New Acyclic and Star Coloring Algorithms with Application to Computing Hessians

Abstract

Acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of Hessians using automatic differentiation or finite differencing, when both sparsity and symmetry are exploited. We present an algorithmic paradigm for finding heuristic solutions for these two NP-hard problems. The underlying common technique is the exploitation of the structure of two-colored induced subgraphs. For a graph $G$ on $n$ vertices and $m$ edges, the time complexity of our star coloring algorithm is $O(n\overline{d}_2)$, where $\overline{d}_k$, a generalization of vertex degree, denotes the average number of distinct paths of length at most $k$ edges starting at a vertex in $G$. The time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of Ackermann’s function. The space complexity of both algorithms is $O(m)$. To the best of our knowledge, our work is the first practical algorithm for the acyclic coloring problem. For the star coloring problem, our algorithm uses fewer colors and is considerably faster than a previously known $O(n\overline{d}_3)$-time algorithm. Computational results from experiments on various large-size test graphs demonstrate that the algorithms are fast and produce highly effective solutions. The use of these algorithms in Hessian computation is expected to reduce overall runtime drastically.