A rainbow q-coloring of a k-uniform hypergraph is a q-coloring of the vertex set such that every hyperedge contains all q colors.We prove that given a rainbow (k − 2⌊ √ k⌋)-colorable k-uniform hypergraph, it is NPhard to find a normal 2-coloring. Previously, this was only known for rainbow ⌊k/2⌋colorable hypergraphs (Guruswami and Lee, SODA 2015).We also study a generalization which we call rainbow (q, p)-coloring, defined as a coloring using q colors such that every hyperedge contains at least p colors. We prove that given a rainbow (k − ⌊ √ kc⌋, k − ⌊3 √ kc⌋)-colorable k uniform hypergraph, it is NP-hard to find a normal c-coloring for any c = o(k).The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.(a variant of the Unique Games Conjecture) it is hard to find a coloring using any constant number of colors [DMR09]. For large constant c, it is known that it is NP-hard to color a ccolorable graph using 2 Ω(c 1/3 ) colors [Hua13], and in general it is known that the chromatic number is NP-hard to approximate within n 1−ǫ for every ǫ > 0 [FK98, Zuc07].In the hypergraph case, stronger hardness results are known: for instance, given a 4colorable 4-uniform hypergraph or a 2-colorable 8-uniform hypergraph, it is quasi-NP-hard 1 to find a coloring using 2 (log n) 1/20−ǫ colors for every ǫ > 0 [Var16] following a series of recent developments [DG13, GHH + 17, Hua15, KS17]. In the 3-uniform case, the current best hardness is that given a 3-colorable 3-uniform hypergraph it is quasi-NP-hard to find a coloring with (log n) γ/ log log log n colors for some γ > 0 [GHH + 17]. Stronger results are known when the hypergraph is only guaranteed to be almost 2-colorable: given an almost 2-colorable 4-uniform hypergraph, it is quasi-NP-hard to find an independent set of relative size 2 − log 1−o(1) n [KS14].Given the strong hardness of hypergraph coloring, it is natural to consider restricted forms of coloring having some additional structure that might make them more amenable to algorithms. One such variant is rainbow colorability which is introduced in [AGH17]. A q coloring of the hypergraph is called a rainbow q-coloring if there exists a coloring of the vertices with q colors such that every hyperedge contains all q colors. More formally,A hypergraph is called rainbow q-colorable if there exists a rainbow q-coloring. If we restrict the uniformity of the hypergraph to k then the definition of q-rainbow coloring is meaningful only when 2 ≤ q ≤ k. It is easy to observe that the property of H being rainbow q-colorable is stronger the larger q is, and that it is always stronger than 2-colorability. We have the following implications on the structure of hypergraphs:

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A seminal result of Håstad (2001) shows that it is NP-hard to find an assignment that satisfies 1 |G | + ε fraction of the constraints of a given k-LIN instance over an abelian group, even if there is an assignment that satisfies (1 − ε) fraction of the constraints, for any constant ε > 0. Engebretsen, Holmerin and Russell (2004) later showed that the same hardness result holds for k-LIN instances over any finite non-abelian group.Unlike the abelian case, where we can efficiently find a solution if the instance is satisfiable, in the non-abelian case, it is NP-complete to decide if a given system of linear equations is satisfiable or not, as shown by Goldmann and Russell (1999).Surprisingly, for certain non-abelian groups G, given a satisfiable k-LIN instance over G, one can in fact do better than just outputting a random assignment using a simple but clever algorithm. The approximation factor achieved by this algorithm varies with the underlying group. In this paper, we show that this algorithm is optimal by proving a tight hardness of approximation of satisfiable k-LIN instance over any non-abelian G, assuming P NP. As a corollary, we also get 3-query probabilistically checkable proofs with perfect completeness over large alphabets with improved soundness.Our proof crucially uses the quasi-random properties of the nonabelian groups defined by Gowers (2008).

We study the following basic problem called Bi-Covering. Given a graph G(V, E), find two (not necessarily disjoint) sets A ⊆ V and B ⊆ V such that A ∪ B = V and that every edge e belongs to either the graph induced by A or to the graph induced by B. The goal is to minimize max{|A|, |B|}. This is the most simple case of the Channel Allocation problem [Gandhi et. al, Networks, 2006]. A solution that outputs V, ∅ gives ratio at most 2. We show that under the similar Strong Unique Game Conjecture by [Bansal-Khot, FOCS, 2009] there is no 2 − ratio algorithm for the problem, for any constant > 0. Given a bipartite graph, Max-bi-clique is a problem of finding largest k × k complete bipartite sub graph. For Max-bi-clique problem, a constant factor hardness was known under random 3-SAT hypothesis of Feige [Feige, STOC, 2002] and also under the assumption that NP ∩ >0 BPTIME(2 n) [Khot, SIAM J. on Comp., 2011]. It was an open problem in [Ambühl et. al., SIAM J. on Comp., 2011] to prove inapproximability of Max-bi-clique assuming weaker conjecture. Our result implies similar hardness result assuming the Strong Unique Games Conjecture. On the algorithmic side, we also give better than 2 approximation for Bi-Covering on numerous special graph classes. In particular, we get 1.876 approximation for Chordal graphs, exact algorithm for Interval Graphs, 1 + o(1) for Minor Free Graph, 2 − 4δ/3 for graphs with minimum degree δn, 2/(1 + δ 2 /8) for δ-vertex expander, 8/5 for Split Graphs, 2 − (6/5) • 1/d for graphs with minimum constant degree d etc. Our algorithmic results are quite non-trivial. In achieving these results, we use various known structural results about the graphs, combined with the techniques that we develop tailored to getting better than 2 approximation.

Given k collections of 2SAT clauses on the same set of variables V , can we find one assignment that satisfies a large fraction of clauses from each collection? We consider such simultaneous constraint satisfaction problems, and design the first nontrivial approximation algorithms in this context.Our main result is that for every CSP F , for k ăÕplog 1 {4 nq, there is a polynomial time constant factor Pareto approximation algorithm for k simultaneous Max-F -CSP instances. Our methods are quite general, and we also use them to give an improved approximation factor for simultaneous Max-w-SAT (for k ăÕplog 1 {3 nq). In contrast, for k " ωplog nq, no nonzero approximation factor for k simultaneous Max-F -CSP instances can be achieved in polynomial time (assuming the Exponential Time Hypothesis).These problems are a natural meeting point for the theory of constraint satisfaction problems and multiobjective optimization. We also suggest a number of interesting directions for future research.