29 research outputs found
Covering skew-supermodular functions by hypergraphs of minimum total size
The paper presents results related to a theorem of Szigeti on covering symmetric skew-supermodular set functions by hypergraphs. We prove the following generalization using a variation of Schrijver´s supermodular colouring theorem: if p(1) and p(2) are skew-supermodular functions with the same maximum value, then it is possible to find in polynomial time a hypergraph of minimum total size that covers both p(1) and p(2). We also give some applications concerning the connectivity augmentation of hypergraphs. (C) 2009 Elsevier B.V. All rights reserved
Covering symmetric skew-supermodular functions with hyperedges
In this paper we give results related to a theorem of Szigeti that concerns
the covering of symmetric skew-supermodular set functions with hyperedges of
minimum total size. In particular, we show the following generalization using a
variation of Schrijver’s supermodular colouring theorem: if p1 and p2 are skewsupermodular
functions whose maximum value is the same, then it is possible to
find in polynomial time a hypergraph of minimum total size that covers both of
them. Note that without the assumption on the maximum values this problem
is NP-hard. The result has applications concerning the local edge-connectivity
augmentation problem of hypergraphs and the global edge-connectivity augmentation
problem of mixed hypergraphs. We also present some results on the case
when the hypergraph must be obtained by merging given hyperedges
A Note on Iterated Rounding for the Survivable Network Design Problem
In this note we consider the survivable network design problem (SNDP) in undirected graphs. We make two contributions. The first is a new counting argument in the iterated rounding based 2-approximation for edge-connectivity SNDP (EC-SNDP) originally due to Jain. The second contribution is to make some connections between hypergraphic version of SNDP (Hypergraph-SNDP) introduced by Zhao, Nagamochi and Ibaraki, and edge and node-weighted versions of EC-SNDP and element-connectivity SNDP (Elem-SNDP). One useful consequence is a 2-approximation for Elem-SNDP that avoids the use of set-pair based relaxation and analysis
Hypergraph Connectivity Augmentation in Strongly Polynomial Time
We consider hypergraph network design problems where the goal is to construct
a hypergraph that satisfies certain connectivity requirements. For graph
network design problems where the goal is to construct a graph that satisfies
certain connectivity requirements, the number of edges in every feasible
solution is at most quadratic in the number of vertices. In contrast, for
hypergraph network design problems, we might have feasible solutions in which
the number of hyperedges is exponential in the number of vertices. This
presents an additional technical challenge in hypergraph network design
problems compared to graph network design problems: in order to solve the
problem in polynomial time, we first need to show that there exists a feasible
solution in which the number of hyperedges is polynomial in the input size.
The central theme of this work is to show that certain hypergraph network
design problems admit solutions in which the number of hyperedges is polynomial
in the number of vertices and moreover, can be solved in strongly polynomial
time. Our work improves on the previous fastest pseudo-polynomial run-time for
these problems. In addition, we develop strongly polynomial time algorithms
that return near-uniform hypergraphs as solutions (i.e., every pair of
hyperedges differ in size by at most one). As applications of our results, we
derive the first strongly polynomial time algorithms for (i) degree-specified
hypergraph connectivity augmentation using hyperedges, (ii) degree-specified
hypergraph node-to-area connectivity augmentation using hyperedges, and (iii)
degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.Comment: arXiv admin note: substantial text overlap with arXiv:2307.0855
Splitting-off in Hypergraphs
The splitting-off operation in undirected graphs is a fundamental reduction
operation that detaches all edges incident to a given vertex and adds new edges
between the neighbors of that vertex while preserving their degrees. Lov\'asz
(1974) and Mader (1978) showed the existence of this operation while preserving
global and local connectivities respectively in graphs under certain
conditions. These results have far-reaching applications in graph algorithms
literature. In this work, we introduce a splitting-off operation in
hypergraphs. We show that there exists a local connectivity preserving complete
splitting-off in hypergraphs and give a strongly polynomial-time algorithm to
compute it in weighted hypergraphs. We illustrate the usefulness of our
splitting-off operation in hypergraphs by showing two applications:
(1) we give a constructive characterization of -hyperedge-connected
hypergraphs and
(2) we give an alternate proof of an approximate min-max relation for max
Steiner rooted-connected orientation of graphs and hypergraphs (due to Kir\'aly
and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the
approximate min-max relation for graphs circumvents the Nash-Williams' strong
orientation theorem and uses tools developed for hypergraphs
Approximating minimum cost connectivity problems
We survey approximation algorithms of connectivity problems.
The survey presented describing various techniques. In the talk the following techniques and results are presented.
1)Outconnectivity: Its well known that there exists a polynomial time algorithm to solve the problems of finding an edge k-outconnected from r subgraph [EDMONDS] and a vertex k-outconnectivity subgraph from r [Frank-Tardos] .
We show how to use this to obtain a ratio 2 approximation for the min cost edge k-connectivity
problem.
2)The critical cycle theorem of Mader: We state a fundamental theorem of Mader and use it to provide a 1+(k-1)/n ratio approximation for the min cost vertex k-connected subgraph, in the metric case.
We also show results for the min power vertex k-connected problem using this lemma.
We show that the min power is equivalent to the min-cost case with respect to approximation.
3)Laminarity and uncrossing: We use the well known laminarity of a BFS solution and show a simple new proof due to Ravi et al for Jain\u27s 2 approximation for Steiner network