Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues
Article No.: 45, Pages 1 - 13
Abstract
Given an n-vertex undirected graph G = (V,E) and positive edge weights {we}e∈E, a linear arrangement is a permutation π : V → {1, 2, …, n}. The value of the arrangement is val(G, π) := 1/n∑ e ={u, v} ∈ E we|π(u) − π (v)|. In the minimum linear arrangement problem, the goal is to find a linear arrangement π * that achieves val(G, π*) = MLA(G) := min π val(G, π).
In this article, we show that for any ϵ > 0 and positive integer r, there is an nO(r/ϵ)-time randomized algorithm that, given a graph G, returns a linear arrangement π, such that
val(G, π) ≤ (1 + 2/(1 − ε)λr(L)) MLA(G) + O(√log n/n ∑ e ∈ E we)
with high probability, where L is the normalized Laplacian of G and λr(L) is the rth smallest eigenvalue of L. Our algorithm gives a constant factor approximation for regular graphs that are weak expanders.
References
[1]
Christoph Ambühl, Monaldo Mastrolilli, and Ola Svensson. 2011. Inapproximability results for maximum edge biclique, minimum linear arrangement, and sparsest cut. SIAM J. Comput. 40, 2 (2011), 567--596.
[2]
Sanjeev Arora, Boaz Barak, and David Steurer. 2015. Subexponential algorithms for unique games and related problems. J. ACM 62, 5 (2015), 42:1--42:25.
[3]
Sanjeev Arora, Alan M. Frieze, and Haim Kaplan. 2002. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program. 92, 1 (2002), 1--36.
[4]
Sanjeev Arora, Subhash Khot, Alexandra Kolla, David Steurer, Madhur Tulsiani, and Nisheeth K. Vishnoi. 2008. Unique games on expanding constraint graphs are easy: Extended abstract. In Proceedings of the 40th Annual ACM Symposium on Theory of computing (STOC’08). 21--28.
[5]
Boaz Barak, Prasad Raghavendra, and David Steurer. 2011. Rounding semidefinite programming hierarchies via global correlation. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’11). 472--481.
[6]
Moses Charikar, Mohammad Taghi Hajiaghayi, Howard J. Karloff, and Satish Rao. 2010. ℓ<sup>2</sup><sub>2</sub> spreading metrics for vertex ordering problems. Algorithmica 56, 4 (2010), 577--604.
[7]
Nikhil R. Devanur, Subhash Khot, Rishi Saket, and Nisheeth K. Vishnoi. 2006. Integrality gaps for sparsest cut and minimum linear arrangement problems. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06). 537--546.
[8]
Devdatt P. Dubhashi and Alessandro Panconesi. 2012. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press.
[9]
Uriel Feige and James R. Lee. 2007. An improved approximation ratio for the minimum linear arrangement problem. Inform. Process. Lett. 101, 1 (2007), 26--29.
[10]
Michael Garey and David S. Johnson. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman.
[11]
Venkatesan Guruswami and Ali Kemal Sinop. 2011. Lasserre hierarchy, higher eigenvalues, and approximation schemes for quadratic integer programming with PSD objectives. In Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS’11). 482--491.
[12]
Venkatesan Guruswami and Ali Kemal Sinop. 2012. Optimal column-based low-rank matrix reconstruction. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’12). 1207--1214.
[13]
Venkatesan Guruswami and Ali Kemal Sinop. 2013a. Approximating non-uniform sparsest cut via generalized spectra. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’13). 295--305.
[14]
Venkatesan Guruswami and Ali Kemal Sinop. 2013b. Rounding Lasserre SDPs using column selection and spectrum-based approximation schemes for graph partitioning and quadratic IPs. CoRR abs/1312.3024 (2013).
[15]
Subhash Khot. 2002. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC’02). 767--775.
[16]
Alexandra Kolla and Madhur Tulsiani. 2007. Playing random and expanding unique games. Unpublished manuscript.
[17]
Jean B. Lasserre. 2002. An explicit equivalent positive semidefinite program for nonlinear 0-1 programs. SIAM J. Optim. 12, 3 (2002), 756--769.
[18]
James R. Lee, Shayan Oveis Gharan, and Luca Trevisan. 2014. Multiway spectral partitioning and higher-order cheeger inequalities. J. ACM 61, 6 (2014), 37:1--37:30.
[19]
Prasad Raghavendra and David Steurer. 2010. Graph expansion and the unique games conjecture. In Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC’10). 755--764.
[20]
Prasad Raghavendra, David Steurer, and Madhur Tulsiani. 2012. Reductions between expansion problems. In Proceedings of the 27th Annual IEEE Conference on Computational Complexity (CCC’12). 64--73.
[21]
Satish Rao and Andréa W. Richa. 2004. New approximation techniques for some linear ordering problems. SIAM J. Comput. 34, 2 (2004), 388--404.
[22]
Madhur Tulsiani. 2009. CSP gaps and reductions in the Lasserre hierarchy. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC’09). 303--312.
Index Terms
- Approximation Guarantees for the Minimum Linear Arrangement Problem by Higher Eigenvalues
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October 2018
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Publication History
Published: 21 August 2018
Accepted: 01 May 2018
Revised: 01 November 2017
Received: 01 February 2014
Published in TALG Volume 14, Issue 4
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