Hardness of approximate hypergraph coloring
2002 (English)In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 31, no 6, 1663-1686 p.Article in journal (Refereed) Published
We introduce the notion of covering complexity of a veri er for probabilistically checkable proofs (PCPs). Such a veri er is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The veri er is also given a random string and decides whether to accept the proof or not, based on the given random string. We de ne the covering complexity of such a veri er, on a given input, to be the minimum number of proofs needed to satisfy the veri er on every random string; i.e., on every random string, at least one of the given proofs must be accepted by the veri er. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and, in particular, (hyper) graph coloring problems. We present a PCP veri er for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this veri er is a simple not-all-equal check on the four bits it reads. This enables us to prove that, for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption.
Place, publisher, year, edition, pages
2002. Vol. 31, no 6, 1663-1686 p.
graph coloring, hypergraph coloring, hardness of approximations, PCP, covering PCP, set splitting, lovasz local lemma, chromatic number, algorithms, graph
IdentifiersURN: urn:nbn:se:kth:diva-22056ISI: 000179328200002OAI: oai:DiVA.org:kth-22056DiVA: diva2:340754
QC 201005252010-08-102010-08-10Bibliographically approved