ECCC-Report TR10-106https://eccc.weizmann.ac.il/report/2010/106Comments and Revisions published for TR10-106en-usFri, 29 Oct 2010 04:04:59 +0200
Revision 1
| Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP |
Yuichi Yoshida
https://eccc.weizmann.ac.il/report/2010/106#revision1Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it.
In this paper, we show that similar results hold for constant-time approximation algorithms in the bounded-degree model. Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution to a basic LP relaxation of the CSP. (ii) Using the oracle, we give a constant-time rounding scheme that achieves an approximation ratio coincident with the integrality gap of the basic LP. (iii) Finally, we give a generic conversion from integrality gaps of basic LPs to hardness results. All of those results are \textit{unconditional}. Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all.
A CSP instance is called $\epsilon$-far from satisfiability if we must remove at least an $\epsilon$-fraction of constraints to make it satisfiable. A CSP is called testable if there is a constant-time algorithm that distinguishes satisfiable instances from $\epsilon$-far instances with probability at least $2/3$. Using the results above, we also derive, under a technical assumption, an equivalent condition under which a CSP is testable in the bounded-degree model.
Fri, 29 Oct 2010 04:04:59 +0200https://eccc.weizmann.ac.il/report/2010/106#revision1
Paper TR10-106
| Optimal Constant-Time Approximation Algorithms and (Unconditional) Inapproximability Results for Every Bounded-Degree CSP |
Yuichi Yoshida
https://eccc.weizmann.ac.il/report/2010/106Raghavendra (STOC 2008) gave an elegant and surprising result: if Khot's Unique Games Conjecture (STOC 2002) is true, then for every constraint satisfaction problem (CSP), the best approximation ratio is attained by a certain simple semidefinite programming and a rounding scheme for it.
In this paper, we show that a similar result holds for constant-time approximation algorithms in the bounded-degree model.
Specifically, we present the followings: (i) For every CSP, we construct an oracle that serves an access, in constant time, to a nearly optimal solution of a basic LP relaxation of the CSP. (ii) Using the oracle, we present a constant-time rounding scheme that achieves an approximation ratio oincident with the integrality gap of the basic LP.
(iii) We give a generic conversion from integrality gaps of basic LPs to hardness results.
All of those results are ``unconditional.'' Therefore, for every bounded-degree CSP, we give the best constant-time approximation algorithm among all.Fri, 02 Jul 2010 14:00:11 +0300https://eccc.weizmann.ac.il/report/2010/106