Semilinear Program Feasibility

We study logical techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems (CSPs). For the fundamental algebraic structure $\Gamma=(\mathbb R; L_1,L_2,\dots)$ where $\mathbb R$ are the real numbers and L 1 ,L 2 ,... is an enumeration of all linear relations with rational coefficients, we prove that a semilinear relation R (i.e., a relation that is first-order definable with linear inequalities) either has a quantifier-free Horn definition in Γ or the CSP for $(\mathbb R; R,L_1,L_2,\dots)$ is NP-hard. The result implies a complexity dichotomy for all constraint languages that are first-order expansions of Γ : the corresponding CSPs are either in P or are NP-complete depending on the choice of allowed relations. We apply this result to two concrete examples (generalised linear programming and metric temporal reasoning) and obtain full complexity dichotomies in both cases.

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