Christian Møller Mikkelstrup

arXiv2026

Fast and Compact Graph Cuts for the Boykov-Kolmogorov Algorithm

Computing a minimum $s$-$t$ cut in a graph is a solution to a wide range of computer vision problems and is often done using the Boykov-Kolmogorov (BK) algorithm. In this paper, we revisit the BK algorithm from both a theoretical and practical point of view. We improve the analysis of the time complexity of the BK algorithm to $O(mn|C|)$ and propose a new algorithm, the fast BK (fBK) algorithm, with a time complexity of $O(m|C|)$, where $m$, $n$, and $|C|$ are the number of edges, the number of vertices, and the capacity of the cut, respectively. We additionally propose a compact graph representation that allows our implementations of the BK and fBK algorithms to find a minimum $s$-$t$ cut in a graph with up to $10^9$ vertices and $10^{10}$ edges on a machine with 128 GB of memory. We denote these as $\texttt{cBK}$ and $\texttt{fcBK}$, respectively. Despite $\texttt{fcBK}$ having the better worst-case guarantee, we find $\texttt{cBK}$ to be the fastest available implementation within the family of BK algorithms when evaluating on a comprehensive set of benchmark datasets, highlighting the importance of memory-efficient implementations. We make our implementations publicly available for further research and implementation development within minimum $s$-$t$ cut algorithms.

cs.CV↗

arXiv2026

Hardness and Approximation for Compressing Deterministic Finite Automata

The delayed deterministic finite automaton ($\text{D}^2\text{FA}$) is a powerful tool for compressing the prohibitively large deterministic finite automata (DFAs) used in modern signature-based deep packet inspection systems. A $\text{D}^2\text{FA}$ augments a DFA with unlabeled default transitions to remove transitions that are similar. This technique works well in practice and has inspired numerous subsequent studies. However, existing approaches do not guarantee a $\text{D}^2\text{FA}$ of optimal size as they restrict the default transitions to form a forest. In this paper, we consider the problem of finding an optimal $\text{D}^2\text{FA}$. We make the key observation that cycles of default transitions can be valid and produce a smaller $\text{D}^2\text{FA}$, provided that for every character in the alphabet, at least one state in the cycle has an outgoing transition labeled by that character. This observation raises fundamental questions concerning the computational complexity of finding an optimal $\text{D}^2\text{FA}$ and the approximation guarantee of the standard (forest-restricted) approach. We answer both questions by showing that it is NP-hard to find an optimal $\text{D}^2\text{FA}$ and by proving that the standard approach provides a $3/2$-approximation to the optimum. To establish NP-hardness, we construct a reduction from set cover using a DFA with gadgets where valid default transition choices correspond to selecting sets. To prove our approximation ratio, we compare the cycles in an optimal solution to a variant where the default transitions form a forest. Finally, we demonstrate that this $3/2$ ratio is tight via a constructed family of DFAs.

cs.DS↗