Relaxed Connected Dominating Set Problem with Application to Secure Power Network Design

TL;DR

This paper introduces the relaxed connected dominating set (RCDS) problem, a variant of the connected dominating set problem motivated by secure power network design, and proposes exact and fixed-parameter algorithms for planar graphs.

Contribution

The paper formulates the RCDS problem, proves its NP-hardness, and develops a fixed-parameter polynomial time algorithm for planar graphs using sphere-cut decomposition and dynamic programming.

Findings

RCDS problem is NP-hard even for planar graphs.

Sphere-cut decomposition approach is efficient for small branchwidth graphs.

Case studies show practical applicability to power networks.

Abstract

This paper investigates a combinatorial optimization problem motived from a secure power network design application in [D\'{a}n and Sandberg 2010]. Two equivalent graph optimization formulations are derived. One of the formulations is a relaxed version of the connected dominating set problem, and hence the considered problem is referred to as relaxed connected dominating set (RCDS) problem. The RCDS problem is shown to be NP-hard, even for planar graphs. A mixed integer linear programming formulation is presented. In addition, for planar graphs a fixed parameter polynomial time solution methodology based on sphere-cut decomposition and dynamic programming is presented. The computation cost of the sphere-cut decomposition based approach grows linearly with problem instance size, provided that the branchwidth of the underlying graph is fixed and small. A case study with IEEE benchmark power networks verifies that small branchwidth are not uncommon in practice. The case study also indicates that the proposed methods show promise in computation efficiency.