PACE Solver Description: twin_width_fmi

Abstract

In this paper we present \texttt{twin\_width\_fmi}'s solver for the heuristic track of PACE's 2025 competition on Minimum Dominating Set. As a baseline, we implement \texttt{greedy-ln}, a standard greedy dominating-set heuristic that repeatedly selects the vertex that newly dominates the largest number of currently undominated vertices. We then use this greedy solution as the starting point for a simulated annealing local search: we attempt vertex removals and exchanges and accept worsening moves with decaying probability, in order to escape local minima while preserving domination. Our best-performing component, which we ultimately submitted, is \texttt{hedom5}. The design of \texttt{hedom5} is inspired by recent iterative-greedy style domination heuristics~\cite{IterativeGreedy22} that alternate between constructive steps, pruning, and focused repair rather than relying on a single pass. In \texttt{hedom5}, the input graph is first stored in a compact CSR structure and simplified using fast reductions such as forcing neighbors of leaves and handling isolates. We then run a lazy gain-based greedy stage using a priority queue: each candidate vertex is scored by how many currently undominated vertices its closed neighborhood would newly dominate, and scores are only recomputed when necessary. After this constructive phase, we perform an aggressive backward pruning pass that iterates over the chosen dominators in reverse insertion order and deletes any vertex whose closed neighborhood is still fully dominated by the remaining set. Finally, we run a budgeted 1-swap local improvement step that attempts to replace a dominator by an alternative vertex that covers all of its uniquely covered vertices, thereby reducing the size of the dominating set. A brief safety patch at the end guarantees full domination.