We consider three variants of the problem of finding a maximum weight restricted $2$-matching in a subcubic graph $G$. (A $2$-matching is any subset of the edges such that each vertex is incident to at most two of its edges.) Depending on the variant a restricted $2$-matching means a $2$-matching that is either triangle-free or square-free or both triangle- and square-free. While there exist polynomial time algorithms for the first two types of $2$-matchings, they are quite complicated or use advanced methodology. For each of the three problems we present a simple reduction to the computation of a maximum weight $b$-matching. The reduction is conducted with the aid of half-edges. A half-edge of edge $e$ is, informally speaking, a half of $e$ containing exactly one of its endpoints. For a subset of triangles of $G$, we replace each edge of such a triangle with two half-edges. Two half-edges of one edge $e$ of weight $w(e)$ may get different weights, not necessarily equal to $\frac{1}{2}w(e)$. In the metric setting when the edge weights satisfy the triangle inequality, this has a geometric interpretation connected to how an incircle partitions the edges of a triangle. Our algorithms are additionally faster than those known before. The running time of each of them is $O(n^2\log{n})$, where $n$ denotes the number of vertices in the graph.
We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths $\frac{5n+n_2}{4}-1$ and $\frac{5n}{4} - 2$ respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm…
The Max-Cut problem is a fundamental NP-hard problem, which is attracting attention in the field of quantum computation these days. Regarding the approximation algorithm of the Max-Cut problem, algorithms based on semidefinite programming have achieved much better approximation ratios than combinatorial algorithms. Therefore, filling the gap is an interesting topic as combinatorial algorithms also have some merits. In sparse graphs, there is a linear-time combinatorial algorithm with the approximation ratio $\frac{1}{2}+\frac{n-1}{4m}$ [Ngoc and Tuza, Comb. Probab. Comput. 1993], which is know…
Every undirected graph $G$ has a (weighted) cut-equivalent tree $T$, commonly named after Gomory and Hu who discovered it in 1961. Both $T$ and $G$ have the same node set, and for every node pair $s,t$, the minimum $(s,t)$-cut in $T$ is also an exact minimum $(s,t)$-cut in $G$. We give the first subcubic-time algorithm that constructs such a tree for a simple graph $G$ (unweighted with no parallel edges). Its time complexity is $\tilde{O}(n^{2.5})$, for $n=|V(G)|$; previously, only $\tilde{O}(n^3)$ was known, except for restricted cases like sparse graphs. Consequently, we obtain the first alg…
In the Single Source Replacement Paths (SSRP) problem we are given a graph $G = (V, E)$, and a shortest paths tree $\widehat{K}$ rooted at a node $s$, and the goal is to output for every node $t \in V$ and for every edge $e$ in $\widehat{K}$ the length of the shortest path from $s$ to $t$ avoiding $e$. We present an $\tilde{O}(m\sqrt{n} + n^2)$ time randomized combinatorial algorithm for unweighted directed graphs. Previously such a bound was known in the directed case only for the seemingly easier problem of replacement path where both the source and the target nodes are fixed. Our new upper …
The family of graphs that can be constructed from isolated vertices by disjoint union and graph join operations are called cographs. These graphs can be represented in a tree-like representation termed parse tree or cotree. In this paper, we study some popular combinatorial problems restricted to cographs. We first present a structural characterization of minimal vertex separators in cographs. Further, we show that listing all minimal vertex separators and the complexity of some constrained vertex separators are polynomial-time solvable in cographs. We propose polynomial-time algorithms for co…
An elastic-degenerate (ED) string is a sequence of $n$ sets of strings of total length $N$, which was recently proposed to model a set of similar sequences. The ED string matching (EDSM) problem is to find all occurrences of a pattern of length $m$ in an ED text. An $O(nm^{1.5}\sqrt{\log m}+N)$-time algorithm for EDSM is known [Aoyama et al., CPM 2018]. The standard assumption in the prior work on this question is that $N$ is substantially larger than both $n$ and $m$, and thus we would like to have a linear dependency on the former. Under this assumption, the natural open problem is whether w…
Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the ra…
