An $O(n^5)$-Time Algorithm for Optimal Broadcast Domination

TL;DR

This paper presents an improved $O(n^5)$-time algorithm for solving the optimal broadcast domination problem on connected unweighted graphs, significantly reducing previous complexity bounds.

Contribution

It introduces a novel $O(n^3)$ path-case algorithm and combines it with existing methods to achieve an overall $O(n^5)$-time solution for general graphs.

Findings

The new algorithm runs in $O(n^3)$ time and space for the path case.

The combined approach solves the problem in $O(n^5)$ time for all connected unweighted graphs.

It resolves the conjecture that the problem can be solved in quintic time.

Abstract

Broadcast domination assigns a nonnegative integer power to every vertex of a graph so that every vertex is within the assigned power of some broadcasting vertex, and the objective is to minimize the sum of the powers. Heggernes and Lokshtanov proved that the problem is polynomial-time solvable on arbitrary connected unweighted graphs by showing that some optimal efficient broadcast has a domination graph that is a path or a cycle, and by reducing the general case to an $O(n^6)$-time algorithm. This paper gives an efficient algorithm of the path-case. Instead of building one auxiliary acyclic graph for every possible left endpoint vertex, we build a single directed acyclic graph whose states are oriented broadcast balls together with their two possible residual sides. The resulting path-case algorithm runs in $O(n^3)$ time and $O(n^3)$ space on an $n$-vertex graph. Combining this routine with the same peel-one-ball reduction of Heggernes and Lokshtanov yields an exact $O(n^5)$-time algorithm for optimal broadcast domination on arbitrary connected unweighted graphs. This resolves the quintic-time conjecture for general graphs attributed to Heggernes and S\ae ther and recorded in subsequent surveys of broadcast domination.