Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Abstract

We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [16] are in this class of problems. We extend a method of Bienstock and Iyengar [4] which was based on ideas from Nesterov [24] to design an algorithm for computing -approximate solutions for this class of semidefinite programs. Our algorithm is in the spirit of Klein and Lu [17], and decreases the dependence of the run-time on from −2 to −1 . For sparse graphs, our method is faster than the best specialized interior point methods. A significant feature of our method is that it treats both the Maxcut and the graph coloring problem in a unified manner.

Feasible and Accurate Algorithms for Covering Semidefinite Programs

“…Our algorithm for covering SDP is not a simple extension of that for packing SDPs [13]. Several steps that were easy for packing SPDs are not so for covering SDPs and give rise to new technical challenges.…”

“…The convergence analysis of this iterative scheme is non-trivial. -Our algorithms use quadratic prox functions as opposed to the more usual logarithmic functions [2,13]. Quadratic prox functions avoid the need to compute matrix exponentials and are numerically more stable.…”

“…(Packing SDPs are analogous to packing linear programs, see [13] for a precise definition). Examples of packing SDPs include those used to solve MAXCUT, COLORING, Shannon capacity and sparse principal component analysis.…”

“…The drawback of these algorithms is the dependence on of at least −2 ; this bound is inherent in these methods [17] and a significant bottleneck in practice on these types of problems [18,5]. A third approach, due to Iyengar, Phillips and Stein [13] also extends ideas from packing linear programs, but starts from the more recent work of Bienstock and Iyengar [6] who build on techniques of Nesterov [21]. Nesterov [22] has also adapted his method to solve the associated saddle point problem with general SDP, although his method does not find feasible solutions.…”

“…These results for packing SDPs have a dependence of only −1 , but slightly larger dependence on the other parameters than the algorithms of Klein and Lu. The work in [13] is also significant in that it explicitly defines and approximately solves all packing SDPs in a unified manner.…”