Parameterized Complexity of CSP for Infinite Constraint Languages

We study parameterized Constraint Satisfaction Problem for infinite constraint languages. The parameters that we study are weight of the satisfying assignment, number of constraints, maximum number of occurrences of a variable in the instance, and maximum number of occurrences of a variable in each constraint. A dichotomy theorem is already known for finite constraint languages with the weight parameter. We prove some general theorems that show, as new results, that some well-known problems are fixed-parameter tractable and some others are in W[1].

💡 Research Summary

The paper investigates the parameterized complexity of the Constraint Satisfaction Problem (CSP) when the underlying constraint language is infinite, focusing on Boolean domains but allowing relations of arbitrary (finite) arity. While the parameterized CSP for finite languages has been extensively studied—most notably the dichotomy theorem for the weight parameter k (the number of variables set to 1 in a solution)—the infinite case has received little attention. The authors address this gap by introducing a mild “fpt‑membership‑checkable” condition: for any relation R in the language, given its index and a tuple, one can decide membership in time f(|tuple|)·poly(log index). This captures many natural infinite languages.

Four parameters are considered:
1. k – the exact weight of a satisfying assignment;
2. k≤ – an upper bound on the weight;
3. u – the total number of constraints;
4. t – the maximum number of occurrences of any variable across the whole instance;
5. e – the maximum number of occurrences of a variable inside a single constraint (the paper often uses t and e together).

The main contributions are organized into two groups.

Group 1 – Fixed‑Parameter Tractability with Additional Parameters.
Theorem 3.2 shows that for any set E⊆ℕ₀, the problem CSP(W_E){k,u,e} is fixed‑parameter tractable (FPT). Moreover, if either E or its complement is finite, the parameter e can be dropped, yielding FPT for CSP(W_E){k,u}. The proof proceeds by encoding an instance as a structure with only unary relations, thereby obtaining a structure of degree 0. By invoking the known FPT result for first‑order model‑checking on bounded‑degree structures (Theorem 3.1), the authors construct a first‑order formula whose size depends only on k, u, and the maximal arity h (derived from e and E). This reduction bypasses the need for a bounded‑arity assumption on the language, which is unavailable for infinite languages.

Concrete corollaries include:

• CSP(W_N){k,u} and CSP(W{1}){k,u} are FPT, where W_N denotes all disjunctions (any weight allowed) and W{1} encodes Weighted Exact CNF.
• CSP(W_odd){k,t} and CSP(W_even){k,u} are FPT, despite earlier results showing that CSP(W_odd∪W_even)_{k,≤} is W