474 research outputs found
Approximating Minimum-Cost k-Node Connected Subgraphs via Independence-Free Graphs
We present a 6-approximation algorithm for the minimum-cost -node
connected spanning subgraph problem, assuming that the number of nodes is at
least . We apply a combinatorial preprocessing, based on the
Frank-Tardos algorithm for -outconnectivity, to transform any input into an
instance such that the iterative rounding method gives a 2-approximation
guarantee. This is the first constant-factor approximation algorithm even in
the asymptotic setting of the problem, that is, the restriction to instances
where the number of nodes is lower bounded by a function of .Comment: 20 pages, 1 figure, 28 reference
Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs
Let be a strongly connected directed graph. We consider the following
three problems, where we wish to compute the smallest strongly connected
spanning subgraph of that maintains respectively: the -edge-connected
blocks of (\textsf{2EC-B}); the -edge-connected components of
(\textsf{2EC-C}); both the -edge-connected blocks and the -edge-connected
components of (\textsf{2EC-B-C}). All three problems are NP-hard, and thus
we are interested in efficient approximation algorithms. For \textsf{2EC-C} we
can obtain a -approximation by combining previously known results. For
\textsf{2EC-B} and \textsf{2EC-B-C}, we present new -approximation
algorithms that run in linear time. We also propose various heuristics to
improve the size of the computed subgraphs in practice, and conduct a thorough
experimental study to assess their merits in practical scenarios
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
Robust Assignments via Ear Decompositions and Randomized Rounding
Many real-life planning problems require making a priori decisions before all
parameters of the problem have been revealed. An important special case of such
problem arises in scheduling problems, where a set of tasks needs to be
assigned to the available set of machines or personnel (resources), in a way
that all tasks have assigned resources, and no two tasks share the same
resource. In its nominal form, the resulting computational problem becomes the
\emph{assignment problem} on general bipartite graphs.
This paper deals with a robust variant of the assignment problem modeling
situations where certain edges in the corresponding graph are \emph{vulnerable}
and may become unavailable after a solution has been chosen. The goal is to
choose a minimum-cost collection of edges such that if any vulnerable edge
becomes unavailable, the remaining part of the solution contains an assignment
of all tasks.
We present approximation results and hardness proofs for this type of
problems, and establish several connections to well-known concepts from
matching theory, robust optimization and LP-based techniques.Comment: Full version of ICALP 2016 pape
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
Approximation Algorithms for Network Design in Non-Uniform Fault Models
Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model [Adjiashvili, 2013; Adjiashvili et al., 2020; Adjiashvili et al., 2022; Boyd et al., 2023], in which the edge set is partitioned into safe and unsafe edges. Given parameters p,q ? 1, the goal is to find a cheap subgraph that remains p-connected even after the failure of q unsafe edges. We also discuss the bulk-robust model [Adjiashvili et al., 2015; Adjiashvili, 2015] and the relative survivable network design model [Dinitz et al., 2022]. While SNDP admits a 2-approximation [K. Jain, 2001], the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions.
Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when p and q are fixed, we consider two special cases. For the s-t case we obtain an approximation ratio that depends only on p,q whenever p+q > pq/2 which includes (p,2) and (2,q) for all p,q ? 1. For the global connectivity case we obtain an O(q) approximation for (2,q), and an O(p) approximation for (p,2) and (p,3) for any p ? 1, and for (p,4) when p is even. These are based on an augmentation framework and decomposing the families of cuts that need to be covered into a small number of uncrossable families.
Our second result is a poly-logarithmic approximation for a generalization of the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. We utilize a recent framework due to Chen et al. [Chen et al., 2022] that was designed for handling group connectivity
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