38,992 research outputs found
Inverse Problems Related to the Wiener and Steiner-Wiener Indices
In a graph, the generalized distance between multiple vertices is the minimum number of edges in a connected subgraph that contains these vertices. When we consider such distances between all subsets of vertices and take the sum, it is called the Steiner -Wiener index and has important applications in Chemical Graph Theory. In this thesis we consider the inverse problems related to the Steiner Wiener index, i.e. for what positive integers is there a graph with Steiner Wiener index of that value
A Characterization of Undirected Graphs Admitting Optimal Cost Shares
In a seminal paper, Chen, Roughgarden and Valiant studied cost sharing
protocols for network design with the objective to implement a low-cost Steiner
forest as a Nash equilibrium of an induced cost-sharing game. One of the most
intriguing open problems to date is to understand the power of budget-balanced
and separable cost sharing protocols in order to induce low-cost Steiner
forests. In this work, we focus on undirected networks and analyze topological
properties of the underlying graph so that an optimal Steiner forest can be
implemented as a Nash equilibrium (by some separable cost sharing protocol)
independent of the edge costs. We term a graph efficient if the above stated
property holds. As our main result, we give a complete characterization of
efficient undirected graphs for two-player network design games: an undirected
graph is efficient if and only if it does not contain (at least) one out of few
forbidden subgraphs. Our characterization implies that several graph classes
are efficient: generalized series-parallel graphs, fan and wheel graphs and
graphs with small cycles.Comment: 60 pages, 69 figures, OR 2017 Berlin, WINE 2017 Bangalor
Minimum Latency Submodular Cover
We study the Minimum Latency Submodular Cover problem (MLSC), which consists
of a metric with source and monotone submodular functions
. The goal is to find a path
originating at that minimizes the total cover time of all functions. This
generalizes well-studied problems, such as Submodular Ranking [AzarG11] and
Group Steiner Tree [GKR00]. We give a polynomial time O(\log \frac{1}{\eps}
\cdot \log^{2+\delta} |V|)-approximation algorithm for MLSC, where
is the smallest non-zero marginal increase of any
and is any constant.
We also consider the Latency Covering Steiner Tree problem (LCST), which is
the special case of \mlsc where the s are multi-coverage functions. This
is a common generalization of the Latency Group Steiner Tree
[GuptaNR10a,ChakrabartyS11] and Generalized Min-sum Set Cover [AzarGY09,
BansalGK10] problems. We obtain an -approximation algorithm for
LCST.
Finally we study a natural stochastic extension of the Submodular Ranking
problem, and obtain an adaptive algorithm with an O(\log 1/ \eps)
approximation ratio, which is best possible. This result also generalizes some
previously studied stochastic optimization problems, such as Stochastic Set
Cover [GoemansV06] and Shared Filter Evaluation [MunagalaSW07, LiuPRY08].Comment: 23 pages, 1 figur
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets
Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set- valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference
for the evaluation of the numerator in the difference quotient.
We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as an embedding of convex sets in this space using Steiner points. In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts
2-Approximation for Prize-Collecting Steiner Forest
Approximation algorithms for the prize-collecting Steiner forest problem
(PCSF) have been a subject of research for over three decades, starting with
the seminal works of Agrawal, Klein, and Ravi and Goemans and Williamson on
Steiner forest and prize-collecting problems. In this paper, we propose and
analyze a natural deterministic algorithm for PCSF that achieves a
-approximate solution in polynomial time. This represents a significant
improvement compared to the previously best known algorithm with a
-approximation factor developed by Hajiaghayi and Jain in 2006.
Furthermore, K{\"{o}}nemann, Olver, Pashkovich, Ravi, Swamy, and Vygen have
established an integrality gap of at least for the natural LP relaxation
for PCSF. However, we surpass this gap through the utilization of a
combinatorial algorithm and a novel analysis technique. Since is the best
known approximation guarantee for Steiner forest problem, which is a special
case of PCSF, our result matches this factor and closes the gap between the
Steiner forest problem and its generalized version, PCSF
Reductions for the Stable Set Problem
One approach to finding a maximum stable set (MSS) in a graph is to try to reduce the size of the problem by transforming the problem into an equivalent problem on a smaller graph. This paper introduces several new reductions for the MSS problem, extends several well-known reductions to the maximum weight stable set (MWSS) problem, demonstrates how reductions for the generalized stable set problem can be used in conjunction with probing to produce powerful new reductions for both the MSS and MWSS problems, and shows how hypergraphs can be used to expand the capabilities of clique projections. The effectiveness of these new reduction techniques are illustrated on the DIMACS benchmark graphs, planar graphs, and a set of challenging MSS problems arising from Steiner Triple Systems
Generalized Steiner selections applied to standard problems of set-valued numerical analysis
Generalized Steiner points and the corresponding selections for set-valued maps share interesting commutation properties with set operations which make them suitable for the set-valued numerical problems presented here. This short overview will present first applications of these selections to standard problems in this area, namely representation of convex, compact sets in R n and set operations, set-valued integration and interpolation as well as the calculation of attainable sets of linear differential inclusions. Hereby, the convergence results are given uniformly for a dense countable representation of generalized Steiner points/selections. To achieve this aim, stronger conditions on the set-valued map F have to be taken into account, e.g. the Lipschitz condition on F has to be satisfied for the Demyanov distance instead of the Hausdorff distance. To establish an overview on several applications, not the strongest available results are formulated in this article
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