DeepAI

# Strong SDP based bounds on the cutwidth of a graph

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.

• 10 publications
• 1 publication
• 2 publications
06/28/2022

### Strengthened Partial-Ordering Based ILP Models for the Vertex Coloring Problem

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### Maximum independent set (stable set) problem: A mathematical programming model with valid inequalities and computational testing

This paper deals with the maximum independent set (M.I.S.) problem, also...
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### Tailored vertex ordering for faster triangle listing in large graphs

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### Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning

Finding tight bounds on the optimal solution is a critical element of pr...
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### Lower bound for the cost of connecting tree with given vertex degree sequence

The optimal connecting network problem generalizes many models of struct...
12/29/2019

### An integer programming approach for solving a generalized version of the Grundy domination number

A sequence of vertices in a graph is called a legal dominating sequence ...
12/11/2019

### Crossing Reduction of Sankey Diagram with Barycentre Ordering via Markov Chain

Sankey diagram is popular for analyzing primary flows in network data. H...