Phase transition for cutting-plane approach to vertex-cover problem

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from “easy” to “hard”. In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an “easy-hard” transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.

💡 Research Summary

The vertex‑cover problem asks for the smallest set of vertices that touches every edge of a given graph. It is a classic NP‑hard optimization problem and serves as a benchmark for studying typical computational hardness on random graph ensembles such as Erdős‑Rényi (ER) graphs. Previous statistical‑mechanical analyses have shown that the solution‑space structure of vertex cover on ER graphs undergoes a phase transition at average degree c = e ≈ 2.7183: for c < e the space is replica‑symmetric (RS), while for c > e it becomes replica‑symmetry‑broken (RSB). Empirically, exact branch‑and‑bound algorithms, which explore the space of feasible 0‑1 configurations, exhibit an “easy‑hard” transition around the same point: running times are modest for c < e and explode for c > e.

In this paper the authors investigate whether this easy‑hard transition is tied to the particular representation used by branch‑and‑bound, or whether it reflects a deeper property of the problem itself. To this end they adopt a completely different algorithmic paradigm: they formulate vertex cover as a 0‑1 integer linear program (ILP), relax it to a linear program (LP), and then iteratively add cutting planes until an integral solution is obtained. The cutting‑plane method operates outside the feasible configuration space for most of its execution; only at the final step does it produce a valid vertex‑cover. The authors refer to this as a “space‑outside” approach.

The experimental protocol is as follows. Random ER graphs with N ≈ 10³–10⁴ vertices are generated for average degrees c ranging from 1.5 to 4.0 in steps of 0.1. For each graph the LP‑cutting‑plane algorithm is run, and two primary metrics are recorded: (i) the number of cutting planes required for convergence, and (ii) the total CPU time. The authors also compare the final vertex‑cover size to the exact optimum (computed by branch‑and‑bound on smaller instances) to verify solution quality.

The results reveal a sharp transition at c ≈ e. Below this threshold the algorithm needs only a few cutting planes and finishes quickly; above it the number of cutting planes grows dramatically and the runtime spikes, mirroring the behavior observed for branch‑and‑bound. This transition persists across different cutting‑plane families (Gomory cuts, clique cuts, mixed strategies) and different LP solvers (CPLEX, Gurobi), indicating that it is not an artifact of a particular implementation. Moreover, even in the “hard” regime the algorithm still returns vertex covers whose size is essentially optimal, showing that the hardness manifests as increased computational effort rather than loss of solution quality.

The authors conclude that the easy‑hard transition at c = e is a fundamental characteristic of the vertex‑cover problem on ER graphs, independent of whether the algorithm explores the interior of the feasible configuration space or works in its exterior via cutting planes. This supports the broader hypothesis that typical computational hardness is rooted in the intrinsic geometry of the solution space, not in the specific representation or search strategy. The findings suggest that similar phase‑transition‑driven hardness should be expected for other NP‑hard combinatorial optimization problems, and that understanding the underlying structural change is crucial for designing algorithms that can either avoid or exploit it.