Search

Since Jacobson [FOCS89] initiated the investigation of succinct graph encodings 35 years ago, there has been a long list of results on balancing the generality of the class, the speed, the succinctness of the encoding, and the query support. Let Cn denote the set consisting of the graphs in a class C that with at most n vertices. A class C is nontrivial if the information-theoretically min number log |Cn| of bits to distinguish the members of Cn is Omega(n). An encoding scheme based upon a single class C is C-opt if it takes a graph G of Cn and produces in deterministic O(n) time an encoded string of at most log |Cn| + o(log |Cn|) bits from which G can be recovered in O(n) time. Despite the extensive efforts in the literature, trees and general graphs were the only nontrivial classes C admitting C-opt encoding schemes that support the degree query in O(1) time. Basing an encoding scheme upon a single class ignores the possibility of a shorter encoded string using additional properties of the graph input. To leverage the inherent structures of individual graphs, we propose to base an encoding scheme upon of multiple classes: An encoding scheme based upon a family F of classes, accepting all graphs in UF, is F-opt if it is C-opt for each C in F. Having a C-opt encoding scheme for each C in F does not guarantee an F-opt encoding scheme. Under this more stringent criterion, we present an F-opt encoding scheme for a family F of an infinite number of classes such that UF comprises all graphs of bounded Hadwiger numbers. F consists of the nontrivial quasi-monotone classes of k-clique-minor-free graphs for each positive integer k. Our F-opt scheme supports queries of degree, adjacency, neighbor-listing, and bounded-distance shortest path in O(1) time per output. We broaden the graph classes admitting opt encoding schemes that also efficiently support fundamental queries., Comment: 35 pages, 6 figures

For an $m$-edge connected simple graph $G$, finding a spanning tree of $G$ with the maximum number of leaves is MAXSNP-complete. The problem remains NP-complete even if $G$ is planar and the maximal degree of $G$ is at most four. Lu and Ravi gave the first known polynomial-time approximation algorithms with approximation factors $5$ and $3$. Later, they obtained a $3$-approximation algorithm that runs in near-linear time. The best known result is Solis-Oba, Bonsma, and Lowski's $O(m)$-time $2$-approximation algorithm. We show an alternative simple $O(m)$-time $2$-approximation algorithm whose analysis is simpler. This paper is dedicated to the cherished memory of our dear friend, Professor Takao Nishizeki., Comment: 10 pages, 4 figures, fixing typos of Equation (3)

Various classes of induced subgraphs are involved in the deepest results of graph theory and graph algorithms. A prominent example concerns the {\em perfection} of $G$ that the chromatic number of each induced subgraph $H$ of $G$ equals the clique number of $H$. The seminal Strong Perfect Graph Theorem confirms that the perfection of $G$ can be determined by detecting odd holes in $G$ and its complement. Chudnovsky et al. show in 2005 an $O(n^9)$ algorithm for recognizing perfect graphs, which can be implemented to run in $O(n^{6+\omega})$ time for the exponent $\omega<2.373$ of square-matrix multiplication. We show the following improved algorithms. 1. The tractability of detecting odd holes was open for decades until the major breakthrough of Chudnovsky et al. in 2020. Their $O(n^9)$ algorithm is later implemented by Lai et al. to run in $O(n^8)$ time, leading to the best formerly known algorithm for recognizing perfect graphs. Our first result is an $O(n^7)$ algorithm for detecting odd holes, implying an $O(n^7)$ algorithm for recognizing perfect graphs. 2. Chudnovsky et al. extend in 2021 the $O(n^9)$ algorithms for detecting odd holes (2020) and recognizing perfect graphs (2005) into the first polynomial algorithm for obtaining a shortest odd hole, which runs in $O(n^{14})$ time. We reduce the time for finding a shortest odd hole to $O(n^{13})$. 3. Conforti et al. show in 1997 the first polynomial algorithm for detecting even holes, running in about $O(n^{40})$ time. It then takes a line of intensive efforts in the literature to bring down the complexity to $O(n^{31})$, $O(n^{19})$, $O(n^{11})$, and finally $O(n^9)$. On the other hand, the tractability of finding a shortest even hole has been open for 16 years until the very recent $O(n^{31})$ algorithm of Cheong and Lu in 2022. We improve the time of finding a shortest even hole to $O(n^{23})$., Comment: 29 pages, 5 figures

