Semidefinite Programming Based Approximation Algorithms Research Articles

MAX CUT in cubic graphs

We present an improved semidefinite programming based approximation algorithm for the MAX CUT problem in graphs of maximum degree at most 3. The approximation ratio of the new algorithm is at least 0.9326. This improves, and also somewhat simplifies, a result of Feige, Karpinski and Langberg. We also observe that results of Hopkins and Staton and of Bondy and Locke yield a simple combinatorial 4/5-approximation algorithm for the problem. Finally, we present a combinatorial 22/27-approximation algorithm for the MAX CUT problem for regular cubic graphs.

An approximation algorithm for scheduling two parallel machines with capacity constraints

We consider the problem of scheduling n independent jobs on two identical parallel machines, with a limit on the number of jobs that can be assigned to each single machine, so as to minimize the total weighted completion time of the jobs. We study a semidefinite programming-based approximation algorithm for solving this problem and prove that the algorithm has a worst case ratio at most 1.1626.

A unified framework for obtaining improved approximation algorithms for maximum graph bisection problems

AbstractWe obtain improved semidefinite programming based approximation algorithms for all the natural maximum bisection problems of graphs. Among the problems considered are: MAX$n\over 2$‐BISECTION—partition the vertices of the graph into two sets of equal size such that the total weight of edges connecting vertices from different sides is maximized; MAX$n\over 2$‐VERTEX‐COVER—find a set containing half of the vertices such that the total weight of edges touching this set is maximized; MAX$n\over 2$‐DENSE‐SUBGRAPH—find a set containing half of the vertices such that the total weight of edges connecting two vertices from this set is maximized; and MAX$n\over 2$UNCUT—partition the vertices into two sets of equal size such that the total weight of edges that do not cross the cut is maximized. We also consider the directed versions of these problems, such as MAX$n\over 2$‐DIRECTED‐BISECTION and MAX$n\over 2$‐DIRECTED‐UNCUT. These results can be used to obtain improved approximation algorithms for the unbalanced versions of the partition problems mentioned above, where we want to partition the graph into two sets of size $k$ and $n - k$, where $k$ is not necessarily $n\over 2$. Our results improve, extend and unify results of Frieze and Jerrum, Feige and Langberg, Ye, and others. All these results may be viewed as extensions of the MAX CUT algorithm of Goemans and Williamson, and the MAX 2‐SAT and MAX DI‐CUT algorithms of Feige and Goemans. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 20:382–402, 2002

Constructing Worst Case Instances for Semidefinite Programming Based Approximation Algorithms

Semidefinite programming based approximation algorithms, such as the Goemans and Williamson approximation algorithm for the MAX CUT problem, are usually shown to have certain performance guarantees using local ratio techniques. Are the bounds obtained in this way tight? This problem was considered before by Karloff [SIAM J. Comput., 29 (1999), pp. 336--350] and by Alon and Sudakov [ Combin. Probab. Comput., 9 (2000), pp. 1--12]. Here we further extend their results and show, for the first time, that the local analyses of the Goemans and Williamson MAX CUT algorithm, as well as its extension by Zwick, are tight for every possible relative size of the maximum cut in the sense that the expected value of the solutions obtained by the algorithms may be as small as the analyses ensure. We also obtain similar results for a related problem. Our approach is quite general and could possibly be applied to some additional problems and algorithms.