arXiv subjects

Svein Høgemo

Publications and source records attributed to Svein Høgemo.

7 recordsLinked to original sources

arXiv2024

Tight Approximation Bounds on a Simple Algorithm for Minimum Average Search Time in Trees

The graph invariant EPT-sum has cropped up in several unrelated fields in later years: As an objective function for hierarchical clustering, as a more fine-grained version of the classical edge ranking problem, and, specifically when the input is a vertex-weighted tree, as a measure of average/expected search length in a partially ordered set. The EPT-sum of a graph $G$ is defined as the minimum sum of the depth of every leaf in an edge partition tree (EPT), a rooted tree where leaves correspond to vertices in $G$ and internal nodes correspond to edges in $G$. A simple algorithm that approximates EPT-sum on trees is given by recursively choosing the most balanced edge in the input tree $G$ to build an EPT of $G$. Due to its fast runtime, this balanced cut algorithm can be used in practice, and has earlier been analysed to give a 1.62-approximation on trees. In this paper, we show that the balanced cut algorithm gives a 1.5-approximation of EPT-sum on trees, which amounts to a tight analysis and answers a question posed by Cicalese et al. in 2014.

cs.DS↗

arXiv2024

Lower Bounds for Leaf Rank of Leaf Powers

Leaf powers and $k$-leaf powers have been studied for over 20 years, but there are still several aspects of this graph class that are poorly understood. One such aspect is the leaf rank of leaf powers, i.e. the smallest number $k$ such that a graph $G$ is a $k$-leaf power. Computing the leaf rank of leaf powers has proved a hard task, and furthermore, results about the asymptotic growth of the leaf rank as a function of the number of vertices in the graph have been few and far between. We present an infinite family of rooted directed path graphs that are leaf powers, and prove that they have leaf rank exponential in the number of vertices (utilizing a type of subtree model first presented by Rautenbach [Some remarks about leaf roots. Discrete mathematics, 2006]). This answers an open question by Brandstädt et al. [Rooted directed path graphs are leaf powers. Discrete mathematics, 2010].

cs.DM↗

arXiv2023

Linear MIM-width of the Square of Trees

Graph parameters measure the amount of structure (or lack thereof) in a graph that makes it amenable to being decomposed in a way that facilitates dynamic programming. Graph decompositions and their associated parameters are important both in practice (as a tool for designing robust algorithms for NP-hard problems) and in theory (relating large classes of problems to the graphs on which they are solvable in polynomial time). Linear MIM-width is a variant of the graph parameter MIM-width, introduced by Vatshelle. MIM-width is a parameter that is constant for many classes of graphs. Most graph classes which have been shown to have constant MIM-width also have constant linear MIM-width. However, computing the (linear) MIM-width of graphs, or showing that it is hard, has proven to be a huge challenge. To date, the only graph class with unbounded linear MIM-width, whose linear MIM-width can be computed in polynomial time, is the trees. In this follow-up, we show that for any tree $T$ with linear MIM-width $k$, the linear MIM-width of its square $T^2$ always lies between $k$ and $2k$, and that these bounds are tight for all $k$.

cs.DM↗

arXiv2022

Recognition of Linear and Star Variants of Leaf Powers is in P

A $k$-leaf power of a tree $T$ is a graph $G$ whose vertices are the leaves of $T$ and whose edges connect pairs of leaves whose distance in $T$ is at most $k$. A graph is a leaf power if it is a $k$-leaf power for some $k$. Over 20 years ago, Nishimura et al. [J. Algorithms, 2002] asked if recognition of leaf powers was in P. Recently, Lafond [SODA 2022] showed an XP algorithm when parameterized by $k$, while leaving the main question open. In this paper, we explore this question from the perspective of two alternative models of leaf powers, showing that both a linear and a star variant of leaf powers can be recognized in polynomial-time.

cs.DS↗

arXiv2021

On Dasgupta's hierarchical clustering objective and its relation to other graph parameters

The minimum height of vertex and edge partition trees are well-studied graph parameters known as, for instance, vertex and edge ranking number. While they are NP-hard to determine in general, linear-time algorithms exist for trees. Motivated by a correspondence with Dasgupta's objective for hierarchical clustering we consider the total rather than maximum depth of vertices as an alternative objective for minimization. For vertex partition trees this leads to a new parameter with a natural interpretation as a measure of robustness against vertex removal. As tools for the study of this family of parameters we show that they have similar recursive expressions and prove a binary tree rotation lemma. The new parameter is related to trivially perfect graph completion and therefore intractable like the other three are known to be. We give polynomial-time algorithms for both total-depth variants on caterpillars and on trees with a bounded number of leaf neighbors. For general trees, we obtain a 2-approximation algorithm.

cs.DS↗

arXiv2020

Hierarchical Clusterings of Unweighted Graphs

We study the complexity of finding an optimal hierarchical clustering of an unweighted similarity graph under the recently introduced Dasgupta objective function. We introduce a proof technique, called the normalization procedure, that takes any such clustering of a graph $G$ and iteratively improves it until a desired target clustering of G is reached. We use this technique to show both a negative and a positive complexity result. Firstly, we show that in general the problem is NP-complete. Secondly, we consider min-well-behaved graphs, which are graphs $H$ having the property that for any $k$ the graph $H(k)$ being the join of $k$ copies of $H$ has an optimal hierarchical clustering that splits each copy of $H$ in the same optimal way. To optimally cluster such a graph $H(k)$ we thus only need to optimally cluster the smaller graph $H$. Co-bipartite graphs are min-well-behaved, but otherwise they seem to be scarce. We use the normalization procedure to show that also the cycle on 6 vertices is min-well-behaved.

cs.CC↗

arXiv2019

Linear MIM-Width of Trees

We provide an $O(n \log n)$ algorithm computing the linear maximum induced matching width of a tree and an optimal layout.

cs.DS↗