Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Simplex partitioning via exponential clocks and the multiway cut problem
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The metric labeling problem is an elegant and powerful mathematical model capturing a wide range of classification problems. The input to the problem consists of a set $L$ of labels and a weighted graph $G=(V,E)$. Additionally, a metric distance function on the labels is defined, and for each label and each vertex, an assignment cost is given. The goal is to find a minimum-cost assignment of the vertices to the labels. The cost of the solution consists of two parts: the assignment costs of the vertices and the separation costs of the edges (where each edge pays its weight times the distance between the two labels to which its endpoints are assigned). Due to the simple structure and the variety of applications, the problem and its special cases (with various distance functions on the labels) have recently received much attention. Metric labeling is known to have a logarithmic approximation, and it has been an open question for some time whether a constant approximation exists. We refute this possibility and prove that no constant factor approximation algorithm exists for metric labeling unless P=NP. Moreover, we prove that the problem is $\Omega((\log |V|)^{1/2-\delta})$-hard to approximate for any constant $\delta: 0