Promise Constraint Satisfaction Problem: Structure and Complexity

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Why is it that some computational problems admit algorithms that always work fast, that is, scale up well with the size of data to be processed, while other computational problems are not like this and (appear to) admit only algorithms that scale up exponentially? Answering this question is one of the fundamental goals of Theoretical Computer Science. Computational complexity theory formalises the two kinds of problems as tractable (or polynomial-time solvable) and NP-hard, respectively. So we can rephrase the above question as follows: What kind of inherent mathematical structure makes a computational problem tractable? This very general question is known to be extremely difficult. The Constraint Satisfaction Problem (CSP) and its variants are extensively used towards answering this question for two reasons: on the one hand, the CSP framework is very general and includes a wide variety of computational problems, and on the other hand, this framework has very rich mathematical structure providing an excellent laboratory both for complexity classification methods and for algorithmic techniques. The so-called algebraic approach to the CSP has been very successful in this quest for understanding tractability. The idea of this approach is that certain algebraic structure (which can viewed roughly as multi-dimensional symmerties) in problem instances leads to tractability, while the absence of such structure leads to NP-hardness. This approach has already provided very deep insights and delivered very strong complexity classification results. In particular, it explained which mathematical features distinguish tractable and NP-hard problems within the class of standard CSPs. The proposed research will aim to extend this understanding to Promise Constraint Satisfaction Problems, which is a much larger class of problems, by uncovering deeper mathematical reasons for tractability and NP-hardness, thus providing stronger evidence that tractable problems share a certain algebraic structure. We will also apply our new theory to resolve long-standing open questions about some classical NP-hard optimisation problems, specifically how much the optimality demand must be relaxed there to guarantee tractability.