Related papers: Approximation Algorithms for $\ell_p$-Shortest Pat…

We obtain hardness of approximation results for the $\ell_p$-Shortest Path problem, a variant of the classic Shortest Path problem with vector costs. For every integer $p \in [2,\infty)$, we show a hardness of $\Omega(p(\log n / \log^2\log…

Consider the following problem: given a graph with edge costs and a subset Q of vertices, find a minimum-cost subgraph in which there are two edge-disjoint paths connecting every pair of vertices in Q. The problem is a failure-resilient…

We introduce and study a novel problem of computing a shortest path tree with a minimum number of non-terminals. It can be viewed as an (unweighted) Steiner Shortest Path Tree (SSPT) that spans a given set of terminal vertices by shortest…

Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…

For a given graph $G=(V,\, E)$ with a terminal set $S$ and a selected root $r\in S$, a positive integer cost and a delay on every edge and a delay constraint $D\in Z^{+}$, the shallow-light Steiner tree (\emph{SLST}) problem is to compute a…

In the Priority Steiner Tree (PST) problem, we are given an undirected graph $G=(V,E)$ with a source $s \in V$ and terminals $T \subseteq V \setminus \{s\}$, where each terminal $v \in T$ requires a nonnegative priority $P(v)$. The goal is…

In the Steiner Tree Augmentation Problem (STAP), we are given a graph $G = (V,E)$, a set of terminals $R \subseteq V$, and a Steiner tree $T$ spanning $R$. The edges $L := E \setminus E(T)$ are called links and have non-negative costs. The…

The Longest Path Problem is a question of finding the maximum length between pairs of vertices of a graph. In the general case, the problem is NP-complete. However, there is a small collection of graph classes for which there exists an…

For a metric graph $G=(V,E)$ and $R\subset V$, the internal Steiner minimum tree problem asks for a minimum weight Steiner tree spanning $R$ such that every vertex in $R$ is not a leaf. This note shows a simple polynomial-time…

The paper revisits the robust $s$-$t$ path problem, one of the most fundamental problems in robust optimization. In the problem, we are given a directed graph with $n$ vertices and $k$ distinct cost functions (scenarios) defined over edges,…

In the classical Steiner tree problem, given an undirected, connected graph $G=(V,E)$ with non-negative edge costs and a set of \emph{terminals} $T\subseteq V$, the objective is to find a minimum-cost tree $E' \subseteq E$ that spans the…

In the \emph{budgeted rooted node-weighted Steiner tree} problem, we are given a graph $G$ with $n$ nodes, a predefined node $r$, two weights associated to each node modelling costs and prizes. The aim is to find a tree in $G$ rooted at $r$…

This paper discusses the shortest path problem in a general directed graph with $n$ nodes and $K$ cost scenarios (objectives). In order to choose a solution, the min-max criterion is applied. The min-max version of the problem is hard to…

In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G=(V, E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k>0; the goal is to find a minimum-cost…

The Restricted Shortest Path (RSP) problem, also known as the Delay-Constrained Least-Cost (DCLC) problem, is an NP-hard bicriteria optimization problem on graphs with $n$ vertices and $m$ edges. In a graph where each edge is assigned a…

The Steiner tree problem is one of the classic and most fundamental $\mathcal{NP}$-hard problems: given an arbitrary weighted graph, seek a minimum-cost tree spanning a given subset of the vertices (terminals). Byrka \emph{et al}. proposed…

We consider the problem of designing sublinear time algorithms for estimating the cost of a minimum metric traveling salesman (TSP) tour. Specifically, given access to a $n \times n$ distance matrix $D$ that specifies pairwise distances…

We prove new results for approximating the graphic TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees. For the graphic TSP itself, we improve the approximation ratio to 7/5. For a…

In this paper we reassess the parameterized complexity and approximability of the well-studied Steiner Forest problem in several graph classes of bounded width. The problem takes an edge-weighted graph and pairs of vertices as input, and…

Let G=(V,E) be a connected graph, where V and E represent, respectively, the node-set and the edge-set. Besides, let Q \subseteq V be a set of terminal nodes, and r \in Q be the root node of the graph. Given a weight c_{ij} \in \mathbb{N}…