2,982 research outputs found
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Traffic-Redundancy Aware Network Design
We consider network design problems for information networks where routers
can replicate data but cannot alter it. This functionality allows the network
to eliminate data-redundancy in traffic, thereby saving on routing costs. We
consider two problems within this framework and design approximation
algorithms.
The first problem we study is the traffic-redundancy aware network design
(RAND) problem. We are given a weighted graph over a single server and many
clients. The server owns a number of different data packets and each client
desires a subset of the packets; the client demand sets form a laminar set
system. Our goal is to connect every client to the source via a single path,
such that the collective cost of the resulting network is minimized. Here the
transportation cost over an edge is its weight times times the number of
distinct packets that it carries.
The second problem is a facility location problem that we call RAFL. Here the
goal is to find an assignment from clients to facilities such that the total
cost of routing packets from the facilities to clients (along unshared paths),
plus the total cost of "producing" one copy of each desired packet at each
facility is minimized.
We present a constant factor approximation for the RAFL and an O(log P)
approximation for RAND, where P is the total number of distinct packets. We
remark that P is always at most the number of different demand sets desired or
the number of clients, and is generally much smaller.Comment: 17 pages. To be published in the proceedings of the Twenty-Third
Annual ACM-SIAM Symposium on Discrete Algorithm
A Constant Factor Approximation for the Single Sink Edge Installation Problem
We present the first constant approximation to the single sink buy-at-bulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a metric. This result improves the previous bound of O(log |R|), where R is the set of sources. We also present a better constant approximation to the related Access Network Design problem. Our algorithms are randomized and combinatorial. As a subroutine in our algorithm, we use an interesting variant of facility location with lower bounds on the amount of demand an open facility needs to serve. We call this variant load balanced facility location and present a constant factor approximation for it, while relaxing the lower bounds by a constant factor
The VPN problems with concave costs
Only recently Goyal, Olver and Shepherd (Proc. STOC, 2008) proved that the symmetric Virtual Private Network Design (sVPN) problem has the tree routing property, namely, that there always exists an optimal solution to the problem whose support is a tree. Combining this with previous results by Fingerhut, Suri and Turner (J. Alg., 1997) and Gupta, Kleinberg, Kumar, Rastogi and Yener (Proc. STOC, 2001), sVPN can be solved in polynomial time. In this paper we investigate an APX-hard generalization of sVPN, where the contribution of each edge to the total cost is proportional to some non-negative, concave and non-decreasing function of the capacity reservation. We show that the tree routing property extends to the new problem, and give a constant-factor approximation algorithm for it. We also show that the undirected uncapacitated single-source minimum concave-cost flow problem has the tree routing property when the cost function has some property of symmetry
Pricing bridges to cross a river.
We consider a Stackelberg pricing problem in directed, uncapacitated networks. Tariffs have to be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs are assumed to be given. There are n clients, the followers, that route their demand independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by problems in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX-hard. Moreover, we show that uniform pricing yields both an mâapproximation, and a (1 + lnD)âapproximation. Here, D is upper bounded by the total demand of all clients. We furthermore discuss some polynomially solvable special cases, and present a short computational study with instances from France TĂ©lĂ©com. In addition, we consider the problem under the additional restriction that the operator must serve all clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless NP = ZPP, and we prove the existence of an nâapproximation algorithm.Pricing; Networks; Tariffs; Costs; Cost; Demand; Problems; Order; Yield; Studies; Approximation; Algorithms; Performance;
Node-weighted Steiner tree and group Steiner tree in planar graphs
We improve the approximation ratios for two optimization problems in planar graphs. For node-weighted Steiner tree, a classical network-optimization problem, the best achievable approximation ratio in general graphs is Î [theta] (logn), and nothing better was previously known for planar graphs. We give a constant-factor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minor-closed graph family, and also generalizes to address other optimization problems such as Steiner forest, prize-collecting Steiner tree, and network-formation games.
The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimum-weight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log3 [superscript 3] n), or O(log2 [superscript 2] n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimum-weight tour that must visit each group
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