Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

Abstract

A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k ≥ 3 and r ≥ 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r = k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r = k − 1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := k−2 2 colors. In particular, for k ≤ 6, it is NP-hard to 2-color (k − 1)-rainbow colorable k-uniform hypergraphs.Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r] n . We prove that any such polymorphism f : [r] n → [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a ∈ [q] S such that fixing x |S = a determines the value of f (x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover.

Combinatorial Gap Theorem and Reductions between Promise CSPs

“…Another direction is exploring combinatorial versions of known improvements of the PCP Theorem, in particular the Smooth Label Cover of Khot [30] (cf. [25]). Finally, the most interesting direction seems to be in exploring combinatorial versions of conjectural improvements of the PCP Theorem, e.g., the d-to-1 Conjecture [31].…”

“…This, and similar such results should not be regarded as heavy hammers that are giving us reductions for free. They rather serve as tools that enable one to disregard the inessential layers and concentrate on the core of the problem, which can then be attacked using various methods (such as algebraic [17,40], topological [21,8,34], or analytic [25]). As such tools, they are indeed useful.…”

“…However, Theorem 6 is very much insufficient for proving NP-hardness of every NP-hard PCSP, e.g., one provably cannot apply it to reduce an NP-hard CSP (such as 3-SAT) to PCSP(NAE 3 2 , NAE 3 n ). More widely applicable sufficient conditions for NP-hardness in terms of polymorphisms have been developed in [8,15,25], or follow from the results in these papers. They are all based on Theorem 2 and its refinements.…”

“…The examples of reductions that are not (known to be) covered include NPhardness proofs in [25,4,26] and reductions that were used in [38] to improve [26]. These examples suggest directions for improving the Main Theorem and thus Theorem 7; we discuss these directions in the Conclusion.…”

Conditional Dichotomy of Boolean Ordered Promise CSPs

“…The fact that we can add the equality constraints follows either by identifying the variables together, or by observing that the polymorphism minion of any PCSP remains the same when we add the equality predicate (see e.g., [BBKO19,GS20]).…”