Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph

In the Densest k-Subgraph (DkS) problem, given an undirected graph G and an integer k, the goal is to find a subgraph of G on k vertices that contains maximum number of edges. Even though Bhaskara et al.'s state-of-the-art algorithm for the problem achieves only O(n1/4 + ϵ) approximation ratio, previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer) and 2O(log2/3 n) (Alon et al.) respectively. In this work, we show, assuming the exponential time hypothesis (ETH), that there is no polynomial-time algorithm that approximates Densest k-Subgraph to within n1/(loglogn)c factor of the optimum, where c > 0 is a universal constant independent of n. In addition, our result has perfect completeness, meaning that we prove that it is ETH-hard to even distinguish between the case in which G contains a k-clique and the case in which every induced k-subgraph of G has density at most 1/n-1/(loglogn)c in polynomial time. Moreover, if we make a stronger assumption that there is some constant ε > 0 such that no subexponential-time algorithm can distinguish between a satisfiable 3SAT formula and one which is only (1 - ε)-satisfiable (also known as Gap-ETH), then the ratio above can be improved to nf(n) for any function f whose limit is zero as n goes to infinity (i.e. f ϵ o(1)).

[1]  SudanMadhu,et al.  Proof verification and the hardness of approximation problems , 1998 .

[2]  Michael Langberg,et al.  Approximation Algorithms for Maximization Problems Arising in Graph Partitioning , 2001, J. Algorithms.

[3]  Takeshi Tokuyama,et al.  Dense subgraph problems with output-density conditions , 2005, TALG.

[4]  Prasad Raghavendra,et al.  A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs , 2016, ICALP.

[5]  Sundar Vishwanathan,et al.  On the Approximability of the Minimum Rainbow Subgraph Problem and Other Related Problems , 2015, Algorithmica.

[6]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[7]  Mohammad Taghi Hajiaghayi,et al.  The prize-collecting generalized steiner tree problem via a new approach of primal-dual schema , 2006, SODA '06.

[8]  Ludek Kucera,et al.  Expected Complexity of Graph Partitioning Problems , 1995, Discret. Appl. Math..

[9]  U. Feige,et al.  On the densest k-subgraph problems , 1997 .

[10]  Aviad Rubinstein Settling the Complexity of Computing Approximate Two-Player Nash Equilibria , 2016, FOCS.

[11]  László Lovász,et al.  Interactive proofs and the hardness of approximating cliques , 1996, JACM.

[12]  Anand Srivastav,et al.  Finding Dense Subgraphs with Semidefinite Programming , 1998, APPROX.

[13]  Alexander Russell,et al.  Minimum Multicolored Subgraph Problem in Multiplex PCR Primer Set Selection and Population Haplotyping , 2006, International Conference on Computational Science.

[14]  Aditya Bhaskara,et al.  Detecting high log-densities: an O(n¼) approximation for densest k-subgraph , 2010, STOC '10.

[15]  John Fearnley,et al.  Inapproximability Results for Approximate Nash Equilibria , 2016, WINE.

[16]  Noga Alon,et al.  Testing subgraphs in large graphs , 2001, Proceedings 2001 IEEE International Conference on Cluster Computing.

[17]  Uriel Feige,et al.  Relations between average case complexity and approximation complexity , 2002, STOC '02.

[18]  V. Sós,et al.  On a problem of K. Zarankiewicz , 1954 .

[19]  Refael Hassin,et al.  Complexity of finding dense subgraphs , 2002, Discret. Appl. Math..

[20]  Russell Impagliazzo,et al.  AM with Multiple Merlins , 2014, 2014 IEEE 29th Conference on Computational Complexity (CCC).

[21]  Elena Tsanko,et al.  Approximating Minimum-Power Degree and Connectivity Problems , 2011, Algorithmica.

[22]  Adrian Vetta,et al.  The Complexity of the Simultaneous Cluster Problem , 2014, J. Graph Algorithms Appl..

[23]  Pasin Manurangsi,et al.  Approximating Dense Max 2-CSPs , 2015, APPROX-RANDOM.

[24]  Guy Kortsarz,et al.  On Choosing a Dense Subgraph (Extended Abstract) , 1993, FOCS 1993.

[25]  Aravindan Vijayaraghavan,et al.  Approximation Algorithms for Label Cover and The Log-Density Threshold , 2017, SODA.

[26]  Siddharth Barman,et al.  Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Caratheodory's Theorem , 2015, STOC.

[27]  Subhash Khot,et al.  Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.

