Related papers: An efficient polynomial-time approximation scheme …
We show how to compute for $n$-vertex planar graphs in $O(n^{11/6}{\rm polylog}(n))$ expected time the diameter and the sum of the pairwise distances. The algorithms work for directed graphs with real weights and no negative cycles. In…
We develop a time-optimal $O(mn^2)$-time algorithm to construct the subtree prune-regraft (SPR) graph on a collection of m phylogenetic trees with n leaves. This improves on the previous bound of $O(mn^3)$. Such graphs are used to better…
Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is…
We consider the problem of embedding the Steiner points of a Steiner tree with given topology into the rectilinear plane. Thereby, the length of the path between a distinguished terminal and each other terminal must not exceed given length…
In this paper we give a linear time algorithm for computing the number of spanninig trees in double nested graphs.
We present an $O(n^2)$-time algorithm to test whether an $n$-vertex directed partial $2$-tree is upward planar. This result improves upon the previously best known algorithm, which runs in $O(n^4)$ time.
The Steiner Forest problem, also known as the Generalized Steiner Tree problem, is a fundamental optimization problem on edge-weighted graphs where, given a set of vertex pairs, the goal is to select a minimum-cost subgraph such that each…
The \emph{Product Structure Theorem} for planar graphs (Dujmovi\'c et al.\ \emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple…
We develop a new algorithmic technique that allows to transfer some constant time approximation algorithms for general graphs into random order streaming algorithms. We illustrate our technique by proving that in random order streams with…
Given a graph G, the {\em maximum internal spanning tree problem} (MIST for short) asks for computing a spanning tree T of G such that the number of internal vertices in T is maximized. MIST has possible applications in the design of…
The Euclidean Steiner Minimal Tree problem takes as input a set $\mathcal P$ of points in the Euclidean plane and finds the minimum length network interconnecting all the points of $\mathcal P$. In this paper, in continuation to the works…
We consider the problem of constructing optimal decision trees: given a collection of tests which can disambiguate between a set of $m$ possible diseases, each test having a cost, and the a-priori likelihood of the patient having any…
In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise…
Search trees on trees (STTs) generalize the fundamental binary search tree (BST) data structure: in STTs the underlying search space is an arbitrary tree, whereas in BSTs it is a path. An optimal BST of size $n$ can be computed for a given…
We study two problems that seek a subtree $T$ of a graph $G=(V,E)$ such that $T$ satisfies a certain property and has minimal maximum degree. - In the Min-Degree Group Steiner Tree problem we are given a collection ${\cal S}$ of groups…
We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a ``semi-duality'' between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling…
The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that $|E(G)|\ge \alpha |V(G)|^2$ for some fixed…
We study the problem of constructing phylogenetic trees for a given set of species. The problem is formulated as that of finding a minimum Steiner tree on $n$ points over the Boolean hypercube of dimension $d$. It is known that an optimal…
We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node $t$ in a given tree $T$. Upon querying a node $v$ of the tree, the strategy receives as a reply an…
We give a $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time for an undirected planar graph $G$ with $n$ vertices and non-negative edge lengths. For $\epsilon>0$ and any two vertices $u$ and $v$ in $G$, our oracle gives a…