8,129 research outputs found
How to Secure Matchings Against Edge Failures
Suppose we are given a bipartite graph that admits a perfect matching and an adversary may delete any edge from the graph with the intention of destroying all perfect matchings. We consider the task of adding a minimum cost edge-set to the graph, such that the adversary never wins. We show that this problem is equivalent to covering a digraph with non-trivial strongly connected components at minimal cost. We provide efficient exact and approximation algorithms for this task. In particular, for the unit-cost problem, we give a log_2 n-factor approximation algorithm and a polynomial-time algorithm for chordal-bipartite graphs. Furthermore, we give a fixed parameter algorithm for the problem parameterized by the treewidth of the input graph. For general non-negative weights we give tight upper and lower approximation bounds relative to the Directed Steiner Forest problem. Additionally we prove a dichotomy theorem characterizing minor-closed graph classes which allow for a polynomial-time algorithm. To obtain our results, we exploit a close relation to the classical Strong Connectivity Augmentation problem as well as directed Steiner problems
Online Directed Spanners and Steiner Forests
We present online algorithms for directed spanners and Steiner forests. These
problems fall under the unifying framework of online covering linear
programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009),
based on primal-dual techniques. Our results include the following:
For the pairwise spanner problem, in which the pairs of vertices to be
spanned arrive online, we present an efficient randomized
-competitive algorithm for graphs with general lengths,
where is the number of vertices. With uniform lengths, we give an efficient
randomized -competitive algorithm, and an
efficient deterministic -competitive algorithm,
where is the number of terminal pairs. These are the first online
algorithms for directed spanners. In the offline setting, the current best
approximation ratio with uniform lengths is ,
due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020).
For the directed Steiner forest problem with uniform costs, in which the
pairs of vertices to be connected arrive online, we present an efficient
randomized -competitive algorithm. The
state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene,
Krishnaswamy, and Panigrahi (SICOMP 2018) and is -competitive. In the offline version, the current best approximation
ratio with uniform costs is , due to Abboud
and Bodwin (SODA 2018).
A small modification of the online covering framework by Buchbinder and Naor
implies a polynomial-time primal-dual approach with separation oracles, which a
priori might perform exponentially many calls. We convert the online spanner
problem and the online Steiner forest problem into online covering problems and
round in a problem-specific fashion
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
A polylogarithmic approximation algorithm for group Steiner tree problem
The group Steiner tree problem is a generalization of the Steiner tree problem where we are given several subsets (groups) of vertices in a weighted graph, and the goal is to find a minimum-weight connected subgraph containing at least one vertex from each group. The problem was introduced by Reich and Widmayer and finds applications in VLSI design. The group Steiner tree problem generalizes the set covering problem, and is therefore at least as hard. We give a randomized -approximation algorithm for the group Steiner tree problem on an -node graph, where is the number of groups.The best previous performance guarantee was (Bateman, Helvig, Robins and Zelikovsky). Noting that the group Steiner problem also models the network design problems with location-theoretic constraints studied by Marathe, Ravi and Sundaram, our results also improve their bicriteria approximation results. Similarly, we improve previous results by Slav{\'\i}k on a tour version, called the errand scheduling problem. We use the result of Bartal on probabilistic approximation of finite metric spaces by tree metrics to reduce the problem to one in a tree metric. To find a solution on a tree, we use a generalization of randomized rounding. Our approximation guarantees improve to in the case of graphs that exclude small minors by using a better alternative to Bartal's result on probabilistic approximations of metrics induced by such graphs (Konjevod, Ravi and Salman) -- this improvement is valid for the group Steiner problem on planar graphs as well as on a set of points in the 2D-Euclidean case
Linear Programming Tools and Approximation Algorithms for Combinatorial Optimization
We study techniques, approximation algorithms, structural properties and lower bounds related to applications of linear programs in combinatorial optimization. The following "Steiner tree problem" is central: given a graph with a distinguished subset of required vertices, and costs for each edge, find a minimum-cost subgraph that connects the required vertices. We also investigate the areas of network design, multicommodity flows, and packing/covering integer programs. All of these problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio.
Overall, we show some new techniques that enhance the already-substantial corpus of LP-based approximation methods, and we also look for limitations of these techniques.
The first half of the thesis deals with linear programming relaxations for the Steiner tree problem. The crux of our work deals with hypergraphic relaxations obtained via the well-known full component decomposition of Steiner trees; explicitly, in this view the fundamental building blocks are not edges, but hyperedges containing two or more required vertices. We introduce a new hypergraphic LP based on partitions. We show the new LP has the same value as several previously-studied hypergraphic ones; when no Steiner nodes are adjacent, we show that the value of the well-known bidirected cut relaxation is also the same. A new partition uncrossing technique is used to demonstrate these equivalences, and to show that extreme points of the new LP are well-structured. We improve the best known integrality gap on these LPs in some special cases. We show that several approximation algorithms from the literature on Steiner trees can be re-interpreted through linear programs, in particular our hypergraphic relaxation yields a new view of the Robins-Zelikovsky 1.55-approximation algorithm for the Steiner tree problem.
The second half of the thesis deals with a variety of fundamental problems in combinatorial optimization. We show how to apply the iterated LP relaxation framework to the problem of multicommodity integral flow in a tree, to get an approximation ratio that is asymptotically optimal in terms of the minimum capacity. Iterated relaxation gives an infeasible solution, so we need to finesse it back to feasibility without losing too much value. Iterated LP relaxation similarly gives an O(k^2)-approximation algorithm for packing integer programs with at most k occurrences of each variable; new LP rounding techniques give a k-approximation algorithm for covering integer programs with at most k variable per constraint. We study extreme points of the standard LP relaxation for the traveling salesperson problem and show that they can be much more complex than was previously known. The k-edge-connected spanning multi-subgraph problem has the same LP and we prove a lower bound and conjecture an upper bound on the approximability of variants of this problem. Finally, we show that for packing/covering integer programs with a bounded number of constraints, for any epsilon > 0, there is an LP with integrality gap at most 1 + epsilon
Hardness results and approximation algorithms for some problems on graphs
This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees.
In the Propagation problem we are given a graph and the goal is to find a minimum-sized set of nodes that covers all of the nodes, where a node is covered if (1) is in , or (2) has a neighbor such that and all of its neighbors except are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of in weighted (general) graphs.
The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem.
We show that it is hard to approximate the optimal value to within a factor of in general graphs. We design and analyze an approximation algorithm with a performance guarantee of on planar graphs.
We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width . Motivated by applications, we introduce a restricted version of the problem that we call the -round General Propagation problem. We give a PTAS for the -round General Propagation problem on planar graphs, for small values of . Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes.
In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph and a subset of terminal nodes . The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans , that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of , given a planar graph that is element-connected on the terminals; in fact, given such a graph the algorithm returns element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio
of in planar graphs
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
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