132,501 research outputs found
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
Distributed Distance-Bounded Network Design Through Distributed Convex Programming
Solving linear programs is often a challenging task in distributed settings.
While there are good algorithms for solving packing and covering linear
programs in a distributed manner (Kuhn et al.~2006), this is essentially the
only class of linear programs for which such an algorithm is known. In this
work we provide a distributed algorithm for solving a different class of convex
programs which we call "distance-bounded network design convex programs". These
can be thought of as relaxations of network design problems in which the
connectivity requirement includes a distance constraint (most notably, graph
spanners). Our algorithm runs in rounds in the
model and finds a -approximation to the optimal
LP solution for any , where is the largest distance
constraint. While solving linear programs in a distributed setting is
interesting in its own right, this class of convex programs is particularly
important because solving them is often a crucial step when designing
approximation algorithms. Hence we almost immediately obtain new and improved
distributed approximation algorithms for a variety of network design problems,
including Basic - and -Spanner, Directed -Spanner, Lowest Degree
-Spanner, and Shallow-Light Steiner Network Design with a spanning demand
graph. Our algorithms do not require any "heavy" computation and essentially
match the best-known centralized approximation algorithms, while previous
approaches which do not use heavy computation give approximations which are
worse than the best-known centralized bounds
Finding Almost Tight Witness Trees
This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs
Choose your witnesses wisely
This paper addresses a graph optimization problem, called the Witness Tree
problem, which seeks a spanning tree of a graph minimizing a certain non-linear
objective function. This problem is of interest because it plays a crucial role
in the analysis of the best approximation algorithms for two fundamental
network design problems: Steiner Tree and Node-Tree Augmentation. We will show
how a wiser choice of witness trees leads to an improved approximation for
Node-Tree Augmentation, and for Steiner Tree in special classes of graphs.Comment: 33 pages, 7 figures, submitted to IPCO 202
Approximating connected facility location problems via Random facility sampling and core detouring
We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00 and for the single-sink rent-or-buy problem from 3.55 to 2.92. We show that our connected facility location algorithms can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems
Approximating Connected Facility Location Problems via Random Facility Sampling and Core Detouring
We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we term core detouring, we show that this approach significantly improves over the previously best known approximation ratios for several NP-hard network design problems. For example, we reduce the approximation ratio for the connected facility location problem from 8.55 to 4.00, and for the single-sink rent-or-buy problem from 3.55 to 2.92. We show that our connected facility location algorithms can be derandomized at the expense of a slightly worse approximation ratio. The versatility of our framework is demonstrated by devising improved approximation algorithms also for other related problems
Improved Local Search Algorithms with Multi-Cycle Reduction for Minimum Concave Cost Network Flow Problems
The minimum concave cost network flow problem (MCCNFP) has many applications in areas such as telecommunication network design, facility location, production and inventory planning, and traffic scheduling and control. However, it is a well known NP-hard problem, and all existing search based exact algorithms are not practical for networks with even moderate numbers of vertices. Therefore, the research community also focuses on approximation algorithms to tackle the problems in practice. In this paper, we present an improved local search algorithm for the minimum concave cost network flow problem based on multi-cycle reduction. The original cycle reduction local search algorithm as proposed by Gallo and Sodini considers only negative cost single cycles; however, we find that such cycle reduction is not complete. We show that negative cost multi-cycles may exist in a network with concave edge costs that has no negative cost cycles, and an existing flow can be reduced to an adjacent neighboring flow with lower cost by redirecting flows along these negative multi-cycles. In this paper, we present an improved local search algorithm based on multi-cycle reduction. We evaluate our proposed algorithm in networks with a simple concave edge cost in different topologies and sizes. The experimental results show that the original cycle reduction algorithms can improve the quality of solutions obtained from a simple minimum cost augmentation approximation heuristic (LDF), and that a multi-cycle reduction yields more improvements; however, it reaches a point of diminished returns when we attempt to reduce more than bicycles
Network Design via Core Detouring for Problems Without a Core
Some of the currently best-known approximation algorithms for network design are based on random sampling. One of the key steps of such algorithms is connecting a set of source nodes to a random subset of them. In a recent work [Eisenbrand,Grandoni,Rothvo\ss,Schäfer-SODA'08], a new technique, \emph{core-detouring}, is described to bound the mentioned connection cost. This is achieved by defining a sub-optimal connection scheme, where paths are detoured through a proper connected subgraph (core). The cost of the detoured paths is bounded against the cost of the core and of the distances from the sources to the core. The analysis then boils down to proving the \emph{existence} of a convenient core. For some problems, such as connected facility location and single-sink rent-or-buy, the choice of the core is obvious (i.e., the Steiner tree in the optimum solution). Other, more complex network design problems do not exhibit any such core. In this paper we show that core-detouring can be nonetheless successfully applied. The basic idea is constructing a convenient core by manipulating the optimal solution in a proper (not necessarily trivial) way. We illustrate that by presenting improved approximation algorithms for two well-studied problems: virtual private network design and single-sink buy-at-bulk
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