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Author : Shyan Akmal, Virginia Vassilevska Williams, Ryan Williams, Zixuan Xu

Abstract : The k-Detour problem is a basic path-finding problem: given a graph G on n vertices, with specified nodes s and t, and a positive integer k, the goal is to determine if G has an st-path of length exactly dist(s,t)+k, where dist(s,t) is the length of a shortest path from s to t. The k-Detour problem is NP-hard when k is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in f(k)poly(n) time, for f as slow-growing as possible. We present faster algorithms for k-Detour in undirected graphs, running in 1.853kpoly(n) randomized and 4.082kpoly(n) deterministic time. The previous fastest algorithms for this problem took 2.746kpoly(n) randomized and 6.523kpoly(n) deterministic time [Bezkov-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a bipartitioned subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length k in undirected graphs [Bjrklund-Husfeldt-Kaski-Koivisto, JCSS 2017]. Our work has direct implications for the k-Longest Detour problem: in this problem, we are given the same input as in k-Detour, but are now tasked with determining if G has an st-path of length at least dist(s,t)+k. Our results for k-Detour imply that we can solve k-Longest Detour in 3.432kpoly(n) randomized and 16.661kpoly(n) deterministic time. The previous fastest algorithms for this problem took 7.539kpoly(n) randomized and 42.549kpoly(n) deterministic time [Fomin et al., STACS 2022].

2.Learning Spanning Forests Optimally using CUT Queries in Weighted Undirected Graphs (arXiv)

Author : Hang Liao, Deeparnab Chakrabarty

Abstract : In this paper we describe a randomized algorithm which returns a maximal spanning forest of an unknown {em weighted} undirected graph making O(n) CUT queries in expectation. For weighted graphs, this is optimal due to a result in [Auza and Lee, 2021] which shows an (n) lower bound for zero-error randomized algorithms. %To our knowledge, it is the only regime of this problem where we have upper and lower bounds tight up to constants. These questions have been extensively studied in the past few years, especially due to the problems connections to symmetric submodular function minimization. We also describe a simple polynomial time deterministic algorithm that makes O(nlognloglogn) queries on undirected unweighted graphs and returns a maximal spanning forest, thereby (slightly) improving upon the state-of-the-art.

See the rest here:
Working with Undirected graphs in Machine Learning part2 - Medium