Matching Interdiction

In the matching interdiction problem, we are given an undirected graph with weights and interdiction costs on the edges and seek to remove a subset of the edges constrained to some budget, such that the weight of a maximum weight matching in the remaining graph is minimized. In this work we introduce the matching interdiction problem and show that it is strongly NP-complete even when the input is restricted to simple, bipartite graphs with unit edge weights and unit interdiction costs. Furthermore, we present a pseudo-polynomial algorithm for solving the matching interdiction problem on graphs with bounded treewidth. The proposed algorithm extends the approach that is typically used for the creation of efficient algorithms on graphs with bounded treewidth to interdiction problems.

💡 Research Summary

The paper introduces the Matching Interdiction problem, a novel combinatorial optimization task where an adversary may delete edges from an undirected weighted graph under a budget constraint, aiming to minimize the weight of a maximum‑weight matching in the residual graph. Formally, a network is given as (G,w,c) with edge weights w(e) and interdiction costs c(e). For a budget B, the objective is ν_B(G)=min_{U⊆E, c(U)≤B} ν(G−U), where ν(·) denotes the weight of a maximum‑weight matching. The decision version MINT(G,w,c,B,K) asks whether ν_B(G)≤K.

The authors first establish strong computational hardness. By a reduction from the classic knapsack problem, they show that even when the graph consists solely of isolated edges, MINT is NP‑complete. This demonstrates that the problem remains hard on the most trivial graph class. To prove strong NP‑completeness under unit weights and unit interdiction costs, they define a variant NMINTU that distinguishes removable from non‑removable edges. They replace each non‑removable edge with a “|E|-gadget”, a complete bipartite subgraph that guarantees the maximum matching size increases by a fixed amount regardless of which edges are removed, thereby encoding the non‑removability using only unit costs. A subsequent reduction from the CLIQUE problem constructs, from any input graph H, a bipartite graph G together with a set N of protected edges. They prove that H contains a clique of size r if and only if NMINTU(G,N,⌈r/2⌉, ν(G)−r(r−3)/2) is a “yes’’ instance. A technical lemma about bipartite graphs without 4‑cycles (Lemma 1) is used to argue the correctness of this reduction. Combining these steps yields Theorem 4: MINTU (the unit‑cost version) is NP‑complete even on simple bipartite graphs.

Having established hardness, the paper turns to tractable cases. For graphs of bounded treewidth k, the authors present a pseudo‑polynomial dynamic programming algorithm. They first compute a nice tree decomposition (X,T) of width k. For each node i of the decomposition they consider the subgraph G_i induced by the bag X_i and its descendants. For any interdiction set U_i within G_i of total cost ≤b (0≤b≤B), they capture the effect of U_i on the rest of the graph by a function a_i: 2^{X_i} → {0,…,C}, where C=∑_{e∈E}c(e). For a subset X⊆X_i, a_i(X) is the weight of a maximum matching in G_i−U_i that leaves exactly the vertices of X unsaturated. The set of all such functions has size (C+1)^{2^{|X_i|}} ≤ (C+1)^{2^{k+1}}. The algorithm proceeds bottom‑up, updating for leaf, introduce, forget, and join nodes by combining the corresponding functions and respecting the budget. For each bag i and each budget b they maintain the set A^i_b of realizable functions; this is stored as a Boolean table Q^i_b of size |A_i|. The transition rules are straightforward: at an introduce node a new vertex v is added to the bag, and the table is updated by considering whether v is matched or left free; at a forget node v is removed, and the table is projected; at a join node the tables from the two children are merged by taking pointwise minima of the function values. After processing the root r, the instance is feasible iff there exists a function a_r∈A^r_B with a_r(∅)≤K.

The running time is O(|V|·(C+1)^{2^{k+1}}·B), which is pseudo‑polynomial because it depends polynomially on the total cost C and the budget B, and exponentially only on the treewidth k (treated as a constant). Consequently, for graphs with small treewidth—such as trees, series‑parallel graphs, or graphs arising from certain network designs—the Matching Interdiction problem can be solved exactly in practice.

The paper concludes by emphasizing that Matching Interdiction is computationally intractable in general, even under severe restrictions, but that the treewidth‑based dynamic programming framework opens the door to exact algorithms for structured networks and can be adapted to other interdiction problems. Future work is suggested on approximation schemes, parameterized complexity beyond treewidth, and empirical evaluation on real‑world network data.