Strong SDP based bounds on the cutwidth of a graph

TL;DR

This work tackles the NP-hard minimum cutwidth problem (MCP) by developing a semidefinite programming (SDP) relaxation that yields strong lower bounds and, from the SDP solution, a feasible ordering to produce upper bounds. The core contribution is a new SDP formulation augmented with a suite of strengthening inequalities, including 3-dicycle equations, triangle inequalities, constraints from the squared linear ordering polytope LO$^{2}$, and liftings from LO, integrated via a cutting-plane procedure with simulated-annealing-based separation. A parallel upper-bound heuristic extracts a feasible vertex ordering from the SDP solution and refines it with simulated annealing. Computational experiments on dense and medium-density graphs show that the SDP-based bounds are competitive and often superior to MILP on denser instances, illustrating robustness to graph density; the authors also outline directions for faster SDP solvers (e.g., ADMM) and potential integration into exact branch-and-bound frameworks. Overall, the paper offers a practical SDP-based toolkit for obtaining tight MCP bounds and highlights promising future applications to related vertex-ordering parameters like treewidth and pathwidth.

Abstract

Given a linear ordering of the vertices of a graph, the cutwidth of a vertex $v$ with respect to this ordering is the number of edges from any vertex before $v$ (including $v$) to any vertex after $v$ in this ordering. The cutwidth of an ordering is the maximum cutwidth of any vertex with respect to this ordering. We are interested in finding the cutwidth of a graph, that is, the minimum cutwidth over all orderings, which is an NP-hard problem. In order to approximate the cutwidth of a given graph, we present a semidefinite relaxation. We identify several classes of valid inequalities and equalities that we use to strengthen the semidefinite relaxation. These classes are on the one hand the well-known 3-dicycle equations and the triangle inequalities and on the other hand we obtain inequalities from the squared linear ordering polytope and via lifting the linear ordering polytope. The solution of the semidefinite program serves to obtain a lower bound and also to construct a feasible solution and thereby having an upper bound on the cutwidth. In order to evaluate the quality of our bounds, we perform numerical experiments on graphs of different sizes and densities. It turns out that we produce high quality bounds for graphs of medium size independent of their density in reasonable time. Compared to that, obtaining bounds for dense instances of the same quality is out of reach for solvers using integer linear programming techniques.

Figures (1)

• Figure 1: Evolution of the bound over the iterations of Algorithm \ref{['alg:CW']}, Erdős-Rényi graph with $n=20$ and $p = 0.8$, $7 \text{ iterations}$

Theorems & Definitions (4)

• Lemma 2.1
• proof
• Lemma 2.2
• proof