Abstract. Many real-life scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this N P-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms. 1

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Abstract—Dynamic spectrum access (DSA), enabled by cognitive radio technologies, has become a promising approach to improve efficiency in spectrum utilization, and the spectrum auction is one important DSA approach, in which secondary users lease some unused bands from primary users. However, spectrum auctions are different from existing auctions studied by economists, because spectrum resources are interference-limited rather than quantity-limited, and it is possible to award one band to multiple secondary users with negligible mutual interference. To accommodate this special feature in wireless communications, in this paper, we present a novel multi-winner spectrum auction game not existing in auction literature. As secondary users may be selfish in nature and tend to be dishonest in pursuit of higher profits, we develop effective mechanisms to suppress their dishonest/collusive behaviors when secondary users distort their valuations about spectrum resources and interference relationships. Moreover, in order to make the proposed game scalable when the size of problem grows, the semi-definite programming (SDP) relaxation is applied to reduce the complexity significantly. Finally, simulation results are presented to evaluate the proposed auction mechanisms, and demonstrate the complexity reduction as well. Index Terms—Cognitive radio, spectrum auction, collusionresistant mechanism, scalable algorithm. I.

facility layout; semidefinite programming; convex programming; global optimisation.

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Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

We analyze the properties of an interior point cutting plane algorithm that is based on a semi-infinite linear formulation of the dual semidefinite program. The cutting plane algorithm approximately solves a linear relaxation of the dual semidefinite program in every iteration and relies on a separation oracle that returns linear cutting planes. We show that the complexity of a variant of the interior point cutting plane algorithm is slightly smaller than that of a direct interior point solver for semidefinite programs where the number of constraints is approximately equal to the dimension of the matrix. Our primary focus in this paper is the design of good separation oracles that return cutting planes that support the feasible region of the dual semidefinite program. Furthermore, we introduce a concept called the tangent space induced by a supporting hyperplane that measures the strength of a cutting plane, characterize the supporting hyperplanes that give higher dimensional tangent spaces, and show how such cutting planes can be found efficiently. Our procedures are analogous to finding facets of an integer polytope in cutting plane methods for integer programming. We illustrate these concepts with two examples in the paper. Finally, we describe separation oracles that return nonpolyhedral cutting surfaces. Recently, Krishnan et al. [41] and Oskoorouchi and Goffin [32] have adopted these separation oracles in conic interior point cutting plane algorithms for solving semidefinite programs.

Abstract. The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph. If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid. Specialized to cuts in graphs, this result solves a problem posed by Lovász. 1.

Many combinatorial optimization problems can be modeled concisely with a system of polynomial equations. Examples include the detection of k-colorings, stable sets, flows, matchings, and satisfiability (see [12] and the references therein). It follows that solving general systems of polynomial equations is at least NP-hard. For this reason, mathematicians have rarely used nonlinear polynomials for practical computation or to provide complexity bounds (although they can be very useful otherwise [1, 10, 30, 18]). In this article, we discuss four iterative algorithms tailored to solve combinatorial systems of polynomial equations. We explain how these algebraic procedures can be applied to integer hull approximation and also the recognition of combinatorial properties such as k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. We report on computational complexity bounds, structural results, and computer experiments. When the field of coefficients is the real numbers our methodology closely resembles other iterative procedures such as Lovász-Schrijver, Sherali-Adams, the Lasserre hierarchy, and others that are used in integer programming and optimization over semialgebraic sets [31, 38, 35, 28]. The algorithms we present are also related to the solvability methods of Laurent, Lasserre and Rostalski [25, 26]. The key difference is that we work over arbitrary fields of coefficients which allows a wider range of modeling.

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