Rajsekar Manokaran
ABSTRACT
The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGC-based hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0-Extension, and and Metric Labeling problems into UGC-based hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (so-called earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semi-definite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semi-definite programming.
- Aaron Archer, Jittat Fakcharoenphol, Chris Harrelson, Robert Krauthgamer, Kunal Talwar, and Éva Tardos. Approximate classification via earthmover metrics. In SODA '04: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1079--1087, 2004. Google Scholar
Digital Library
- Per Austrin. Balanced max 2-sat might not be the hardest. In STOC '07: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pages 189--197, 2007. Google ScholarDigital Library
- Per Austrin. Towards sharp inapproximability for any 2-csp. In FOCS '07: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pages 307--317, 2007. Google ScholarDigital Library
- Yair Bartal. On approximating arbitrary metrices by tree metrics. In STOC '98: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 161--168, 1998. Google ScholarDigital Library
- Gruia Calinescu, Howard Karloff, and Yuval Rabani. Approximation algorithms for the 0-extension problem. In SODA '01: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 8--16, 2001. Google ScholarDigital Library
- Chandra Chekuri, Sanjeev Khanna, Joseph (Seffi) Naor, and Leonid Zosin. Approximation algorithms for the metric labeling problem via a new linear programming formulation. SIAM J. Disc. Math., 18:608--625, 2005. Google ScholarDigital Library
- Julia Chuzhoy and Joseph (Seffi) Naor. The hardness of metric labeling. SIAM J. Comput., 36(5):1376--1386, 2006. Google ScholarDigital Library
- Gruia C\ualinescu, Howard Karloff, and Yuval Rabani. An improved approximation algorithm for multiway cut. In STOC '98: Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 48--52, 1998. Google ScholarDigital Library
- Elias Dahlhaus, David Johnson, Christos Papadimitriou, Paul Seymour, and Mihalis Yannakakis. The complexity of multiway cuts (extended abstract). In STOC '92: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pages 241--251, 1992. Google ScholarDigital Library
- Jittat Fakcharoenphol, Chris Harrelson, Satish Rao, and Kunal Talwar. An improved approximation algorithm for the 0-extension problem. In SODA '03: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 257--265, 2003. Google ScholarDigital Library
- Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. Approximating metrics by tree metrics. SIGACT News, 35(2):60--70, 2004. Google ScholarDigital Library
- Anupam Gupta and Éva Tardos. A constant factor approximation algorithm for a class of classification problems. In STOC '00: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pages 652--658, 2000. Google ScholarDigital Library
- David R. Karger, Philip Klein, Cliff Stein, Mikkel Thorup, and Neal E. Young. Rounding algorithms for a geometric embedding of minimum multiway cut. In STOC '99: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, pages 668--678, 1999. Google ScholarDigital Library
- Howard Karloff, Subhash Khot, Aranyak Mehta, and Yuval Rabani. On earthmover distance, metric labeling, and 0-extension. In STOC '06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 547--556, 2006. Google ScholarDigital Library
- Alexander Karzanov. Minimum 0-extension of graph metrics. Europ. J. Combinat., 19:71--101, 1998. Google ScholarDigital Library
- Alexander Karzanov. A combinatorial algorithm for the minimum $(2,r)$-metric problem and some generalizations. Combinatorica, 18:549--569, 1999.Google ScholarCross Ref
- Subhash Khot. On the power of unique 2-prover 1-round games. In STOC '02: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 767--775, 2002. Google ScholarDigital Library
- Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? In FOCS '04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 146--154, 2004. Google ScholarDigital Library
- Subhash Khot and Ryan O'Donnell. Sdp gaps and ugc-hardness for maxcutgain. In FOCS '06: Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 217--226, 2006. Google ScholarDigital Library
- Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2-ε. J. Comput. Syst. Sci., 74(3):335--349, 2008. Google ScholarDigital Library
- Subhash A. Khot and Nisheeth K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into $\ell _1$. In FOCS '05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 53--62, 2005. Google ScholarDigital Library
- Jon Kleinberg and Éva Tardos. Approximation algorithms for classification problems with pairwise relationships: metric labeling and markov random fields. J. ACM, 49(5):616--639, 2002. Google ScholarDigital Library
- Robert Krauthgamer, James R. Lee, Manor Mendel, and Assaf Naor. Measured descent: A new embedding method for finite metrics. In FOCS '04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 434--443, 2004. Google ScholarDigital Library
- James R. Lee and Assaf Naor. Extending lipschitz functions via random metric partitions. Inventiones Mathematicae, 160(1):59--95, 2005.Google ScholarCross Ref
- Elchanan Mossel, Ryan O'Donnell, and Krzysztof Oleszkiewicz. Noise stability of functions with low influences: Invariance and optimality. In FOCS '05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 21--30, 2005. Google ScholarDigital Library
- Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In STOC '08: Proceedings of the 40th ACM Symposium on Theory of Computing, 2008. Google ScholarDigital Library
Index Terms
- Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling
Recommendations
Integrality Gaps for Strong SDP Relaxations of UNIQUE GAMES
FOCS '09: Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer ScienceWith the work of Khot and Vishnoi (FOCS 2005) as a starting point, we obtain integrality gaps for certain strong SDP relaxations of unique games. Specifically, we exhibit a gap instance for the basic semidefinite program strengthened by all valid linear ...
On multiway cut parameterized above lower bounds
We introduce a concept of parameterizing a problem above the optimum solution of its natural linear programming relaxation and prove that the node multiway cut problem is fixed-parameter tractable (FPT) in this setting. As a consequence we prove that ...
On Earthmover Distance, Metric Labeling, and 0-Extension
We study the fundamental classification problems 0-Extension and Metric Labeling. A generalization of Multiway Cut, 0-Extension is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization ...
Comments