An $O(n^5)$-Time Algorithm for Optimal Broadcast Domination
This resolves a conjecture about the quintic-time complexity for broadcast domination, providing a faster exact algorithm for a fundamental graph problem.
The authors present an O(n^5)-time algorithm for optimal broadcast domination on arbitrary connected unweighted graphs, improving on the previous O(n^6) algorithm. The key innovation is an O(n^3) algorithm for the path case, which combined with existing reductions yields the overall improvement.
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 Sæther and recorded in subsequent surveys of broadcast domination.