[28]  Prasad Raghavendra,et al.  Graph expansion and the unique games conjecture , 2010, STOC '10.

[29]  Euiwoong Lee,et al.  Partitioning a graph into small pieces with applications to path transversal , 2016, Mathematical Programming.

[30]  Carsten Lund,et al.  Proof verification and hardness of approximation problems , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[31]  Rico Zenklusen,et al.  Hardness and approximation for network flow interdiction , 2015, Networks.

[32]  Prasad Raghavendra,et al.  Reductions between Expansion Problems , 2010, 2012 IEEE 27th Conference on Computational Complexity.

[33]  Kumar Chellapilla,et al.  Finding Dense Subgraphs with Size Bounds , 2009, WAW.

[34]  Piotr Faliszewski,et al.  Finding a collective set of items: From proportional multirepresentation to group recommendation , 2014, Artif. Intell..

[35]  Guy Kortsarz,et al.  On choosing a dense subgraph , 1993, Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science.

[36]  Michael Dinitz,et al.  The Densest k-Subhypergraph Problem , 2016, APPROX-RANDOM.

[37]  Mohammad Taghi Hajiaghayi,et al.  Improved Approximation Algorithms for Label Cover Problems , 2011, Algorithmica.

[38]  Ola Svensson,et al.  Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling , 2007, 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07).

[39]  Mark Braverman,et al.  Approximating the best Nash Equilibrium in no(log n)-time breaks the Exponential Time Hypothesis , 2015, Electron. Colloquium Comput. Complex..

[40]  Uriel Feige,et al.  The Dense k -Subgraph Problem , 2001, Algorithmica.

[41]  Carsten Lund,et al.  Efficient probabilistically checkable proofs and applications to approximations , 1993, STOC.

[42]  Shuichi Miyazaki,et al.  The Hospitals/Residents Problem with Quota Lower Bounds , 2011, ESA.

[43]  Shi Li,et al.  On the Hardness of Approximating the k-Way Hypergraph Cut problem , 2020, Theory Comput..

[44]  Euiwoong Lee Partitioning a graph into small pieces with applications to path transversal , 2017, SODA 2017.

[45]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[46]  Irit Dinur,et al.  Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover , 2016, Electron. Colloquium Comput. Complex..

[47]  Robert Krauthgamer,et al.  How hard is it to approximate the best Nash equilibrium? , 2009, SODA.

[48]  László Lovász,et al.  Approximating clique is almost NP-complete , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[49]  U. Feige,et al.  On the Densest K-subgraph Problem , 1997 .

[50]  Michael Dinitz,et al.  Everywhere-Sparse Spanners via Dense Subgraphs , 2012, 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science.

[51]  Stavros G. Kolliopoulos,et al.  Partially-Ordered Knapsack and Applications to Scheduling , 2002, ESA.

[52]  Russell Impagliazzo,et al.  Complexity of k-SAT , 1999, Proceedings. Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317).

[53]  Ran Raz,et al.  Two Query PCP with Sub-Constant Error , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[54]  Sanjeev Arora,et al.  Inapproximabilty of Densest κ-Subgraph from Average Case Hardness , 2011 .

[55]  Aditya Bhaskara,et al.  Polynomial integrality gaps for strong SDP relaxations of Densest k-subgraph , 2011, SODA.

[56]  Yuan Zhou,et al.  Approximation Algorithms and Hardness of the k-Route Cut Problem , 2011, TALG.

[57]  Aviad Rubinstein,et al.  ETH-Hardness for Signaling in Symmetric Zero-Sum Games , 2015, ArXiv.

[58]  Omri Weinstein,et al.  ETH Hardness for Densest-k-Subgraph with Perfect Completeness , 2015, SODA.

[59]  Yakov Babichenko,et al.  Can Almost Everybody be Almost Happy? , 2015, ITCS.

[60]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[61]  Mark Jerrum,et al.  Large Cliques Elude the Metropolis Process , 1992, Random Struct. Algorithms.

[62]  Wei Chen,et al.  Combining Traditional Marketing and Viral Marketing with Amphibious Influence Maximization , 2015, EC.

[63]  Rishi Saket,et al.  Quasi-Random PCP and Hardness of 2-Catalog Segmentation , 2010, FSTTCS.

[64]  David Pisinger,et al.  The quadratic knapsack problem - a survey , 2007, Discret. Appl. Math..

[65]  Pasin Manurangsi,et al.  On approximating projection games , 2015 .

[66]  Eli Ben-Sasson,et al.  Short PCPs with Polylog Query Complexity , 2008, SIAM J. Comput..