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Front Matter, Table of Contents, Preface, Conference Organization, LIPIcs, Vol. 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), pages 0:i-0:xxii

LIPIcs, Volume 254, STACS 2023, Complete Volume, LIPIcs, Vol. 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), pages 1-1026

Dynamic programming (DP) is one of the fundamental paradigms in algorithm design. However, many DP algorithms have to fill in large DP tables, represented by two-dimensional arrays, which causes at least quadratic running times and space usages. This has led to the development of improved algorithms for special cases when the DPs satisfy additional properties like, e.g., the Monge property or total monotonicity. In this paper, we consider a new condition which assumes (among some other technical assumptions) that the rows of the DP table are monotone. Under this assumption, we introduce a novel data structure for computing $(1+\varepsilon)$-approximate DP solutions in near-linear time and space in the static setting, and with polylogarithmic update times when the DP entries change dynamically. To the best of our knowledge, our new condition is incomparable to previous conditions and is the first which allows to derive dynamic algorithms based on existing DPs. Instead of using two-dimensional arrays to store the DP tables, we store the rows of the DP tables using monotone piecewise constant functions. This allows us to store length-$n$ DP table rows with entries in $[0,W]$ using only polylog$(n,W)$ bits, and to perform operations, such as $(\min,+)$-convolution or rounding, on these functions in polylogarithmic time. We further present several applications of our data structure. For bicriteria versions of $k$-balanced graph partitioning and simultaneous source location, we obtain the first dynamic algorithms with subpolynomial update times, as well as the first static algorithms using only near-linear time and space. Additionally, we obtain the currently fastest algorithm for fully dynamic knapsack., Comment: Abstract shortened to comply with arxiv formatting rules. To appear at STACS'23

In 1982, Papadimitriou and Yannakakis introduced the Exact Matching (EM) problem where given an edge colored graph, with colors red and blue, and an integer k, the goal is to decide whether or not the graph contains a perfect matching with exactly k red edges. Although they conjectured it to be NP-complete, soon after it was shown to be solvable in randomized polynomial time in the seminal work of Mulmuley et al., placing it in the complexity class RP. Since then, all attempts at finding a deterministic algorithm for EM have failed, thus leaving it as one of the few natural combinatorial problems in RP but not known to be contained in P, and making it an interesting instance for testing the hypothesis RP = P. Progress has been lacking even on very restrictive classes of graphs despite the problem being quite well known as evidenced by the number of works citing it. In this paper we aim to gain more insight into EM by studying a new optimization problem we call Top-k Perfect Matching (TkPM) which we show to be polynomially equivalent to EM. By virtue of being an optimization problem, it is more natural to approximate TkPM so we provide approximation algorithms for it. Some of the approximation algorithms rely on a relaxation of EM on bipartite graphs where the output is required to be a perfect matching with a number of red edges differing from k by at most k/2, which is of independent interest and generalizes to the Exact Weight Perfect Matching (EWPM) problem. We also consider parameterized algorithms and show that TkPM can be solved in FPT time parameterized by k and the independence number of the graph. This result again relies on new tools developed for EM which are also of independent interest., Leibniz International Proceedings in Informatics (LIPIcs), 254, ISSN:1868-8969, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), ISBN:978-3-95977-266-2

In a strongly connected graph G = (V,E), a cut arc (also called strong bridge) is an arc e ∈ E whose removal makes the graph no longer strongly connected. Equivalently, there exist u,v ∈ V, such that all u-v walks contain e. Cut arcs are a fundamental graph-theoretic notion, with countless applications, especially in reachability problems. In this paper we initiate the study of cut paths, as a generalisation of cut arcs, which we naturally define as those paths P for which there exist u,v ∈ V, such that all u-v walks contain P as subwalk. We first prove various properties of cut paths and define their remainder structures, which we use to present a simple O(m)-time verification algorithm for a cut path (|V| = n, |E| = m). Secondly, we apply cut paths and their remainder structures to improve several reachability problems from bioinformatics, as follows. A walk is called safe if it is a subwalk of every node-covering closed walk of a strongly connected graph. Multi-safety is defined analogously, by considering node-covering sets of closed walks instead. We show that cut paths provide simple O(m)-time algorithms verifying if a walk is safe or multi-safe. For multi-safety, we present the first linear time algorithm, while for safety, we present a simple algorithm where the state-of-the-art employed complex data structures. Finally we show that the simultaneous computation of remainder structures of all subwalks of a cut path can be performed in linear time, since they are related in a structured way. These properties yield an O(mn)-time algorithm outputting all maximal multi-safe walks, improving over the state-of-the-art algorithm running in time O(m²+n³). The results of this paper only scratch the surface in the study of cut paths, and we believe a rich structure of a graph can be revealed, considering the perspective of a path, instead of just an arc., LIPIcs, Vol. 254, 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023), pages 17:1-17:17