On approximating complex quadratic optimization problems via semidefinite programming relaxations

  • Authors:

  • Anthony Man–Cho So;Jiawei Zhang;Yinyu Ye

  • Affiliations:

  • Department of Computer Science, Stanford University, Stanford, CA;Department of Information, Operations, and Management Sciences, Stern School of Business, New York University, New York, NY;Department of Management Science and Engineering and, by courtesy, Electrical Engineering, Stanford University, Stanford, CA

  • Venue:
  • IPCO'05 Proceedings of the 11th international conference on Integer Programming and Combinatorial Optimization
  • Year:
  • 2005

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Abstract

In this paper we study semidefinite programming (SDP) models for a class of discrete and continuous quadratic optimization problems in the complex Hermitian form. These problems capture a class of well–known combinatorial optimization problems, as well as problems in control theory. For instance, they include Max–3–Cut where the Laplacian matrix is positive semidefinite (in particular, some of the edge weights can be negative). We present a generic algorithm and a unified analysis of the SDP relaxations which allow us to obtain good approximation guarantees for our models. Specifically, we give an $(k sin(\frac{\pi}{k}))^{2}/(4\pi)$ –approximation algorithm for the discrete problem where the decision variables are k–ary and the objective matrix is positive semidefinite. To the best of our knowledge, this is the first known approximation result for this family of problems. For the continuous problem where the objective matrix is positive semidefinite, we obtain the well–known π/4 result due to [2], and independently, [12]. However, our techniques simplify their analyses and provide a unified framework for treating these problems. In addition, we show for the first time that the integrality gap of the SDP relaxation is precisely π/4. We also show that the unified analysis can be used to obtain an O(1/log n)–approximation algorithm for the continuous problem in which the objective matrix is not positive semidefinite.