Promise Constraint Satisfaction and Width

We study the power of the bounded-width consistency algorithm in the context of the fixed-template Promise Constraint Satisfaction Problem (PCSP). Our main technical finding is that the template of every PCSP that is solvable in bounded width satisfies a certain structural condition implying that its algebraic closure-properties include weak near unanimity polymorphisms of all large arities. While this parallels the standard (non-promise) CSP theory, the method of proof is quite different and applies even to the regime of sublinear width. We also show that, in contrast with the CSP world, the presence of weak near unanimity polymorphisms of all large arities does not guarantee solvability in bounded width. The separating example is even solvable in the second level of the Sherali-Adams (SA) hierarchy of linear programming relaxations. This shows that, unlike for CSPs, linear programming can be stronger than bounded width. A direct application of these methods also show that the problem of $q$-coloring $p$-colorable graphs is not solvable in bounded or even sublinear width, for any two constants $p$ and $q$ such that $3 \leq p \leq q$. Turning to algorithms, we note that Wigderson's algorithm for $O(\sqrt{n})$-coloring $3$-colorable graphs with $n$ vertices is implementable in width $4$. Indeed, by generalizing the method we see that, for any $\epsilon > 0$ smaller than $1/2$, the optimal width for solving the problem of $O(n^\epsilon)$-coloring $3$-colorable graphs with $n$ vertices lies between $n^{1-3\epsilon}$ and $n^{1-2\epsilon}$. The upper bound gives a simple $2^{\Theta(n^{1-2\epsilon}\log(n))}$-time algorithm that, asymptotically, beats the straightforward $2^{\Theta(n^{1-\epsilon})}$ bound that follows from partitioning the graph into $O(n^\epsilon)$ many independent parts each of size $O(n^{1-\epsilon})$.