DeepAI

# A General Framework for Multi-level Subsetwise Graph Sparsifiers

Given an undirected weighted graph $G(V,E)$, a subsetwise sparsifier over a terminal set $T\subset V$ is a subgraph $G'$ having a certain structure which connects the terminals. Examples are Steiner trees (minimal-weight trees spanning $T$) and subsetwise spanners (subgraphs $G'(V',E')$ such that for given $\alpha,\beta\geq1$, $d_{G'}(u,v)\leq \alpha d_{G}(u,v)+\beta$ for $u,v\in T$). Multi-level subsetwise sparsifiers are generalizations in which terminal vertices require different levels or grades of service. This paper gives a flexible approximation algorithm for several multi-level subsetwise sparsifier problems, including multi-level graph spanners, Steiner trees, and $k$--connected subgraphs. The algorithm relies on computing an approximation to the single level instance of the problem% and an efficient approach to obtain a multi-level solution. For the subsetwise spanner problem, there are few existing approximation algorithms for even a single level; consequently we give a new polynomial time algorithm for computing a subsetwise spanner for a single level. Specifically, we show that for $k\in\N$, $\eps>0$, and $T\subset V$, there is a subsetwise $(2k-1)(1+\eps)$--spanner with total weight $O(|T|^\frac1kW(\ST(G,T)))$, where $W(\ST(G,T))$ is the weight of the Steiner tree of $G$ over the subset $T$. This is the first algorithm and corresponding weight guarantee for a multiplicative subsetwise spanner for nonplanar graphs. We also generalize a result of Klein to give a constant approximation to the multi-level subsetwise spanner problem for planar graphs. Additionally, we give a polynomial-size ILP for optimally computing pairwise spanners of arbitrary distortion (beyond linear distortion functions), and provide experiments to illustrate the performance of our algorithms.

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04/01/2019

### Approximation algorithms and an integer program for multi-level graph spanners

Given a weighted graph G(V,E) and t > 1, a subgraph H is a t--spanner of...
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### Computing Vertex-Weighted Multi-Level Steiner Trees

In the classical vertex-weighted Steiner tree problem (VST), one is give...
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### Kruskal-based approximation algorithm for the multi-level Steiner tree problem

We study the multi-level Steiner tree problem: a generalization of the S...
01/23/2023

### A New Approximation Algorithm for Minimum-Weight (1,m)–Connected Dominating Set

Consider a graph with nonnegative node weight. A vertex subset is called...
02/11/2021