Lower bound for the cost of connecting tree with given vertex degree sequence

TL;DR

This paper establishes a lower bound for the cost of connecting trees with a specified degree sequence using semidefinite programming, aiding in network design and evaluation of heuristic algorithms.

Contribution

It introduces a novel semidefinite programming-based lower bound for tree connection costs with given degree sequences, applicable to various network optimization problems.

Findings

Lower bound computed via semidefinite programming.

Heuristic algorithms evaluated against the lower bound.

Method applied to real-life and synthetic data sets.

Abstract

The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, structure identification from data, etc. For the case of connecting trees with the given sequence of vertex degrees, the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear matrix constraints, which is reduced to the solution of a series of convex programs with linear matrix inequality constraints. The proposed lower bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life data sets. Keywords: Optimal communication network, generalized Wiener index, origin-destination matrix, semidefinite programming, quadratic matrix inequality.