For vertices $u$ and $v$ of an $n$-vertex graph $G$, a $uv$-trail of $G$ is an induced $uv$-path of $G$ that is not a shortest $uv$-path of $G$. Berger, Seymour, and Spirkl [Discrete Mathematics 2021] gave the previously only known polynomial-time algorithm, running in $O(n^{18})$ time, to either output a $uv$-trail of $G$ or ensure that $G$ admits no $uv$-trail. We reduce the complexity to the time required to perform a poly-logarithmic number of multiplications of $n^2\times n^2$ Boolean matrices, leading to a largely improved $O(n^{4.75})$-time algorithm., Comment: 18 pages, 6 figures, a preliminary version appeared in STACS 2022

An even (respectively, odd) hole in a graph is an induced cycle with even (respectively, odd) length that is at least four. Bienstock [DM 1991 and 1992] proved that detecting an even (respectively, odd) hole containing a given vertex is NP-complete. Conforti, Chornu\'ejols, Kappor, and Vu\v{s}kovi\'{c} [FOCS 1997] gave the first known polynomial-time algorithm to determine whether a graph contains even holes. Chudnovsky, Kawarabayashi, and Seymour [JGT 2005] estimated that Conforti et al.'s algorithm runs in $O(n^{40})$ time on an $n$-vertex graph and reduced the required time to $O(n^{31})$. Subsequently, da~Silva and Vu\v{s}kovi\'{c}~[JCTB 2013], Chang and Lu [JCTB 2017], and Lai, Lu, and Thorup [STOC 2020] improved the time to $O(n^{19})$, $O(n^{11})$, and $O(n^9)$, respectively. The tractability of determining whether a graph contains odd holes has been open for decades until the algorithm of Chudnovsky, Scott, Seymour, and Spirkl [JACM 2020] that runs in $O(n^9)$ time, which Lai et al. also reduced to $O(n^8)$. By extending Chudnovsky et al.'s techniques for detecting odd holes, Chudnovsky, Scott, and Seymour [Combinatorica 2020 to appear] (respectively, [arXiv 2020]) ensured the tractability of finding a long (respectively, shortest) odd hole. They also ensured the NP-hardness of finding a longest odd hole, whose reduction also works for finding a longest even hole. Recently, Cook and Seymour ensured the tractability of finding a long even hole. An intriguing missing piece is the tractability of finding a shortest even hole, left open for at least 15 years by, e.g., Chudnovsky et al. [JGT 2005] and Johnson [TALG 2005]. We resolve this long-standing open problem by giving the first known polynomial-time algorithm, running in $O(n^{31})$ time, for finding a shortest even hole in an $n$-vertex graph that contains even holes., Comment: 8 pages, 1 figure

The three-in-a-tree problem is to determine if a simple undirected graph contains an induced subgraph which is a tree connecting three given vertices. Based on a beautiful characterization that is proved in more than twenty pages, Chudnovsky and Seymour [Combinatorica 2010] gave the previously only known polynomial-time algorithm, running in $O(mn^2)$ time, to solve the three-in-a-tree problem on an $n$-vertex $m$-edge graph. Their three-in-a-tree algorithm has become a critical subroutine in several state-of-the-art graph recognition and detection algorithms. In this paper we solve the three-in-a-tree problem in $\tilde{O}(m)$ time, leading to improved algorithms for recognizing perfect graphs and detecting thetas, pyramids, beetles, and odd and even holes. Our result is based on a new and more constructive characterization than that of Chudnovsky and Seymour. Our new characterization is stronger than the original, and our proof implies a new simpler proof for the original characterization. The improved characterization gains the first factor $n$ in speed. The remaining improvement is based on dynamic graph algorithms., Comment: 46 pages, 12 figures, accepted to STOC 2020

Let $G$ be an $n$-node simple directed planar graph with nonnegative edge weights. We study the fundamental problems of computing (1) a global cut of $G$ with minimum weight and (2) a~cycle of $G$ with minimum weight. The best previously known algorithm for the former problem, running in $O(n\log^3 n)$ time, can be obtained from the algorithm of \Lacki, Nussbaum, Sankowski, and Wulff-Nilsen for single-source all-sinks maximum flows. The best previously known result for the latter problem is the $O(n\log^3 n)$-time algorithm of Wulff-Nilsen. By exploiting duality between the two problems in planar graphs, we solve both problems in $O(n\log n\log\log n)$ time via a divide-and-conquer algorithm that finds a shortest non-degenerate cycle. The kernel of our result is an $O(n\log\log n)$-time algorithm for computing noncrossing shortest paths among nodes well ordered on a common face of a directed plane graph, which is extended from the algorithm of Italiano, Nussbaum, Sankowski, and Wulff-Nilsen for an undirected plane graph., Comment: 25 pages, 14 figures