Analysis of a Random Local Search Algorithm for Dominating Set

TL;DR

The paper analyzes Random Local Search (RLS) for finding a minimum dominating set on cycles, establishing a rigorous upper bound $O(n^4 \log^2 n)$ on the expected time to reach an optimum. It develops cycle-specific domination models and leverages reversible Markov-chain techniques, along with an electrical-network view via effective resistance, to bound key hitting times. A sequence of representations—private neighborhoods, adjacency graphs, and adjacency-particle systems—facilitates a level-wise analysis that decomposes the search into tractable stochastic steps. The results provide theoretical runtime guarantees for a classical NP-hard problem and introduce modeling tools that could extend to other randomized heuristics on graph classes.

Abstract

Dominating Set is a well-known combinatorial optimization problem which finds application in computational biology or mobile communication. Because of its $\mathrm{NP}$-hardness, one often turns to heuristics for good solutions. Many such heuristics have been empirically tested and perform rather well. However, it is not well understood why their results are so good or even what guarantees they can offer regarding their runtime or the quality of their results. For this, a strong theoretical foundation has to be established. We contribute to this by rigorously analyzing a Random Local Search (RLS) algorithm that aims to find a minimum dominating set on a graph. We consider its performance on cycle graphs with $n$ vertices. We prove an upper bound for the expected runtime until an optimum is found of $\mathcal{O}\left(n^4\log^2(n)\right)$. In doing so, we introduce several models to represent dominating sets on cycles that help us understand how RLS explores the search space to find an optimum. For our proof we use techniques which are already quite popular for the analysis of randomized algorithms. We further apply a special method to analyze a reversible Markov Chain, which arises as a result of our modeling. This method has not yet found wide application in this kind of runtime analysis.

Figures (5)

• Figure 1: The Triangle Grid graph $T_4$. The origin is marked $o$. The vertices in $\partial(o,4)$ are highlighted.
• Figure 2: Left: The set $S \coloneq \{A,B,C\}$. $E$ is an $S$-private neighbor of $C$. $S$ is redundant because $A$ is a redundant vertex. Right: The set $S' \coloneq \{B,C\}$. $B$ is now an $S$-private neighbor of itself. Both $B$ and $C$ have private neighbors. $S'$ is thus irredundant.
• Figure 3: An irredundant and non-optimal dominating set of $C_6$.
• Figure 4: A dominating set of $C_6$. $C$ is clockwise, but not counterclockwise movable.
• Figure 5: Left: A dominating set $D$ on $C_8$. Middle: The intermediate adjacency graph of $D$ with corresponding weight function $w$. Right: The Edge-to-Vertex Dual Graph of the intermediate adjacency graph, the adjacency graph $\mathcal{A}\mathopen{}\mathclose{\left( D \right)$.

Theorems & Definitions (53)

• Definition 1: Cycle
• Proposition 1
• Definition 2: Stochastic Process
• Theorem 1: Multiplicative Drift
• Definition 3: Markov Chain
• Theorem 2: Wald's Equation
• Definition 4: Network
• Definition 5: Weighted random walk on a network
• Definition 6: Reversibility
• Definition 7: Flow