Polynomial-time Computation: Opening the Blackboxes in Constraint Problems

The class P of polynomial-time computable computational problems is the most important and robust complexity class for the study of efficient computation. Answering what problems belong to P will lead to groundbreaking applications in science and society. Moreover, P is a relatively recent mathematical object and radically different from classical notions studied for centuries; thus, capturing it promises the discovery of new fundamental theorems in mathematics.Our current understanding of P is limited; for instance, the P=NP millenium problem is wide open. There neither exists a uniform reduction technique, nor a single algorithmic scheme capturing the power of P, nor a description of P in purely logical terms. We intend to provide these in a context which is so rich and vast that it requires the unification of some of the most important techniques, and will enhance our general understanding of P.Within the microcosm of finite-domain constraint satisfaction problems (CSPs), the recent resolution of the Feder-Vardi conjecture by Bulatov and by Zhuk provides a satisfactory picture of P. Our goal is a vast and uniform generalisation of this result in three directions: towards approximation via Promise CSPs, towards optimisation via Valued CSPs, and towards infinite domains via countably categorical CSPs and CSPs over numeric domains. In particular, our setting includes the linear programming problem as a numeric Valued CSP, the approximate graph coloring problem as a Promise CSP, and many problems from qualitative reasoning as infinite-domain CSPs. Our methods range from universal algebra, model theory, Ramsey theory, to complexity theory. Building on cross-connections between these extensions, we will provide a uniform description of P within this diverse and applicable universe, thus making a revolutionary leap in the resolution of the general problem.