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.

翻译：广播支配为图的每个顶点分配一个非负整数功率，使得每个顶点均处于某个广播顶点所分配功率的覆盖范围内，其目标是最小化功率之和。Heggernes与Lokshtanov通过证明某些最优有效广播的支配图是路径或环，并将一般情形归约为$O(n^6)$时间复杂度算法，证明了该问题在任意连通无权图上可在多项式时间内求解。本文提出了一种路径情形的高效算法。我们无需为每个可能的左端点顶点构建一个辅助无环图，而是构建单一有向无环图，其状态由定向广播球及其两个可能的残留侧面组成。由此得到的路径情形算法在$n$顶点图上运行时间为$O(n^3)$，空间复杂度为$O(n^3)$。将该方法与Heggernes与Lokshtanov的逐球剥离归约相结合，可在任意连通无权图上得到最优广播支配的精确$O(n^5)$时间复杂度算法。这解决了归因于Heggernes与Sæther并在后续广播支配综述中记载的关于一般图的五次方时间猜想。