Conditional Hardness for Approximate Coloring

We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our hardness result on a certain `fish shaped' variant of his conjecture. We also prove that the problem almost coloring is hard for any constant \eps>0, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all but a \eps fraction of the vertices without monochromatic edges, and the case where the graph contains no independent set of relative size at least \eps. Our result is based on bounding various generalized noise-stability quantities using the invariance principle of Mossel et al [MOO'05]

Conditional Hardness for Approximate Coloring

We study the AprxColoring(q,Q) problem: Given a graph G, decide whether Χ(G) ≤ q or Χ(G)≥Q. We present hardness results for this problem for any constants 3 ≤ q \u3c Q. For q ≥ 4, our result is base on Khot\u27s 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775]. For q=3, we base our hardness result on a certain “⋉-shaped variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O\u27Donnell, and Oleszkiewicz [ Noise stability of functions with low influences: Invariance and optimality, Ann. of Math. (2), to appear] and should have wider applicability

Improved Hardness of Approximating Chromatic Number

We prove that for sufficiently large K, it is NP-hard to color K-colorable graphs with less than 2^{K^{1/3}} colors. This improves the previous result of K versus K^{O(log K)} in Khot [14]

Approximately coloring graphs without long induced paths

It is an open problem whether the 3-coloring problem can be solved in polynomial time in the class of graphs that do not contain an induced path on $t$ vertices, for fixed $t$. We propose an algorithm that, given a 3-colorable graph without an induced path on $t$ vertices, computes a coloring with $\max\{5,2\lceil{\frac{t-1}{2}}\rceil-2\}$ many colors. If the input graph is triangle-free, we only need $\max\{4,\lceil{\frac{t-1}{2}}\rceil+1\}$ many colors. The running time of our algorithm is $O((3^{t-2}+t^2)m+n)$ if the input graph has $n$ vertices and $m$ edges

A Hypergraph Dictatorship Test with Perfect Completeness

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based \PCP construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are $o(1).$ Their test makes $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ but has an inherent loss of perfect completeness. In this paper we give an adaptive hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity $1+O(\frac{\log q}{q})$.Comment: Some minor correction

Improved Inapproximability Results for Maximum k-Colorable Subgraph

We study the maximization version of the fundamental graph coloring problem. Here the goal is to color the vertices of a k-colorable graph with k colors so that a maximum fraction of edges are properly colored (i.e. their endpoints receive different colors). A random k-coloring properly colors an expected fraction 1-1/k of edges. We prove that given a graph promised to be k-colorable, it is NP-hard to find a k-coloring that properly colors more than a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2) was known. Our result pins down the correct asymptotic dependence of the approximation factor on k. Along the way, we prove that approximating the Maximum 3-colorable subgraph problem within a factor greater than 32/33 is NP-hard. Using semidefinite programming, it is known that one can do better than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of edges of a k-colorable graph.Comment: 16 pages, 2 figure

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