A Quasipolynomial $(2+\varepsilon)$-Approximation for Planar Sparsest Cut

The (non-uniform) sparsest cut problem is the following graph-partitioning problem: given a “supply” graph, and demands on pairs of vertices, delete some subset of supply edges to minimize the ratio of the supply edges cut to the total demand of the pairs separated by this deletion. Despite much effort, there are only a handful of nontrivial classes of supply graphs for which constant-factor approximations are known. We consider the problem for planar graphs, and give a (2 + ε)-approximation algorithm that runs in quasipolynomial time. Our approach defines a new structural decomposition of an optimal solution using a “patching” primitive. We combine this decomposition with a Sherali-Adams-style linear programming relaxation of the problem, which we then round. This should be compared with the polynomial-time approximation algorithm of Rao (1999), which uses the metric linear programming relaxation and `1-embeddings, and achieves an O( √ log n)-approximation in polynomial time.

[1]  James R. Lee,et al.  On the geometry of graphs with a forbidden minor , 2009, STOC '09.

[2]  Philip N. Klein,et al.  New hardness results for planar graph problems in p and an algorithm for sparsest cut , 2020, STOC.

[3]  Chandra Chekuri,et al.  Flow-cut gaps for integer and fractional multiflows , 2010, SODA '10.

[4]  James R. Lee,et al.  Euclidean distortion and the sparsest cut , 2005, STOC '05.

[5]  Yuval Rabani,et al.  An O(log k) Approximate Min-Cut Max-Flow Theorem and Approximation Algorithm , 1998, SIAM J. Comput..

[6]  Anupam Gupta,et al.  Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut , 2005, SODA '05.

[7]  Éva Tardos,et al.  Improved bounds on the max-flow min-cut ratio for multicommodity flows , 1993, Comb..

[8]  Farhad Shahrokhi,et al.  Sparsest cuts and bottlenecks in graphs , 1990, Discret. Appl. Math..

[9]  Prasad Raghavendra,et al.  Coarse Differentiation and Multi-flows in Planar Graphs , 2007, APPROX-RANDOM.

[10]  Satish Rao,et al.  An approximate max-flow min-cut relation for undirected multicommodity flow, with applications , 1995, Comb..

[11]  Sanjeev Arora,et al.  Nearly linear time approximation schemes for Euclidean TSP and other geometric problems , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[12]  Viresh Patel,et al.  Determining Edge Expansion and Other Connectivity Measures of Graphs of Bounded Genus , 2010, SIAM J. Comput..

[13]  Satish Rao.  Faster algorithms for finding small edge cuts in planar graphs , 1992, STOC '92.

[14]  A KhotSubhash,et al.  The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative-Type Metrics into ℓ1 , 2015 .

[15]  Ittai Abraham,et al.  Cops, robbers, and threatening skeletons: padded decomposition for minor-free graphs , 2013, STOC.

[16]  R. Ravi,et al.  Approximation through multicommodity flow , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[17]  Prasad Raghavendra,et al.  Approximating Sparsest Cut in Graphs of Bounded Treewidth , 2010, APPROX-RANDOM.

[18]  Cynthia A. Phillips,et al.  Finding minimum-quotient cuts in planar graphs , 1993, STOC.

[19]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[20]  Ittai Abraham,et al.  Strong-Diameter Decompositions of Minor Free Graphs , 2007, SPAA '07.

[21]  Anupam Gupta,et al.  Embedding k-outerplanar graphs into ℓ1 , 2003, SODA '03.

[22]  Assaf Naor,et al.  Vertical perimeter versus horizontal perimeter , 2017, Annals of Mathematics.

[23]  Nathan Linial,et al.  The geometry of graphs and some of its algorithmic applications , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[24]  Satish Rao,et al.  Small distortion and volume preserving embeddings for planar and Euclidean metrics , 1999, SCG '99.

[25]  James R. Lee,et al.  Embeddings of Topological Graphs: Lossy Invariants, Linearization, and 2-Sums , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[26]  Satish Rao,et al.  Expander flows, geometric embeddings and graph partitioning , 2004, STOC '04.

[27]  James R. Lee,et al.  Lp metrics on the Heisenberg group and the Goemans-Linial conjecture , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[28]  Lee-Ad Gottlieb,et al.  The traveling salesman problem: low-dimensionality implies a polynomial time approximation scheme , 2011, STOC '12.

[29]  Yuval Rabani,et al.  ON THE HARDNESS OF APPROXIMATING MULTICUT AND SPARSEST-CUT , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[30]  Guy Kindler,et al.  Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[31]  Assaf Naor,et al.  The integrality gap of the Goemans-Linial SDP relaxation for sparsest cut is at least a constant multiple of √log n , 2017, STOC.

[32]  Anupam Gupta,et al.  Cuts, Trees and ℓ1-Embeddings of Graphs* , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[33]  Refael Hassin,et al.  On multicommodity flows in planar graphs , 1984, Networks.

[34]  Philip N. Klein,et al.  Excluded minors, network decomposition, and multicommodity flow , 1993, STOC.

[35]  Amit Chakrabarti,et al.  When the cut condition is enough: a complete characterization for multiflow problems in series-parallel networks , 2012, STOC '12.