An initial study of time complexity in infinite-domain constraint satisfaction

Abstract

The constraint satisfaction problem (CSP) is a widely studied problem with numerous applications in computer science and artificial intelligence. For infinite-domain CSPs, there are many results separating tractable and NP-hard cases while upper and lower bounds bounds on the time complexity of hard cases are virtually unexplored. Hence, we initiate a study of the worst-case time complexity of such CSPs. We analyse backtracking algorithms and determine upper bounds on their time complexity. We present asymptotically faster algorithms based on enumeration techniques and we show that these algorithms are applicable to well-studied problems in, for instance, temporal reasoning. Finally, we prove non-trivial lower bounds applicable to many interesting CSPs, under the assumption that certain complexity-theoretic assumptions hold. The gap between upper and lower bounds is in many cases surprisingly small, which suggests that our upper bounds cannot be significantly improved.

Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

“…However, to the best of our knowledge, all concrete lower bounds using SETH for exponential-time algorithms falls into one of the following cases: either the lower bound matches the running time of a trivial algorithm, as in the case of H S , NAE-SAT, and Π 2 -3-SAT, showing that no improvement is possible; or the lower bounds are with respect to a much more permissive complexity parameter than n, such as treewidth [41]. The one other example we are aware of is from the study of infinite-domain CSPs by Jonsson and Lagerkvist [30], who obtained upper bounds of the form O * (2 f (n) ) for non-linear functions f and a lower bound stating that the CSPs are not solvable in O(c n ) time for any constant c. These bounds are therefore in a sense closer to non-subexponentiality results usually obtained from the ETH. SETH and other conjectures have also seen significant applications over recent years in producing conditional lower bounds for polynomial-time solvable problems, but these are only tangentially relevant here.…”

A Multivariate Complexity Analysis of Qualitative Reasoning Problems

“…2 1) time [22] and is thus in E. • CSP(B ∨ω ), where B is a set of binary, jointly exhaustive and pairwise disjoint relations containing the equality relation over a countably infinite domain, and where B ∨ω = k≥1 B ∨k , is not in E under the SETH [20].…”

“…Unfortunately, neither approach seem fit for qualitative reasoning problems. On the one hand, they are solvable in 2 O(n 2 ) time or 2 O(n•log n) time in certain cases [20], but we can currently only rule out subexponential 2 o(n) algorithms under ETH [23]. On the other hand, despite the immense success of parameterized complexity, there is a lack of natural FPT algorithms for qualitative reasoning, and we are only aware of a handful of less surprising examples such as tree-width [6,10].…”

Fine-Grained Time Complexity of Constraint Satisfaction Problems

“…However, for arbitrary constraint languages over infinite domains there is reason to believe that the situation is overall much more complex. For example, there are NP-complete problems CSP(Γ) not solvable in O(c n ) time for any c ≥ 0, assuming the SETH is true [34], which is in stark contrast to finite-domain CSPs which are always solvable in O(|D| n ) time for every finite domain D. In contrast to Theorem 13, the results by Jonsson & Lagerkvist [34] were not obtained by ppinterpretability, and it would be interesting to investigate if such strong, lower bounds could be obtained using algebraically informed approaches.…”