Document

# Approximating Unique Games Using Low Diameter Graph Decomposition

## File

LIPIcs.APPROX-RANDOM.2017.18.pdf
• Filesize: 0.51 MB
• 15 pages

## Cite As

Vedat Levi Alev and Lap Chi Lau. Approximating Unique Games Using Low Diameter Graph Decomposition. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.18

## Abstract

We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints. Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.
##### Keywords
• Unique Games
• Low Diameter Graph Decomposition
• Bounded Genus Graphs
• Fixed Minor Free Graphs
• Approximation Algorithms
• Linear Programming

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## References

1. Ittai Abraham, Cyril Gavoille, Anupam Gupta, Ofer Neiman, and Kunal Talwar. Cops, robbers, and threatening skeletons: Padded decomposition for minor-free graphs. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing. ACM, 2014.
2. Amit Agarwal, Moses Charikar, Konstantin Makarychev, and Yury Makarychev. O(sqrt(log n)) approximation algorithms for min uncut. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 2005.
3. Naman Agarwal. Unique games conjecture: The boolean hypercube and connections to graph lifts. Master’s thesis, University of Illinois at Urbana-Champaign, 2014.
4. Naman Agarwal, Guy Kindler, Alexandra Kolla, and Luca Trevisan. Unique games on the hypercube. Chicago Journal of Theoretical Computer Science, 2015, 2015.
5. Sanjeev Arora, Boaz Barak, and David Steurer. Subexponential algorithms for unique games and related problems. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2010.
6. Sanjeev Arora, Subhash A. Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K. Vishnoi. Unique games on expanding constraint graphs are easy. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. ACM, 2008.
7. Brenda S. Baker. Approximation algorithms for np-complete problems on planar graphs. Journal of the ACM, 41, 1994.
8. Nikhil Bansal, Uriel Feige, Robert Krauthgamer, Konstantin Makarychev, Viswanath Nagarajan, Joseph Naor, and Roy Schwartz. Min-max graph partitioning and small set expansion. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2011.
9. Boaz Barak, Prasad Raghavendra, and David Steurer. Rounding semidefinite programming hierarchies via global correlation. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2011.
10. Punyashloka Biswal, James R. Lee, and Satish Rao. Eigenvalue bounds, spectral partitioning, and metrical deformations via flows. Journal of the ACM, 57, 2010.
11. Moses Charikar, Konstantin Makarychev, and Yury Makarychev. Near-optimal algorithms for unique games. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing. ACM, 2006.
12. Eden Chlamtac, Konstantin Makarychev, and Yury Makarychev. How to play unique games using embeddings. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2006.
13. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Ken ichi Kawarabayashi. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005.
14. Erik D. Demaine and MohammadTaghi Hajiaghayi. The bidimensionality theory and its algorithmic applications. The Computer Journal, 51, 2008.
15. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing. ACM, 2003.
16. Jittat Fakcharoenphol and Kunal Talwar. An improved decomposition theorem for graphs excluding a fixed minor. Approximation, Randomization, and Combinatorial Optimization: Algorithms and Techniques. Springer Berlin Heidelberg, 2003.
17. Jean Fonlupt, Ali Ridha Mahjoub, and J. P. Uhry. Compositions in the bipartite subgraph polytope. Discrete mathematics, 105, 1992.
18. Anupam Gupta and Kunal Talwar. Approximating unique games. In Proceedings of the 70th Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2006.
19. Venkatesan Guruswami and Ali Kemal Sinop. Lasserre hierarchy and approximation schemes for graph partitioning and quadratic integer programming with psd objectives. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2011.
20. Frank Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing, 4, 1975.
21. Jonathan A. Kelner. Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus. SIAM Journal on Computing, 35, 2006.
22. Jonathan A. Kelner, James R. Lee, Gregory N. Price, and Shang-Hua Teng. Metric uniformization and spectral bounds for graphs. Geometric and Functional Analysis, 21, 2011.
23. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of computing. ACM, 2002.
24. Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O'Donnell. Optimal inapproximability results for max-cut and other 2-variable csps? SIAM Journal on Computing, 37, 2007.
25. Subhash Khot and Oded Regev. Vertex cover might be hard to approximate to within 2- ε. Journal of Computer and System Sciences, 74, 2008.
26. Philip Klein, Serge A. Plotkin, and Satish Rao. Excluded minors, network decomposition, and multicommodity flow. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, 1993.
27. Alexandra Kolla. Spectral algorithms for unique games. Computational Complexity, 20, 2011.
28. Lap Chi Lau, Ramamoorthi Ravi, and Mohit Singh. Iterative methods in combinatorial optimization, volume 46. Cambridge University Press, 2011.
29. James R. Lee and Anastasios Sidiropoulos. Genus and the geometry of the cut graph. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM, 2010.
30. Lászlo Lovász, Martin Grötschel, and Alexander Schrijver. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1988.
31. Prasad Raghavendra. Optimal algorithms and inapproximability results for every csp? In Proceedings of the 40th Annual ACM Symposium on Theory of Computing. ACM, 2008.
32. Prasad Raghavendra and David Steurer. Graph expansion and the unique games conjecture. In Proceedings of the 42nd ACM Symposium on Theory of Computing. ACM, 2010.
33. Prasad Raghavendra, David Steurer, and Madhur Tulsiani. Reductions between expansion problems. In Proceedings of the 27th IEEE Annual Conference on Computational Complexity. IEEE, 2012.
34. Daniel A. Spielman and Shang-Hua Teng. Spectral partitioning works: Planar graphs and finite element meshes. Linear Algebra and its Applications, 421, 2007.
35. David Steurer and Nisheeth K. Vishnoi. Connections between unique games and multicut. Electronic Colloquium on Computational Complexity, 16, 2009.
36. Luca Trevisan. Approximation algorithms for unique games. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science. IEEE, 2005.
X

Feedback for Dagstuhl Publishing