Topology and Adjunction in Promise Constraint Satisfaction

Abstract

The approximate graph coloring problem, whose complexity is unresolved in most cases, concerns finding a c-coloring of a graph that is promised to be k-colorable, where c \geq k. This problem naturally generalizes to promise graph homomorphism problems and further to promise constraint satisfaction problems. The complexity of these problems has recently been studied through an algebraic approach. In this paper, we introduce two new techniques to analyze the complexity of promise CSPs: one is based on topology and the other on adjunction. We apply these techniques, together with the previously introduced algebraic approach, to obtain new unconditional NP-hardness results for a significant class of approximate graph coloring and promise graph homomorphism problems.

1-in-3 vs. Not-All-Equal: Dichotomy of a broken promise

“…A consequence of this fact is that the universal-algebraic tools that allow generating an infinite set of new identities from a single polymorphic identity fails for PCSPs. This, in turn, stimulated the use of different tools to study PCSP polymorphisms, including Boolean function analysis [BGS23a], topology [KOWŽ23], matrix and tensor theory [CŽ23c,CŽ23a,CŽ23b], and Fourier analysis [HMS23].…”

Boolean symmetric vs. functional PCSP dichotomy

“…For PCSPs, even the case of graphs and structures on Boolean domains is widely open; these two were established for CSPs 979-8-3503-3587-3/23/$31.00 ©2023 IEEE a long time ago [23], [28] and constituted important evidence for conjecturing a dichotomy. Following the important work of Barto et al [6] on extending the algebraic framework from the realms of CSPs to the world of PCSPs, there have been several recent works on complexity classifications of fragments of PCSPs [20], [22], [8], [12], [5], [2], [9], [13], [27], [26], hardness conditions [6], [12], [7], [29], and power of algorithms [6], [10], [3], [17]. Nevertheless, a classification of more concrete fragments of PCSPs is needed for making progress with the general theory, such as finding hardness and tractability criteria, as well as with resolving longstanding open questions, such as approximate graph colouring.…”