Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs

This work revisits the PCP Verifiers used in the works of Hastad [Has01], Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable 4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to prove the following improved hardness results: Assuming that NP\not\subseteq DTIME(N^{O(loglog N)}), There is no polynomial time algorithm that, given an n-vertex 2-colorable 4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for some constant c > 0. There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for some constant c > 0. For any fixed k >= 4, there is no polynomial time algorithm that finds a partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0. Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs is an exponential improvement over the previous results of Guruswami et al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log n)^{-c} beyond the random assignment threshold for Max-E3-SAT and Max-Ek-Set-Splitting is an exponential improvement over the previous bounds proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results avoid the use of a variable bias parameter used in previous works, which leads to the improved hardness thresholds in addition to simplifying the analysis substantially. Apart from standard techniques from Fourier Analysis, for the first mentioned result we use a mixing estimate of Markov Chains based on uniform reverse hypercontractivity over general product spaces from the work of Mossel et al.[MOS13].Comment: 23 Page

A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover

Given a $k$-uniform hyper-graph, the E$k$-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyper-edge. We present a new multilayered PCP construction that extends the Raz verifier. This enables us to prove that E$k$-Vertex-Cover is NP-hard to approximate within factor $(k-1-\epsilon)$ for any $k \geq 3$ and any $\epsilon>0$. The result is essentially tight as this problem can be easily approximated within factor $k$. Our construction makes use of the biased Long-Code and is analyzed using combinatorial properties of $s$-wise $t$-intersecting families of subsets

Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs

We give a new method for analysing the mixing time of a Markov chain using path coupling with stopping times. We apply this approach to two hypergraph problems. We show that the Glauber dynamics for independent sets in a hypergraph mixes rapidly as long as the maximum degree Delta of a vertex and the minimum size m of an edge satisfy m>= 2Delta+1. We also show that the Glauber dynamics for proper q-colourings of a hypergraph mixes rapidly if m>= 4 and q > Delta, and if m=3 and q>=1.65Delta. We give related results on the hardness of exact and approximate counting for both problems.Comment: Simpler proof of main theorem. Improved bound on mixing time. 19 page

On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values