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

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.

Figures (5)

• Figure 1: Example of the construction from the proof of \ref{['thm:paths-hardness-DAGs']}. The undirected graph on the left depicts an instance of Vertex Cover. The digraph resulting from the construction is depicted on the right. The arc set $\{(s, u_e), (v_{2,1}, t), \dots, (v_{2,\ell},t)\}$ is a minimum-size identifying set for the $s$-$t$-paths in the digraph, corresponding to the vertex cover $\{v_2\}$ for the undirected graph.
• Figure 2: Example showing that the bound given in \ref{['thm:paths-flow-gap']} is tight with $k = 3$. The marked edges $(v_1, v_2), (v_3, v_4), (v_5, v_6)$ form an identifying set for the $s$-$t$-paths $X^G_{s, t}$. Any minimum-size identifying set for the set of $s$-$t$-flows of value $1$, however, is the complement of a spanning tree in the underlying undirected graph and thus has cardinality $|E| - (|V| - 1) = k(k+1)/2 = 6$.
• Figure 3: Example for the construction in the proof of \ref{['thm:sigma-2-p-forbidden-pairs-identification']}. The left figure shows the digraph $D$ and the right figure shows the corresponding digraph $D'$ where $s' = s_0 = s_1$, $t_0 = t_1 = s_2$, and $t' = t_2$. The set of forbidden pairs $\mathcal{F} = \{ \{a, e\} \}$ in $D$ results in the forbidden pairs $\mathcal{F}' = \{\{a_2, e_2\}, \{a_0, b_2\}, \{a_1, b_2\}, \{b_0, a_2\}, \{b_1, a_2\}, \{c_0, d_2\}, \{c_1, d_2\}, \{d_0, c_2\}, \{d_1, c_2\} \}$ in $D'$.
• Figure 4: Gadget $G_i$ used in the proof of \ref{['thm:sigma-2-p-fixed-arcs']}. This is a simplified version of the "switch" gadget used by fortune1980directed. Dashed arcs represent arcs in $\bar{E}$.
• Figure 5: Overview of the construction for proving \ref{['thm:sigma-2-p-fixed-arcs']}. The arcs $(v_i, w_i), (v'_i, w'_i)$ connected by a line represent a switch gadget $G_i$, where the nodes $v_i, v'_i, w_i, w'_i$ incident to arcs from the digraph $D$; the connection of the nodes $s_i, t_i, s'_i, t'_i$ from $G_i$ to nodes in other gadgets are depicted in the two boxed labeled $G_i$ on each side of the figure.

Theorems & Definitions (71)

• definition 1
• lemma 1
• proof
• theorem 1
• proof : Proof of \ref{['thm:convex']}
• corollary 1
• lemma 2
• proof
• theorem 2
• proof