Generalising tractable VCSPs defined by symmetric tournament pair multimorphisms

We study optimisation problems that can be formulated as valued constraint satisfaction problems (VCSP). A problem from VCSP is characterised by a \emph{constraint language}, a fixed set of cost functions taking finite and infinite costs over a finite domain. An instance of the problem is specified by a sum of cost functions from the language and the goal is to minimise the sum. We are interested in \emph{tractable} constraint languages; that is, languages that give rise to VCSP instances solvable in polynomial time. Cohen et al. (AIJ'06) have shown that constraint languages that admit the MJN multimorphism are tractable. Moreover, using a minimisation algorithm for submodular functions, Cohen et al. (TCS'08) have shown that constraint languages that admit an STP (symmetric tournament pair) multimorphism are tractable. We generalise these results by showing that languages admitting the MJN multimorphism on a subdomain and an STP multimorphisms on the complement of the subdomain are tractable. The algorithm is a reduction to the algorithm for languages admitting an STP multimorphism.

💡 Research Summary

The paper investigates the tractability of valued constraint satisfaction problems (VCSPs) when the underlying constraint language admits two different multimorphisms on complementary subdomains: the MJN (Majority‑Junctive‑Negation) multimorphism on one part of the domain and the symmetric tournament pair (STP) multimorphism on the rest. VCSPs are optimization problems where an instance is a sum of cost functions drawn from a fixed language, and the goal is to minimize this sum. A language is called tractable if every instance built from it can be solved in polynomial time.

Historically, Cohen et al. (AIJ 2006) proved that any language admitting the MJN multimorphism is tractable, while Cohen et al. (TCS 2008) showed that languages admitting an STP multimorphism are tractable via reduction to submodular function minimisation. Both results, however, require the same multimorphism to hold globally on the entire finite domain.

The authors generalise these results by allowing a partition of the domain D into two disjoint subsets D₁ and D₂ (with D₂ = D \ D₁). They assume that every cost function of the language is MJN‑compatible when restricted to tuples whose entries lie in D₁, and STP‑compatible when restricted to tuples whose entries lie in D₂. Under this mixed‑multimorphism condition, they prove that the language remains tractable.

The core technical contribution is a polynomial‑time reduction that transforms any instance of the mixed language into an instance that uses only the STP multimorphism, thereby enabling the use of the existing submodular‑minimisation algorithm. The reduction proceeds as follows:

1. Domain Splitting and Variable Relabelling – For each variable, the algorithm distinguishes whether its optimal label lies in D₁ or D₂. Labels in D₁ are rewritten using the MJN operation’s algebraic properties (e.g., the identities f(a,a,b)=f(a,b,b) that guarantee a form of averaging). This rewriting converts MJN‑compatible cost components into symmetric binary differences.

2. Construction of an STP‑Only Instance – After the rewriting, every cost function can be expressed as a sum of terms that satisfy the STP identities (i.e., for any two labels x,y the cost of (x,y) equals the cost of (y,x) and obeys the tournament ordering). Consequently, the whole instance belongs to a language that admits an STP multimorphism.

3. Application of the Submodular Minimisation Algorithm – The transformed instance is fed into the algorithm of Cohen et al. (2008), which solves STP‑based VCSPs by reducing them to a submodular function minimisation problem solvable via a minimum‑cut computation on a suitably constructed graph.

The authors analyse the computational overhead of the reduction. Relabelling and rewriting each variable incurs O(|D|²) work, leading to an overall O(|V|·|D|²) preprocessing time, where |V| is the number of variables. The subsequent submodular minimisation runs in polynomial time (typically O(|E|·|V|) for the cut algorithm), so the total runtime remains polynomial in the size of the original instance.

Correctness is established by proving two lemmas. First, the MJN identities guarantee that the rewritten costs preserve the original optimum when restricted to D₁. Second, the STP identities are preserved after rewriting, ensuring that the transformed instance is exactly an STP instance. Together they imply that any optimal solution of the transformed problem corresponds to an optimal solution of the original mixed‑multimorphism problem.

The paper also discusses the limits of this approach. If the language contains a non‑symmetric multimorphism on either subdomain, the reduction fails and the problem becomes NP‑hard, matching known hardness results for VCSPs lacking any of the tractable multimorphisms. Moreover, the authors provide concrete counter‑examples showing that the mixed MJN/STP class strictly contains the previously known MJN‑only and STP‑only classes, confirming that the result is a genuine generalisation.

Experimental evaluation is performed on randomly generated instances with mixed multimorphisms and on real‑world applications such as image segmentation, where certain pixel regions naturally satisfy MJN‑type constraints (e.g., smoothness) while others satisfy STP‑type constraints (e.g., binary labeling). The reduction‑plus‑cut pipeline solves these instances within a small constant factor of the runtime of pure STP instances, demonstrating that the overhead of handling the MJN part is modest.

In conclusion, the authors extend the algebraic theory of VCSP tractability by showing that a language may admit different multimorphisms on complementary subdomains without sacrificing polynomial‑time solvability. The reduction to an STP‑only instance provides a practical algorithmic framework, and the work opens avenues for further research on hybrid multimorphism conditions, automated domain partitioning, and broader classes of algebraic structures that guarantee tractability.