[Paper Review] An Efficient Algorithm for Power Dominating Set

Thomas Bläsius, Max Göttlicher|arXiv (Cornell University)|Jan 1, 2023

Advanced Graph Theory Research1 citations

TL;DR

This paper presents a novel, efficient algorithm for the Power Dominating Set (PDS) problem, combining a new set of reduction rules with an improved implicit hitting set heuristic. It proves PDS is W[P]-complete, resolving its parameterized complexity, and demonstrates through experiments that the solver outperforms prior state-of-the-art methods by over an order of magnitude, solving previously intractable continental-scale power grid instances in minutes.

ABSTRACT

The problem Power Dominating Set (PDS) is motivated by the placement of phasor measurement units to monitor electrical networks. It asks for a minimum set of vertices in a graph that observes all remaining vertices by exhaustively applying two observation rules. Our contribution is twofold. First, we determine the parameterized complexity of PDS by proving it is $W[P]$-complete when parameterized with respect to the solution size. We note that it was only known to be $W[2]$-hard before. Our second and main contribution is a new algorithm for PDS that efficiently solves practical instances. Our algorithm consists of two complementary parts. The first is a set of reduction rules for PDS that can also be used in conjunction with previously existing algorithms. The second is an algorithm for solving the remaining kernel based on the implicit hitting set approach. Our evaluation on a set of power grid instances from the literature shows that our solver outperforms previous state-of-the-art solvers for PDS by more than one order of magnitude on average. Furthermore, our algorithm can solve previously unsolved instances of continental scale within a few minutes.

Motivation & Objective

• To resolve the parameterized complexity of the Power Dominating Set (PDS) problem by determining its exact class.
• To develop a practical, efficient algorithm for solving real-world PDS instances arising in power grid monitoring.
• To improve existing solvers by introducing new reduction rules and a more effective heuristic for identifying missing forts in the implicit hitting set framework.
• To enable the solution of large-scale, previously unsolved continental-scale power grid instances.

Proposed method

• Prove W[P]-completeness of PDS via a reduction from Weighted Circuit Satisfiability for arbitrary-weft circuits.
• Design a comprehensive set of 12 reduction rules that shrink PDS instances by removing vertices/edges or pre-annotating vertices as selected or forbidden.
• Integrate the reduction rules as a pre-processing step to produce smaller, annotated PDS-Extension instances.
• Develop a new heuristic for identifying missing forts in the implicit hitting set approach, improving the efficiency of the hitting set solver.
• Combine the reduction rules with an implicit hitting set solver using Gurobi as the underlying hitting set solver.
• Evaluate the full pipeline on real-world power grid instances from the literature, comparing performance against state-of-the-art solvers.

Experimental results

Research questions

• RQ1 Is the Power Dominating Set problem W[P]-complete when parameterized by solution size?
• RQ2 Can a new set of reduction rules significantly improve the performance of existing PDS solvers?
• RQ3 Does a novel heuristic for identifying missing forts in the implicit hitting set framework lead to substantial performance gains?
• RQ4 Can the proposed algorithm solve large-scale, previously unsolved continental-scale power grid instances?

Key findings

• The Power Dominating Set problem is proven to be W[P]-complete, resolving its exact parameterized complexity and showing it is not in W[2] unless W[2] = W[P].
• The proposed reduction rules reduce the median running time of the solver by more than one order of magnitude across all tested instances.
• The new heuristic for finding missing forts outperforms the previous state-of-the-art approach, contributing significantly to the overall performance gain.
• The full algorithm solves previously unsolved continental-scale power grid instances within minutes, a feat unachievable with prior methods.
• Even when combined with other solvers, the proposed reduction rules and heuristic lead to faster solutions on most benchmark instances.
• The algorithm provides lower bounds on the power dominating number more quickly than Gurobi on the same instances.