When to Identify Is to Control: On the Controllability of Combinatorial Optimization Problems

2026-02-18

AI Summary

This study investigates how to manipulate a subset of variables in combinatorial optimization so that any prescribed target solution becomes optimal for a given objective function. Focusing on the equivalence between control sets and identification sets, the work integrates tools from convex analysis, matroid theory, and computational complexity to develop efficient algorithms for computing minimum-weight control sets in both convex and binary vector settings. It establishes that the family of control sets forms a matroid in the convex case and proves, for the first time, that deciding identification sets for s–t paths in directed acyclic graphs is Σ₂^P-complete. Furthermore, the paper derives tight bounds on the approximation gap between this problem and flow polyhedra.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Consider a finite ground set $E$, a set of feasible solutions $X \subseteq \mathbb{R}^{E}$, and a class of objective functions $\mathcal{C}$ defined on $X$. We are interested in subsets $S$ of $E$ that control $X$ in the sense that we can induce any given solution $x \in X$ as an optimum for any given objective function $c \in \mathcal{C}$ by adding linear terms to $c$ on the coordinates corresponding to $S$. This problem has many applications, e.g., when $X$ corresponds to the set of all traffic flows, the ability to control implies that one is able to induce all target flows by imposing tolls on the edges in $S$. Our first result shows the equivalence between controllability and identifiability. If $X$ is convex, or if $X$ consists of binary vectors, then $S$ controls $X$ if and only if the restriction of $x$ to $S$ uniquely determines $x$ among all solutions in $X$. In the convex case, we further prove that the family of controlling sets forms a matroid. This structural insight yields an efficient algorithm for computing minimum-weight controlling sets from a description of the affine hull of $X$. While the equivalence extends to matroid base families, the picture changes sharply for other discrete domains. We show that when $X$ is equal to the set of $s$-$t$-paths in a directed graph, deciding whether an identifying set of a given cardinality exists is $Σ\mathsf{_2^P}$-complete. The problem remains $\mathsf{NP}$-hard even on acyclic graphs. For acyclic instances, however, we obtain an approximation guarantee by proving a tight bound on the gap between the smallest identifying sets for $X$ and its convex hull, where the latter corresponds to the $s$-$t$-flow polyhedron.

Problem

• controllability
• combinatorial optimization
• identifiability
• controlling set
• optimization control

Innovation

• controllability
• identifiability
• matroid
• combinatorial optimization
• computational complexity