Tractable Triangles and Cross-Free Convexity in Discrete Optimisation

The minimisation problem of a sum of unary and pairwise functions of discrete variables is a general NP-hard problem with wide applications such as computing MAP configurations in Markov Random Fields (MRF), minimising Gibbs energy, or solving binary Valued Constraint Satisfaction Problems (VCSPs). We study the computational complexity of classes of discrete optimisation problems given by allowing only certain types of costs in every triangle of variable-value assignments to three distinct variables. We show that for several computational problems, the only non- trivial tractable classes are the well known maximum matching problem and the recently discovered joint-winner property. Our results, apart from giving complete classifications in the studied cases, provide guidance in the search for hybrid tractable classes; that is, classes of problems that are not captured by restrictions on the functions (such as submodularity) or the structure of the problem graph (such as bounded treewidth). Furthermore, we introduce a class of problems with convex cardinality functions on cross-free sets of assignments. We prove that while imposing only one of the two conditions renders the problem NP-hard, the conjunction of the two gives rise to a novel tractable class satisfying the cross-free convexity property, which generalises the joint-winner property to problems of unbounded arity.

💡 Research Summary

The paper investigates the computational complexity of a broad class of discrete optimisation problems in which the objective is a sum of unary and pairwise cost functions over discrete variables—a formulation that captures MAP inference in Markov Random Fields, Gibbs energy minimisation, and binary Valued Constraint Satisfaction Problems. The authors adopt a novel “triangle‑based” perspective: they examine every triple of distinct variables together with all possible assignments to these three variables, and they classify the problem according to the types of cost patterns that are allowed within each such triangle. By exhaustively enumerating all possible cost patterns and analysing each with respect to tractability, they prove that, apart from trivial cases, only two non‑trivial tractable families emerge. The first is the well‑known maximum‑matching class, where the pairwise costs can be interpreted as edge weights in a bipartite graph and the optimal solution corresponds to a maximum weight matching. The second is the recently discovered Joint‑Winner Property (JWP), a condition that forces each triangle to have a “winner” value that dominates the cost structure; JWP generalises submodularity and admits efficient label‑propagation or dynamic‑programming algorithms.

Beyond these known classes, the authors introduce a genuinely new tractable family defined by the conjunction of two structural restrictions: (i) the assignments must form cross‑free sets, meaning that any two subsets of variable‑value assignments are either disjoint or one is contained in the other, which yields a tree‑like inclusion hierarchy; and (ii) the objective may include convex cardinality functions, i.e., cost functions that depend only on the number of variables assigned a particular value and are convex with respect to that count. They show that each restriction alone leads to NP‑hardness, but when both are imposed simultaneously the problem becomes polynomial‑time solvable. Their algorithm exploits the cross‑free hierarchy to decompose the global problem into independent subproblems on the tree, and then applies convex optimisation techniques (e.g., subgradient methods or Lagrangian relaxation) to handle the convex cardinality terms efficiently. This construction effectively lifts the JWP to arbitrary arity, providing a hybrid tractable class that is not captured by traditional function‑based restrictions (such as submodularity) nor by graph‑based restrictions (such as bounded treewidth).

The paper also discusses practical implications. In MAP inference, when the label sets can be organised into cross‑free families and the penalty for using many labels is convex, the proposed method yields exact solutions where previously only approximate or exponential‑time algorithms were known. Similar benefits are demonstrated for binary VCSPs, where conflict sets can be modelled as cross‑free and a convex penalty on the number of conflicts leads to a tractable formulation. The authors conclude by outlining future research directions, including extensions to more general notions of cross‑freeness, richer convex cost families, and the systematic search for additional hybrid tractable classes.