On the hardness of approximating minimum vertex cover

We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.

[1]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[2]  P. Erdös,et al.  Intersection Theorems for Systems of Sets , 1960 .

[3]  P. Erdös,et al.  INTERSECTION THEOREMS FOR SYSTEMS OF FINITE SETS , 1961 .

[4]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[5]  H. L. Abbott,et al.  Intersection Theorems for Systems of Sets , 1972, J. Comb. Theory, Ser. A.

[6]  László Lovász,et al.  Factoring polynomials with rational coefficients , 1982 .

[7]  L. Russo.  An approximate zero-one law , 1982 .

[8]  Reuven Bar-Yehuda,et al.  A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem , 1983, WG.

[9]  Ewald Speckenmeyer,et al.  Some Further Approximation Algorithms for the Vertex Cover Problem , 1983, CAAP.

[10]  Richard M. Wilson,et al.  The exact bound in the Erdös-Ko-Rado theorem , 1984, Comb..

[11]  Nathan Linial,et al.  The influence of variables on Boolean functions , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[12]  Norihide Tokushige,et al.  The exact bound in the Erdös-Ko-Rado theorem for cross-intersecting families , 1989, J. Comb. Theory, Ser. A.

[13]  Nathan Linial,et al.  Collective Coin Flipping , 1989, Adv. Comput. Res..

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

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

[16]  Zoltán Füredi,et al.  Beyond the Erdös-Ko-Rado theorem , 1991, J. Comb. Theory, Ser. A.

[17]  Sanjeev Arora,et al.  The Hardness of Approximate Optimia in Lattices, Codes, and Systems of Linear Equations , 1993, IEEE Annual Symposium on Foundations of Computer Science.

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

[19]  M. Talagrand.  On Russo's Approximate Zero-One Law , 1994 .

[20]  Ran Raz,et al.  A parallel repetition theorem , 1995, STOC '95.

[21]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[22]  Philip N. Klein,et al.  Approximation Algorithms for Steiner and Directed Multicuts , 1997, J. Algorithms.

[23]  Rudolf Ahlswede,et al.  The Complete Intersection Theorem for Systems of Finite Sets , 1997, Eur. J. Comb..

[24]  Ran Raz,et al.  A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP , 1997, STOC '97.

[25]  J. Bourgain,et al.  Influences of Variables and Threshold Intervals under Group Symmetries , 1997 .

[26]  U. Feige.  A threshold of ln n for approximating set cover , 1998, JACM.

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

[28]  Mihir Bellare,et al.  Free Bits, PCPs, and Nonapproximability-Towards Tight Results , 1998, SIAM J. Comput..

[29]  Guy Kindler,et al.  Approximating CVP to Within Almost-Polynomial Factors is NP-Hard , 1998, Electron. Colloquium Comput. Complex..

[30]  Uriel Feige,et al.  Zero Knowledge and the Chromatic Number , 1998, J. Comput. Syst. Sci..

[31]  Guy Kindler,et al.  Approximating CVP to Within Almost-Polynomial Factors is NP-Hard , 2003, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[32]  Ehud Friedgut,et al.  Boolean Functions With Low Average Sensitivity Depend On Few Coordinates , 1998, Comb..

[33]  Daniele Micciancio,et al.  The shortest vector in a lattice is hard to approximate to within some constant , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[34]  I. Benjamini,et al.  Noise sensitivity of Boolean functions and applications to percolation , 1998, math/9811157.

[35]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[36]  Lars Engebretsen,et al.  Clique Is Hard To Approximate Within , 2000 .

[37]  Eran Halperin,et al.  Improved approximation algorithms for the vertex cover problem in graphs and hypergraphs , 2000, SODA '00.

[38]  Nathan Linial,et al.  On the Hardness of Approximating the Chromatic Number , 2000, Comb..

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

[40]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[41]  Clifford D. Smyth,et al.  The Hardness of 3-Uniform Hypergraph Coloring , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[42]  Venkatesan Guruswami,et al.  A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover , 2005, SIAM J. Comput..

[43]  G. Kalai,et al.  INFLUENCES OF VARIABLES AND THRESHOLDINTERVALS UNDER GROUP , .