6,656 research outputs found
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
Approximating minimum-power edge-multicovers
Given a graph with edge costs, the {\em power} of a node is themaximum cost
of an edge incident to it, and the power of a graph is the sum of the powers of
its nodes. Motivated by applications in wireless networks, we consider the
following fundamental problem in wireless network design. Given a graph
with edge costs and degree bounds , the {\sf
Minimum-Power Edge-Multi-Cover} ({\sf MPEMC}) problem is to find a
minimum-power subgraph of such that the degree of every node in
is at least . We give two approximation algorithms for {\sf MPEMC}, with
ratios and , where is the maximum
degree bound. This improves the previous ratios and , and
implies ratios for the {\sf Minimum-Power -Outconnected
Subgraph} and for the {\sf Minimum-Power
-Connected Subgraph} problems; the latter is the currently best known ratio
for the min-cost version of the problem
Approximating minimum power covers of intersecting families and directed edge-connectivity problems
AbstractGiven a (directed) graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Let G=(V,E) be a graph with edge costs {c(e):e∈E} and let k be an integer. We consider problems that seek to find a min-power spanning subgraph G of G that satisfies a prescribed edge-connectivity property. In the Min-Powerk-Edge-Outconnected Subgraph problem we are given a root r∈V, and require that G contains k pairwise edge-disjoint rv-paths for all v∈V−r. In the Min-Powerk-Edge-Connected Subgraph problem G is required to be k-edge-connected. For k=1, these problems are at least as hard as the Set-Cover problem and thus have an Ω(ln|V|) approximation threshold. For k=Ω(nε), they are unlikely to admit a polylogarithmic approximation ratio [15]. We give approximation algorithms with ratio O(kln|V|). Our algorithms are based on a more general O(ln|V|)-approximation algorithm for the problem of finding a min-power directed edge-cover of an intersecting set-family; a set-family F is intersecting if X∩Y,X∪Y∈F for any intersecting X,Y∈F, and an edge set I covers F if for every X∈F there is an edge in I entering X
Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs
In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ? 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem
On rooted -connectivity problems in quasi-bipartite digraphs
We consider the directed Rooted Subset -Edge-Connectivity problem: given a
set of terminals in a digraph with edge costs and
an integer , find a min-cost subgraph of that contains edge disjoint
-paths for all . The case when every edge of positive cost has
head in admits a polynomial time algorithm due to Frank, and the case when
all positive cost edges are incident to is equivalent to the -Multicover
problem. Recently, [Chan et al. APPROX20] obtained ratio for
quasi-bipartite instances, when every edge in has an end in . We give
a simple proof for the same ratio for a more general problem of covering an
arbitrary -intersecting supermodular set function by a minimum cost edge
set, and for the case when only every positive cost edge has an end in
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