A new multilayered PCP and the hardness of hypergraph vertex cover
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Approximate coloring of uniform hypergraphs
Journal of Algorithms
A new PCP outer verifier with applications to homogeneous linear equations and max-bisection
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Is constraint satisfaction over two variables always easy?
Random Structures & Algorithms
Guest column: inapproximability results via Long Code based PCPs
ACM SIGACT News
Hardness of Approximating the Closest Vector Problem with Pre-Processing
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
The complexity of properly learning simple concept classes
Journal of Computer and System Sciences
On hardness of learning intersection of two halfspaces
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
SDP gaps for 2-to-1 and other label-cover variants
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
On the hardness of learning intersections of two halfspaces
Journal of Computer and System Sciences
On unique games with negativeweights
COCOA'11 Proceedings of the 5th international conference on Combinatorial optimization and applications
Bypassing UGC from some optimal geometric inapproximability results
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
2log1-ε n hardness for the closest vector problem with preprocessing
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation resistance on satisfiable instances for predicates with few accepting inputs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Hi-index | 0.02 |
In this paper, we consider the problem of coloring a 3-colorable 3-uniform hypergraph. In the minimization version of this problem, given a 3-colorable 3-uniform hypergraph, one seeks an algorithm to color the hypergraph with as few colors as possible. We show that it is NP-hard to color a 3-colorable 3-uniform hypergraph with constantly many colors. In fact, we show a stronger result that it is NP-hard to distinguish whether a 3-uniform hypergraph with nvertices is 3-colorable or it contains no independent set of size \delta n for an arbitrarily small constant \delta 0. In the maximization version of the problem, given a 3-uniform hypergraph,the goal is to color the vertices with 3 colors so as to maximize the number of non-monochromatic edges. We show that it is NP-hard to distinguish whether a 3-uniformhypergraph is 3-colorable or any coloring of the vertices with 3 colors has at most \frac{8}{9} + \varepsilonfraction of the edges non-monochromatic where \varepsilon 0 is an arbitrarily small constant. This result is tight since assigning a random color independently to every vertex makes \frac{8}{9} fraction of the edges non-monochromatic.These results are obtained via a new construction of a probabilistically checkable proof system (PCP) for NP. We develop a new construction of the PCP Outer Verifier. An important feature of this construction is smoothening of the projection maps.Dinur, Regev and Smyth [6] independently showed that it is NP-hard to color a 2-colorable 3-uniform hypergraph with constantly many colors. In the "good case", the hypergraph they construct is 2-colorable and hence their result is stronger. In the "bad case" however, the hypergraph we construct has a stronger property, namely, it does not even contain an independent set of size \delta n.