Optimal Long Code Test with One Free Bit

For arbitrarily small constants epsilon, delta ≫ 0$, we present a long code test with one free bit, completeness 1-epsilon and soundness delta. Using the test, we prove the following two inapproximability results:1. Assuming the Unique Games Conjecture of Khot, given an n-vertex graph that has two disjoint independent sets of size (1/2-epsilon)n each, it is NP-hard to find an independent set of size delta n.2. Assuming a (new) stronger version of the Unique Games Conjecture, the scheduling problem of minimizing weighted completion time with precedence constraints is inapproximable within factor 2-epsilon.

[1]  Subhash Khot.  Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique , 2004, FOCS.

[2]  Mihir Bellare,et al.  Free bits, PCPs and non-approximability-towards tight results , 1995, Proceedings of IEEE 36th Annual Foundations of Computer Science.

[3]  Elchanan Mossel,et al.  Noise stability of functions with low influences: Invariance and optimality , 2005, IEEE Annual Symposium on Foundations of Computer Science.

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

[5]  Sanjeev Arora,et al.  Probabilistic checking of proofs; a new characterization of NP , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[6]  Gerhard J. Woeginger.  On the approximability of average completion time scheduling under precedence constraints , 2003, Discret. Appl. Math..

[7]  Irit Dinur,et al.  The importance of being biased , 2002, STOC '02.

[8]  Nisheeth K. Vishnoi,et al.  Unique games on expanding constraint graphs are easy: extended abstract , 2008, STOC.

[9]  Monaldo Mastrolilli,et al.  Single Machine Precedence Constrained Scheduling Is a Vertex Cover Problem , 2006, ESA.

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

[11]  José R. Correa,et al.  Single-Machine Scheduling with Precedence Constraints , 2005, Math. Oper. Res..

[12]  Luca Trevisan,et al.  Approximation algorithms for unique games , 2005, 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05).

[13]  Nicolai N. Pisaruk,et al.  A fully combinatorial 2-approximation algorithm for precedence-constrained scheduling a single machine to minimize average weighted completion time , 2003, Discret. Appl. Math..

[14]  Andreas S. Schulz.  Scheduling to Minimize Total Weighted Completion Time: Performance Guarantees of LP-Based Heuristics and Lower Bounds , 1996, IPCO.

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

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

[17]  Subhash Khot.  On the power of unique 2-prover 1-round games , 2002, STOC '02.

[18]  Jeffrey B. Sidney,et al.  Decomposition Algorithms for Single-Machine Sequencing with Precedence Relations and Deferral Costs , 1975, Oper. Res..

[19]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[20]  Maurice Queyranne,et al.  Decompositions, Network Flows, and a Precedence Constrained Single-Machine Scheduling Problem , 2003, Oper. Res..

[21]  E. Lawler.  Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints , 1978 .

[22]  O. Svensson,et al.  Inapproximability Results for Sparsest Cut, Optimal Linear Arrangement, and Precedence Constrained Scheduling , 2007, FOCS 2007.

[23]  C. N. Potts,et al.  An algorithm for the single machine sequencing problem with precedence constraints , 1980 .

[24]  David P. Williamson,et al.  Two-dimensional Gantt charts and a scheduling algorithm of Lawler , 1999, SODA '99.

[25]  Rajeev Motwani,et al.  Precedence Constrained Scheduling to Minimize Sum of Weighted Completion Times on a Single Machine , 1999, Discret. Appl. Math..

[26]  Martin E. Dyer,et al.  Formulating the single machine sequencing problem with release dates as a mixed integer program , 1990, Discret. Appl. Math..

[27]  Fabián A. Chudak,et al.  A half-integral linear programming relaxation for scheduling precedence-constrained jobs on a single machine , 1999, Oper. Res. Lett..

[28]  David B. Shmoys,et al.  Scheduling to minimize average completion time: off-line and on-line algorithms , 1996, SODA '96.

[29]  Jan Karel Lenstra,et al.  Complexity of Scheduling under Precedence Constraints , 1978, Oper. Res..

[30]  GoldwasserShafi,et al.  Interactive proofs and the hardness of approximating cliques , 1996 .

[31]  Gerhard J. Woeginger,et al.  Polynomial time approximation algorithms for machine scheduling: ten open problems , 1999 .

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