On Subexponential and FPT-Time Inapproximability

Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithm design. While each of them being very active in its own, there is an increasing attention to the connection between these different frameworks. In particular, whether Independent Set would be better approximable once endowed with subexponential-time or FPT-time is a central question. In this article, we provide new insights to this question using two complementary approaches; the former makes a strong link between the linear PCP conjecture and inapproximability; the latter builds a class of equivalent problems under approximation in subexponential time.

[1]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[2]  Ran Raz,et al.  Two Query PCP with Sub-Constant Error , 2008, FOCS.

[3]  Marcin Pilipczuk,et al.  Exact and approximate bandwidth , 2009, Theor. Comput. Sci..

[4]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[5]  Michael R. Fellows,et al.  Parameterized approximation of dominating set problems , 2008, Inf. Process. Lett..

[6]  Vangelis Th. Paschos,et al.  Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms , 2011, Discret. Appl. Math..

[7]  Liming Cai,et al.  On Fixed-Parameter Tractability and Approximability of NP Optimization Problems , 1997, J. Comput. Syst. Sci..

[8]  Liming Cai,et al.  Fixed-Parameter Approximation: Conceptual Framework and Approximability Results , 2010, Algorithmica.

[9]  Danupon Nanongkai,et al.  Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[10]  Mihalis Yannakakis,et al.  Optimization, approximation, and complexity classes , 1991, STOC '88.

[11]  Vangelis Th. Paschos,et al.  Parameterized (in)approximability of subset problems , 2013, Oper. Res. Lett..

[12]  Mohammad Taghi Hajiaghayi,et al.  Fixed-Parameter and Approximation Algorithms: A New Look , 2013, IPEC.

[13]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[14]  Yijia Chen,et al.  On Parameterized Approximability , 2006, IWPEC.

[15]  David Zuckerman,et al.  Electronic Colloquium on Computational Complexity, Report No. 100 (2005) Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER , 2005 .

[16]  Zvi Galil,et al.  Explicit Constructions of Linear-Sized Superconcentrators , 1981, J. Comput. Syst. Sci..

[17]  Andreas Björklund,et al.  Set Partitioning via Inclusion-Exclusion , 2009, SIAM J. Comput..

[18]  Ryan Williams,et al.  Improved Parameterized Algorithms for above Average Constraint Satisfaction , 2011, IPEC.

[19]  Liming Cai,et al.  On the existence of subexponential parameterized algorithms , 2003, J. Comput. Syst. Sci..

[20]  Stefan Kratsch,et al.  Safe Approximation and Its Relation to Kernelization , 2011, IPEC.

[21]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[22]  Mohammad Taghi Hajiaghayi,et al.  The Foundations of Fixed Parameter Inapproximability , 2013, ArXiv.

[23]  Dániel Marx,et al.  Parameterized Complexity and Approximation Algorithms , 2008, Comput. J..

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

[25]  Michael R. Fellows,et al.  Parameterized Approximation Problems , 2006, IWPEC.

[26]  Luke Mathieson,et al.  A Proof Checking View of Parameterized Complexity , 2012, ArXiv.

[27]  Sunil Arya,et al.  Space-time tradeoffs for approximate nearest neighbor searching , 2009, JACM.

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

[29]  Vangelis Th. Paschos,et al.  Laboratoire D'analyse Et Modélisation De Systèmes Pour L'aide À La Décision Cahier Du Lamsade 280 Efficient Approximation of Min Coloring by Moderately Exponential Algorithms Efficient Approximation of Min Coloring by Moderately Exponential Algorithms , 2008 .

[30]  Hans Ulrich Simon,et al.  On Approximate Solutions for Combinatorial Optimization Problems , 1990, SIAM J. Discret. Math..

[31]  Avi Wigderson,et al.  Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

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

[33]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[34]  Carsten Lund,et al.  On the hardness of approximating minimization problems , 1994, JACM.

[35]  Ge Xia,et al.  Strong computational lower bounds via parameterized complexity , 2006, J. Comput. Syst. Sci..

[36]  Serge Gaspers,et al.  An exponential time 2-approximation algorithm for bandwidth , 2009, Theor. Comput. Sci..

[37]  Madhav V. Marathe,et al.  On Approximation Algorithms for the Minimum Satisfiability Problem , 1996, Inf. Process. Lett..

[38]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .