TY - GEN

T1 - On hardness of approximating the parameterized clique problem

AU - Khot, Subhash

AU - Shinkar, Igor

PY - 2016/1/14

Y1 - 2016/1/14

N2 - In the Gap-clique(k; k/2 ) problem, the input is an n-vertex graph G, and the goal is to decide whether G contains a clique of size k or contains no clique of size k/2 . It is an open question in the study of fixed parameterized tractability whether the Gap-clique(k; k/2 ) problem is fixed parameter tractable, i.e., whether it has an algorithm that runs in time f(k) n, where f(k) is an arbitrary function of the parameter k and the exponent is a constant independent of k. In this paper, we give some evidence that the problem Gap-clique(k; k/2 ) is not fixed parameter tractable. Speciff-cally, we define a constraint satisfaction problem, which we call Deg-2-sat, where the input is a system of k0 quadratic equations in k0 variables over a finite field F of size n0, and the goal is to decide whether there is a solution in F that satisfies all the equations simultaneously. The main result in this paper is an "FPT-reduction" from Deg-2-sat to the Gap-clique(k; k/2 ) problem. If one were to hypothesize that the Deg-2-sat problem is not fixed parameter tractable, then our reduction would imply that the Gap-clique(k; k/2 ) problem is not fixed parameter tractable either. The reduction relies on the algebraic techniques used in proof of the PCP theorem.

AB - In the Gap-clique(k; k/2 ) problem, the input is an n-vertex graph G, and the goal is to decide whether G contains a clique of size k or contains no clique of size k/2 . It is an open question in the study of fixed parameterized tractability whether the Gap-clique(k; k/2 ) problem is fixed parameter tractable, i.e., whether it has an algorithm that runs in time f(k) n, where f(k) is an arbitrary function of the parameter k and the exponent is a constant independent of k. In this paper, we give some evidence that the problem Gap-clique(k; k/2 ) is not fixed parameter tractable. Speciff-cally, we define a constraint satisfaction problem, which we call Deg-2-sat, where the input is a system of k0 quadratic equations in k0 variables over a finite field F of size n0, and the goal is to decide whether there is a solution in F that satisfies all the equations simultaneously. The main result in this paper is an "FPT-reduction" from Deg-2-sat to the Gap-clique(k; k/2 ) problem. If one were to hypothesize that the Deg-2-sat problem is not fixed parameter tractable, then our reduction would imply that the Gap-clique(k; k/2 ) problem is not fixed parameter tractable either. The reduction relies on the algebraic techniques used in proof of the PCP theorem.

KW - Clique

KW - Fixed parameter tractability

KW - Hardness of approximation

KW - Parameterized complexity

UR - http://www.scopus.com/inward/record.url?scp=84966534239&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84966534239&partnerID=8YFLogxK

U2 - 10.1145/2840728.2840733

DO - 10.1145/2840728.2840733

M3 - Conference contribution

AN - SCOPUS:84966534239

T3 - ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

SP - 37

EP - 45

BT - ITCS 2016 - Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

PB - Association for Computing Machinery, Inc

T2 - 7th ACM Conference on Innovations in Theoretical Computer Science, ITCS 2016

Y2 - 14 January 2016 through 16 January 2016

ER -