Hardness of Bipartite Expansion
We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U,V,E) and a parameter beta, the goal is to find a subset V' subseteq V containing beta fraction of the vertices of V which minimizes the size of N(V'), the neighborhood of V'. This problem, which we call Bipartite Expansion, is a special case of submodular minimization subject to a cardinality constraint, and is also related to other problems in graph partitioning and expansion. Previous to this work, there was no hardness of approximation known for Bipartite Expansion.
In this paper we show the following strong inapproximability for Bipartite Expansion: for any constants tau, gamma > 0
there is no algorithm which, given a constant beta > 0 and a bipartite graph G(U,V,E), runs in polynomial time and decides whether
- (YES case) There is a subset S^* subseteq V s.t. |S^*| >= beta*|V| satisfying |N(S^*)| <= gamma |U|, or
- (NO case) Any subset S subseteq V s.t. |S| >= tau*beta*|V| satisfies |N(S)| >= (1 - gamma)|U|, unless
NP subseteq intersect_{epsilon > 0}{DTIME}(2^{n^epsi;on}) i.e. NP has subexponential time algorithms.
We note that our hardness result stated above is a vertex expansion analogue of the Small Set (Edge) Expansion Conjecture of
Raghavendra and Steurer 2010.
inapproximability
bipartite expansion
PCP
submodular minimization
55:1-55:17
Regular Paper
Subhash
Khot
Subhash Khot
Rishi
Saket
Rishi Saket
10.4230/LIPIcs.ESA.2016.55
A. Agarwal, M. Charikar, K. Makarychev, and Y. Makarychev. O(sqrt(log n)) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In Proceedings of the ACM Symposium on the Theory of Computing, pages 573-581, 2005.
N. Alon. Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica, 6(3):207-219, 1986.
C. Ambühl, M. Mastrolilli, and O. Svensson. Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut. SIAM Journal of Computing, 40(2):567-596, 2011.
B. Applebaum, B. Barak, and A. Wigderson. Public-key cryptography from different assumptions. In Proceedings of the ACM Symposium on the Theory of Computing, pages 171-180, 2010.
S. Arora, S. Rao, and U. V. Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM, 56(2):1-37, 2009.
P. Austrin and S. Khot. A simple deterministic reduction for the gap minimum distance of code problem. In Proceedings of ICALP, pages 474-485, 2011.
S. Chawla, R. Krauthgamer, R. Kumar, Y. Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15(2):94-114, 2006.
Q. Cheng and D. Wan. A deterministic reduction for the gap minimum distance problem: [extended abstract]. In Proceedings of the ACM Symposium on the Theory of Computing, pages 33-38, 2009.
I. Dumer, D. Micciancio, and M. Sudan. Hardness of approximating the minimum distance of a linear code. IEEE Trans. Information Theory, 49(1):22-37, 2003.
U. Feige. Relations between average case complexity and approximation complexity. In Proceedings of the ACM Symposium on the Theory of Computing, pages 534-543, 2002.
U. Feige, M. Hajiaghayi, and J. R. Lee. Improved approximation algorithms for minimum weight vertex separators. SIAM Journal of Computing, 38(2):629-657, 2008.
V. Guruswami, C. Umans, and S. P. Vadhan. Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes. Journal of the ACM, 56(4), 2009.
S. Khot. On the power of unique 2-prover 1-round games. In Proceedings of the ACM Symposium on the Theory of Computing, pages 767-775, 2002.
S. Khot. Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal of Computing, 36(4):1025-1071, 2006.
S. Khot and N. K. Vishnoi. The unique games conjecture, integrality gap for cut problems and embeddability of negative type metrics into 𝓁₁. In Proceedings of the Annual Symposium on Foundations of Computer Science, pages 53-62, 2005.
F. T. Leighton and S. Rao. Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46(6):787-832, 1999.
N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15(2):215-245, 1995.
R. J. Lipton and R. E. Tarjan. Applications of a planar separator theorem. SIAM Journal of Computing, 9(3):615-627, 1980.
A. Louis, P. Raghavendra, and S. Vempala. The complexity of approximating vertex expansion. In Proceedings of the Annual Symposium on Foundations of Computer Science, pages 360-369, 2013.
A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-227, 1988.
R. Müller and D. Wagner. α-vertex separator is NP-hard even for 3-regular graphs. Computing, 46:343-353, 1991.
N. Pippenger. Sorting and selecting in rounds. SIAM Journal of Computing, 16(6):1032-1038, 1987.
P. Raghavendra and D. Steurer. Graph expansion and the unique games conjecture. In Proceedings of the ACM Symposium on the Theory of Computing, pages 755-764, 2010.
P. Raghavendra, D. Steurer, and M. Tulsiani. Reductions between expansion problems. In Proceedings of the Annual IEEE Conference on Computational Complexity, pages 64-73, 2012.
A. Rao. Randomness Extractors for Independent Sources and Applications. PhD thesis, University of Texas at Austin, 2007.
M. Sipser and D. A. Spielman. Expander codes. IEEE Transactions on Information Theory, 42(6):1710-1722, 1996.
Z. Svitkina and L. Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. SIAM Journal of Computing, 40(6):1715-1737, 2011.
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