Approximation for Maximum Surjective Constraint Satisfaction Problems

Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are computational problems where we are given a set of variables denoting values from a finite domain B and a set of constraints on the variables. A solution to such a problem is a surjective mapping from the set of variables to B such that the number of satisfied constraints is maximized. We study the approximation performance that can be acccchieved by algorithms for these problems, mainly by investigating their relation with Max-CSPs (which are the corresponding problems without the surjectivity requirement). Our work gives a complexity dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the assumption that there is a complexity dichotomy for Max-CSP(B) between PO and APX-complete, which has already been proved on the Boolean domain and 3-element domains.

💡 Research Summary

The paper investigates the approximation complexity of Maximum Surjective Constraint Satisfaction Problems (Max‑Sur‑CSP). A Max‑Sur‑CSP instance consists of a finite relational structure A (the variables) and a fixed template structure B (the domain), together with a set of constraints. The goal is to find a surjective mapping h : A → B that maximizes the number of satisfied constraints. This problem combines two well‑studied directions: the maximization version of CSP (Max‑CSP) and the surjectivity requirement (Sur‑CSP).

The authors first establish that Max‑Sur‑CSP belongs to the class APX for any finite domain B. They present a simple randomized algorithm: order the variables by degree, fix a bijection from the first |B| variables to the elements of B (ensuring surjectivity), and assign the remaining variables uniformly at random. By analyzing the expected number of satisfied k‑tuples, they obtain a constant approximation ratio r = min_i |R_Bi| / |B|^{k_i}. Using the method of conditional expectations, this randomized algorithm can be derandomized into a deterministic polynomial‑time r‑approximation algorithm.

Next, the paper explores the relationship between the approximation hardness of Max‑CSP and Max‑Sur‑CSP. A reduction is given that, for any instance of Max‑CSP, adds |B| fresh variables (with no additional constraints) to obtain an instance of Max‑Sur‑CSP. The optimal values of the two instances coincide, and any r‑approximate solution for the surjective version yields an r‑approximate solution for the original Max‑CSP. Consequently, if Max‑CSP is not r‑approximable in polynomial time, neither is Max‑Sur‑CSP. Moreover, if Max‑CSP is APX‑hard, the same reduction shows that Max‑Sur‑CSP is also APX‑hard (the AP‑reduction constant α equals 1). As a concrete example, the authors consider the structure C₆ (the 6‑cycle modulo graph). Max‑CSP(C₆) is exactly Max‑Cut, known to be APX‑hard and, under the Unique Games Conjecture, to have an optimal approximation ratio α_GW ≈ 0.878. By the reduction, Max‑Sur‑CSP(C₆) inherits the same hardness and cannot be approximated beyond α_GW in polynomial time.

The third major contribution is a positive result: if Max‑CSP(B) admits a polynomial‑time r‑approximation algorithm (or even an exact polynomial‑time algorithm, i.e., belongs to PO), then Max‑Sur‑CSP(B) admits a PTAS. The authors design a two‑stage algorithm. First, they run the given r‑approximation algorithm for Max‑CSP to obtain a mapping h. If h is already surjective, they output it. Otherwise, they iteratively replace values of variables that cause missing domain elements by randomly selecting a variable that shares its current image with another variable and reassigning it to a missing element. The probability that any particular satisfied constraint is destroyed during this fixing phase is bounded by k·|B|/(|A|−|B|), which becomes arbitrarily small as the instance size grows. By choosing ε > 0 and ensuring |A| is large enough, the final mapping h* satisfies at least (r − ε)·OPT constraints, yielding a PTAS for Max‑Sur‑CSP whenever Max‑CSP is in PO.

Combining the hardness and the PTAS results, the authors obtain a dichotomy theorem: assuming the known dichotomy for Max‑CSP (that each finite B yields either PO or APX‑complete), the same dichotomy holds for Max‑Sur‑CSP—each problem is either PTAS‑solvable or APX‑complete. This dichotomy is already established for Boolean domains and for all three‑element domains, because the Max‑CSP dichotomy has been proved for those cases.

The paper also highlights that some Max‑Sur‑CSPs are NP‑hard yet admit PTAS, a phenomenon not observed for Max‑CSPs. An example is the Minimum‑Asymmetric‑Cut problem, which corresponds to Max‑Sur‑CSP on a Boolean template with equality and disjunction relations.

In the concluding section, the authors discuss open questions, such as extending the dichotomy to all finite domains, characterizing precisely which templates B make Max‑Sur‑CSP(B) tractable (in PO), and exploring the impact of the surjectivity requirement on other optimization problems. Overall, the work bridges the gap between maximization and surjectivity in CSPs, providing both hardness and algorithmic results that parallel the well‑studied Max‑CSP landscape.