Related papers: A $1.75$ LP approximation for the Tree Augmentatio…
In Part I, we study a special case of the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. In the special case, we forbid so-called stems; these are a particular type of subtree configuration. For…
In Connectivity Augmentation problems we are given a graph $H=(V,E_H)$ and an edge set $E$ on $V$, and seek a min-size edge set $J \subseteq E$ such that $H \cup J$ has larger edge/node connectivity than $H$. In the Edge-Connectivity…
We describe a $\frac{3}{2}$-approximation algorithm for the Forest Augmentation Problem (\textsf{FAP}), which is a special case of the Weighted 2-Edge-Connected Spanning Subgraph Problem (\textsf{Weighted 2-ECSS}). This significantly…
Increasing the connectivity of a graph is a pivotal challenge in robust network design. The weighted connectivity augmentation problem is a common version of the problem that takes link costs into consideration. The problem is then to find…
We study the Weighted Tree Augmentation Problem for general link costs. We show that the integrality gap of the ODD-LP relaxation for the (weighted) Tree Augmentation Problem for a $k$-level tree instance is at most $2 - \frac{1}{2^{k-1}}$.…
We study the unweighted Tree Augmentation Problem (TAP) via the Lasserre (Sum of Squares) system. We prove an approximation guarantee of ($1.8+\epsilon$) relative to an SDP relaxation, which matches the combinatorial approximation guarantee…
We consider connectivity augmentation problems in the Steiner setting, where the goal is to augment the edge-connectivity between a specified subset of terminal nodes. In the Steiner Augmentation of a Graph problem ($k$-SAG), we are given a…
Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation…
Graph Balancing is the problem of orienting the edges of a weighted multigraph so as to minimize the maximum weighted in-degree. Since the introduction of the problem the best algorithm known achieves an approximation ratio of $1.75$ and it…
In the laminar-constrained spanning tree problem, the goal is to find a minimum-cost spanning tree which respects upper bounds on the number of times each cut in a given laminar family is crossed. This generalizes the well-studied…
The capacitated tree cover problem with edge loads is a variant of the tree cover problem, where we are given facility opening costs, edge costs and loads, as well as vertex loads. We try to find a tree cover of minimum cost such that the…
As a variant of the routing and wavelength assignment problem (RWAP), the RWAP with partial path protection (RWAP-PPP) designs a reliable optical-fiber network for telecommunications. It assigns paths and wavelengths to meet communication…
We study budget constrained network upgradeable problems. We are given an undirected edge weighted graph $G=(V,E)$ where the weight an edge $e \in E$ can be upgraded for a cost $c(e)$. Given a budget $B$ for improvement, the goal is to find…
We present an approximation algorithm for Weighted Tree Augmentation with approximation factor $1+\ln 2 + \varepsilon < 1.7$. This is the first algorithm beating the longstanding factor of $2$, which can be achieved through many standard…
This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large…
The Virtual Network Embedding Problem (VNEP) considers the efficient allocation of resources distributed in a substrate network to a set of request networks. Many existing works discuss either heuristics or exact algorithms, resulting in 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 present a $\frac53$-approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning…
Given a vertex-weighted connected graph $G = (V, E)$, the maximum weight internal spanning tree (MwIST for short) problem asks for a spanning tree $T$ of $G$ such that the total weight of the internal vertices in $T$ is maximized. The…
Consider a random graph model where each possible edge $e$ is present independently with some probability $p_e$. Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we…