We discuss the use of semidefinite programming for combinatorial optimization problems. The main topics covered include (i) the Lovfisz theta function and its applications to stable sets, perfect graphs, and coding theory. (it) the automatic generation of strong valid inequalities, (iii) the maximum cut problem and related problems, and (iv) the embedding of finite metric spaces and its relatio...

We establish more efficient methods for solving certain classes of NP-hard problems exactly, as well as methods for proving limitations on how quickly the same problems can be solved. For example, in the Max-Cut problem, one is given a graph G = (V,E) and integer K, and one wishes to determine if G has a subset of vertices such that the number of edges leaving the subset is at least K. The triv...

We investigate the approximability properties of several weighted problems, by comparing them with the respective unweighted problems. For an appropriate (and very general) deenition of niceness, we show that if a nice weighted problem is hard to approximate within r, then its polynomially bounded weighted version is hard to approximate within r ? o(1). Then we turn our attention to speciic pro...

The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. Although some aspects generalize , surprising counterexamples are found. We speculate that the phenomena revealed may be of interest in both spin systems in condensed matter and in quantum gravity.

This work is partially funded by the German Research Foundation (RO 2497/6-1). [1] Boykov, Y., Kolmogorov, V.: An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision, TPAMI 2004 [2] Boykov, Y., Jolly, M.: Interactive graph cuts for optimal boundary & region segmentation of objects in nd images, ICCV 2001 [3] Rother, C., Kolmogorov, V., Blake, A.: Grabcut: In...

We propose to solve the break minimization problem in sports scheduling by transforming it into a maximum cut problem in an undirected graph and applying a branch-and-cut algorithm. Our approach outperforms previous approaches with constraint programming and integer programming techniques.

A novel web image annotation method by candidate annotations clustering and parallel graph bipartition is proposed in this paper. Firstly, surrounding texts and other textual information in the hosting pages are extracted as the candidate annotations. For Web images, the candidate annotation sets of which are usually fairly large. Therefore, we cluster candidate annotations to reduce computatio...

We consider the max-cut and max-k-cut problems under graph-based constraints. Our approach can handle any constraint specified using monadic second-order (MSO) logic on graphs of constant treewidth. We give a 1 2 -approximation algorithm for this class of problems.