Approximating Node Connectivity Problems via Set Covers (original) (raw)
Related papers
An Improved Approximation Algorithm for the Minimum Cost Subset k-Connected Subgraph Problem
Algorithmica, 2014
The minimum cost subset k-connected subgraph problem is a cornerstone problem in the area of network design with vertex connectivity requirements. In this problem, we are given a graph G = (V, E) with costs on edges and a set of terminals T. The goal is to find a minimum cost subgraph such that every pair of terminals are connected by k openly (vertex) disjoint paths. In this paper, we present an approximation algorithm for the subset k-connected subgraph problem which improves on the previous best approximation guarantee of O(k 2 log k) by Nutov (FOCS 2009). Our approximation guarantee, α(|T |), depends upon the number of terminals: α(|T |) = O(k log 2 k) if 2k ≤ |T | < k 2 O(k log k) if |T | ≥ k 2 So, when the number of terminals is large enough, the approximation guarantee improves significantly. Moreover, we show that, given an approximation algorithm for |T | = k, we can obtain almost the same approximation guarantee for any instances with |T | > k. This suggests that the hardest instances of the problem are when |T | ≈ k.
Approximating minimum cost connectivity problems
2000
(GSN) problem in which we seek to find a low cost subgraph (where the cost of a subgraph is the sum of the costs of its edges) that satisfies prescribed connectivity requirements. These problems include the following well known problems: min-cost k-flow, min-cost spanning tree, traveling salesman, directed/undirected Steiner Tree, Steiner forest, k-edge/node-connected spanning subgraph, and others.
Lecture Notes in Computer Science, 1997
The problem of finding a minimum weight k-vertex connected spanning subgraph is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusion-minimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive polynomial time approximation algorithms for the cases k = 3, 4, 5. * Up to 1990, E. A. Dinic, Moscow.
A 2-Approximation Algorithm for Finding an Optimum 3-Vertex-Connected Spanning Subgraph
Journal of Algorithms, 1999
The problem of finding a minimum weight k-vertex connected spanning subgraph in a graph G = (V, E) is considered. For k ≥ 2, this problem is known to be NP-hard. Combining properties of inclusionminimal k-vertex connected graphs and of k-out-connected graphs (i.e., graphs which contain a vertex from which there exist k internally vertex-disjoint paths to every other vertex), we derive an auxiliary polynomial time algorithm for finding a ( k 2 + 1)-connected subgraph with a weight at most twice the optimum to the original problem. In particular, we obtain a 2-approximation algorithm for the case k = 3 of our problem. This improves the best previously known approximation ratio 3. The complexity of the algorithm is O(|V | 3 |E|) = O(|V | 5 ). * Up to 1990,
Tree-Approximations for the Weighted Cost-Distance Problem
Lecture Notes in Computer Science, 2001
We generalize the Cost-Distance problem: Given a set of sites in-dimensional Euclidean space and a weighting over pairs of sites, construct a network that minimizes the cost (i.e. weight) of the network and the weighted distances between all pairs of sites. It turns out that the optimal solution can contain Steiner points as well as cycles. Furthermore, there are instances where crossings optimize the network. We then investigate how trees can approximate the weighted Cost-Distance problem. We show that for any given set of sites and a non-negative weighting of pairs, provided the sum of the weights is polynomial, one can construct in polynomial time a tree that approximates the optimal network within a factor of. Finally, we show that better approximation rates are not possible for trees. We prove this by giving a counterexample. Thus, we show that for this instance that every tree solution differs from the optimal network by a factor .
Approximating some network design problems with node costs
Theoretical Computer Science, 2011
We study several multi-criteria undirected network design problems with node costs and lengths. All these problems are related to the Multicommodity Buy at Bulk (MBB) problem in which we are given a graph G = (V , E), demands {d st : s, t ∈ V }, and a family {c v : v ∈ V } of subadditive cost functions. For every s, t ∈ V we seek to send d st flow units from s to t, so
Approximation algorithm for the Multicovering Problem
2021
Let mathcalH=(V,mathcalE)\mathcal{H}=(V,\mathcal{E})mathcalH=(V,mathcalE) be a hypergraph with maximum edge size ell\ellell and maximum degree Delta\DeltaDelta. For given numbers bvinmathbbNgeq2b_v\in \mathbb{N}_{\geq 2}bvinmathbbNgeq2, vinVv\in VvinV, a set multicover in mathcalH\mathcal{H}mathcalH is a set of edges CsubseteqmathcalEC \subseteq \mathcal{E}CsubseteqmathcalE such that every vertex vvv in VVV belongs to at least bvb_vbv edges in CCC. Set Multicover is the problem of finding a minimum-cardinality set multicover. Peleg, Schechtman and Wool conjectured that for any fixed Delta\DeltaDelta and b:=minvinVbvb:=\min_{v\in V}b_{v}b:=minvinVbv, the problem of \sbmultcov is not approximable within a ratio less than delta:=Delta−b+1\delta:=\Delta-b+1delta:=Delta−b+1, unless mathcalP=mathcalNP\mathcal{P} =\mathcal{NP}mathcalP=mathcalNP. Hence it's a challenge to explore for which classes of hypergraph the conjecture doesn't hold. We present a polynomial time algorithm for the Set Multicover problem which combines a deterministic threshold algorithm with conditioned randomized rounding steps. Our algorithm yields an approximation ratio of $ \max\left\{ \frac{148}{149}\delta, \left(1- \frac{ (b-1)e...
A 3-Approximation Algorithm for Finding Optimum 4,5-Vertex-Connected Spanning Subgraphs
Journal of Algorithms, 1999
The problem of nding a minimum weight k-vertex connected spanning subgraph in a graph G = (V; E) is considered. For k 2, this problem is known to be NP-hard. Based on the paper of Auletta, Dinitz, Nutov and Parente in this issue, we derive a 3-approximation algorithm for k 2 f4; 5g. This improves the best previously known approximation ratios 4 1 6 and 4 17 30 , respectively. The complexity of the suggested algorithm is O(jV j 5 ) for the deterministic and O(jV j 4 log jV j)
A 4/3-Approximation Algorithm for the Minimum 2-Edge Connected Subgraph Problem
ACM Transactions on Algorithms, 2019
We present a factor 4/3 approximation algorithm for the problem of finding a minimum 2-edge connected spanning subgraph of a given undirected multigraph. The algorithm is based upon a reduction to a restricted class of graphs. In these graphs, the approximation algorithm constructs a 2-edge connected spanning subgraph by modifying the smallest 2-edge cover.