Symmetric Submodular Minimization from Comparisons

2026-10-07

AI Summary

This study addresses the minimization of symmetric submodular functions using only comparison information, without access to exact function values, thereby resolving the open problem of polynomial-time comparison algorithms for weighted graph cuts. To this end, it proposes a novel framework that integrates minimum-capacity ordering with contraction techniques, specifically optimized for down-closed families and integer-weighted graphs. The primary contribution is a deterministic polynomial-time algorithm requiring only O(n³) comparisons, matching the optimal value-query complexity while achieving superior performance in small-weight regimes. Furthermore, this work establishes an Ω(n log n) comparison lower bound for randomized algorithms, significantly advancing existing theoretical boundaries.

Abstract

Given value-oracle access to a symmetric submodular function $f:2^V\to\mathbb{R}$ with $|V|=n$, a nontrivial minimizer can be found using $O(n^3)$ value queries. We study the weaker comparison model, in which a query on $S,T\subseteq V$ reveals only whether $f(S)$ is smaller than, equal to, or larger than $f(T)$. We give a deterministic polynomial-time algorithm that finds a nontrivial minimizer of any symmetric submodular function using $O(n^3)$ comparisons, matching the best-known deterministic value-oracle bound despite not knowing the function values. More generally, the same $O(n^3)$-comparison bound holds for minimization over the nonempty members of any downward-closed family. Our algorithm combines the minimum-capacity ordering recently introduced by Iwata and Konno with the contraction framework of Goemans and Soto. Applying this result to weighted graph cut functions resolves the main open question of Cohen-Addad et al., who gave an $\widetilde{O}(n^3)$-comparison algorithm that runs in exponential time and asked whether a weighted minimum cut can be found in polynomial time using comparisons. For graphs with $m$ edges of integer weight at most $B$, we also give a deterministic polynomial-time algorithm that finds a minimum cut using \[ \widetilde{O}\!\left(n^2+\min\!\left\{mB,\,nB^2\right\}\right) \] comparisons, improving on the $O(n^3)$ bound when $B$ is small. Finally, we show that every randomized algorithm that outputs a minimum cut with probability at least $2/3$ makes $Ω(n \log n)$ expected comparisons in the worst case. Under the stronger assumption that all edge weights are polynomially bounded integers, we obtain an $Ω(n \log \log n)$ expected comparison lower bound. These bounds contrast with the value-oracle model, where no $ω(n)$ lower bound is known even for deterministic algorithms.

Problem

• Symmetric Submodular Minimization
• Comparison Model
• Minimum Cut
• Query Complexity
• Lower Bound

Innovation

• symmetric submodular minimization
• comparison model
• minimum cut
• deterministic polynomial-time algorithm
• query complexity