On earthmover distance, metric labeling, and 0-extension

• Authors: Howard Karloff;Subhash Khot;Aranyak Mehta;Yuval Rabani
• Venue: Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
• Year: 2006

We study the fundamental classification problems O-EXTENSION and METRIC LABELING. MINIMUM WEIGHT TRIANGULATION is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization METRIC LABELING is motivated by applications in computer vision. Researchers had proposed using earthmover metrics to get polynomial time-solvable relaxations for these problems. A conjecture that has attracted much attention recently is that the integrality ratio for these relaxations is constant.We prove that the integrality ratio of the earthmover relaxation for METRIC LABELING is Ω(log n) (which is asymptotically tight), k being the number of labels, whereas the best previous lower bound on the integrality ratio was only constant; that the integrality ratio of the earthmover relaxation for O-EXTENSION is Ω(√log k), k being the number of terminals (it was known to be O((log k)/log log k)), whereas the best previous lower bound was only constant; that for no ε0 is there a polynomial-time O((log n)1/4-ε)-approximation algorithm for O-EXTENSION, n being the number of vertices, unless NP ⊆ DTIME(npoly(log n)), whereas the strongest inapproximability result known before was only MAX SNP-hardness; and that there is a polynomial-time approximation algorithm for O-EXTENSION with performance ratio O(√diam(d)), where diam(d) is the ratio of the largest to smallest nonzero distances in the terminal metric.