policy” that maximizes the expected value of items successfully placed in the knapsack, where the final overflowing item contributes no value. We consider both nonadaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). An important facet of our work lies in characterizing the benefit of adaptivity. For this purpose we advocate the use of a measure called the adaptivity gap: the ratio of the expected value obtained by an optimal adaptive policy to that obtained by an optimal nonadaptive policy. We bound the adaptivity gap of the stochastic knapsack problem by demonstrating a polynomial-time algorithm that computes a nonadaptive policy whose expected value approximates that of an optimal adaptive policy to within a factor of four. We also devise a polynomial-time adaptive policy that approximates the optimal adaptive policy to within a factor of 3 + ε for any constant ε > 0.

Submodular functions are a key concept in combinatorial optimization. Algorithms that involve submodular functions usually assume that they are given by a (value) oracle. Many interesting problems involving submodular functions can be solved using only polynomially many queries to the oracle, e.g., exact minimization or approximate maximization.In this paper, we consider the problem of approximating a non-negative, monotone, submodular function f on a ground set of size n everywhere, after only poly(n) oracle queries. Our main result is a deterministic algorithm that makes poly(n) oracle queries and derives a functionf such that, for every set S,f (S) approximates f (S) within a factor α(n), where α(n) = √ n + 1 for rank functions of matroids and α(n) = O( √ n log n) for general monotone submodular functions. Our result is based on approximately finding a maximum volume inscribed ellipsoid in a symmetrized polymatroid, and the analysis involves various properties of submodular functions and polymatroids.Our algorithm is tight up to logarithmic factors. Indeed, we show that no algorithm can achieve a factor better than Ω( √ n/ log n), even for rank functions of a matroid.

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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 Feage-Lovdsz (STOC92) semidefinite programming relaxation of one-round two-prover proof systems, together with rounding techniques for the solutions of semidefinite progmms, as introduced by Goemans and Williamson (STO C94). As a consequence of our approach, we present improved approximation algorithmsfor M A X 2SAT and MAX DICUT. The algorithms are guamnteed 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.

We consider the Asymmetric Traveling Salesman problem for costs satisfying the triangle inequality. We derive a randomized algorithm which delivers a solution within a factor O(log n/ log log n) of the optimum with high probability.