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 $O(n^{6+ω})$ time for the exponent $ω<2.373$ 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 $O(n^{23})$.
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in pol…
Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect $(A+B)$-path packing problem is: given an undirected graph $G$ and two disjoint node subsets $A,B$ with even cardinalities, find a shortest $|A|/2+|B|/2$ disjoint paths whose ends are both in $A$ or both in $B$. Besides its NP…
We consider the computational problem of finding short paths in the skeleton of the perfect matching polytope of a bipartite graph. We prove that unless $P=NP$, there is no polynomial-time algorithm that computes a path of constant length between two vertices at distance two of the perfect matching polytope of a bipartite graph. Conditioned on $P\neq NP$, this disproves a conjecture by Ito, Kakimura, Kamiyama, Kobayashi and Okamoto [SIAM Journal on Discrete Mathematics, 36(2), pp. 1102-1123 (2022)]. Assuming the Exponential Time Hypothesis we prove the stronger result that there exists no poly…
In this paper, we study a set of combinatorial optimization problems on weighted graphs: the shortest path problem with negative weights, the weighted perfect bipartite matching problem, the unit-capacity minimum-cost maximum flow problem and the weighted perfect bipartite $b$-matching problem under the assumption that $\Vert b\Vert_1=O(m)$. We show that each one of these four problems can be solved in $\tilde{O}(m^{10/7}\log W)$ time, where $W$ is the absolute maximum weight of an edge in the graph, which gives the first in over 25 years polynomial improvement in their sparse-graph time compl…
Given an undirected graph and two disjoint vertex pairs $s_1,t_1$ and $s_2,t_2$, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting $s_1$ with $t_1$, and $s_2$ with $t_2$, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs…
Forcing a unique minimum spanning tree and a unique shortest path
Hokkaido University（北海道大学）；Institute of Informatics, University of Warsaw（华沙大学信息学研究所）；LAMSADE, Université Paris-Dauphine, PSL University, CNRS UMR7243（LAMSADE，巴黎第九大学，PSL大学，法国国家科学研究中心UMR7243）；Université Paris Cité, IRIF, CNRS（巴黎西岱大学，IRIF，法国国家科学研究中心）；Nagoya University（名古屋大学）
A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. Th…
We give an $\tilde{O}(n^{7/5} \log (nC))$-time algorithm to compute a minimum-cost maximum cardinality matching (optimal matching) in $K_h$-minor free graphs with $h=O(1)$ and integer edge weights having magnitude at most $C$. This improves upon the $\tilde{O}(n^{10/7}\log{C})$ algorithm of Cohen et al. [SODA 2017] and the $O(n^{3/2}\log (nC))$ algorithm of Gabow and Tarjan [SIAM J. Comput. 1989]. For a graph with $m$ edges and $n$ vertices, the well-known Hungarian Algorithm computes a shortest augmenting path in each phase in $O(m)$ time, yielding an optimal matching in $O(mn)$ time. The Hop…
The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomi…
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-…
The 2-Wasserstein distance (or RMS distance) is a useful measure of similarity between probability distributions that has exciting applications in machine learning. For discrete distributions, the problem of computing this distance can be expressed in terms of finding a minimum-cost perfect matching on a complete bipartite graph given by two multisets of points $A,B \subset \mathbb{R}^2$, with $|A|=|B|=n$, where the ground distance between any two points is the squared Euclidean distance between them. Although there is a near-linear time relative $\varepsilon$-approximation algorithm for the c…
