Hardness of Approximate Coloring

The graph coloring problem is a notoriously hard problem, for which we do not have efficient algorithms. A coloring of a graph is an assignment of colors to its vertices such that the end points of every edge have different colors. A k-coloring is a coloring that uses at most k distinct colors. The graph coloring problem is to find a coloring that uses the minimum number of colors. Given a 3-colorable graph, the best known efficient algorithms output an n0.199···-coloring. It is known that efficient algorithms cannot find a 4-coloring, assuming P ̸=NP (such results are commonly known as hardness results). Hence there is a large gap (n0.199··· vs 4) between what current algorithms can achieve and the hardness results known. In this thesis, we narrow the aforesaid gap for some generalizations of graph coloring, by giving improved hardness results (for exponentially better parameters in some cases). Some of our main results are as follows: 1. For the case of almost 3-colorable graphs, we show hardness of finding a 2poly(log coloring, assuming a variant of the Unique Games Conjecture (UGC). 2. For the case of 3-colorable 3-uniform hypergraphs, we show quasi-NP-hardness of finding a 2 logn/ log log -coloring. 3. For the case of 4-colorable 4-uniform hypergraphs, we show quasi-NP-hardness of finding a 2(logn)1/21 -coloring. 4. For the problem of the approximating the covering number of CSPs with nonodd predicates, we show hardness of approximation to any constant factor, assuming a variant of UGC.