Framework For Approximation Algorithms Research Articles

Two-stage Robust Network Design with Exponential Scenarios

We study two-stage robust variants of combinatorial optimization problems on undirected graphs, like Steiner tree, Steiner forest, and uncapacitated facility location. Robust optimization problems, previously studied by Dhamdhere et al. (Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005), Golovin et al. (Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006), and Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007), are two-stage planning problems in which the requirements are revealed after some decisions are taken in Stage 1. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust k-Steiner tree problem, for example, one buys some edges in Stage 1. Then k terminals are revealed in Stage 2 and one has to buy more edges, at a higher cost, to complete the Stage 1 solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for k-Steiner tree), or any subset of k terminal-pairs (for k-Steiner forest), or any subset of k clients (for facility location). Feige et al. (Proc. of the 12th International Integer Programming and Combinatorial Optimization Conference, pp. 439---453, 2007) give an LP-based general framework for approximation algorithms for a class of two stage robust problems. Their framework cannot be used for network design problems like k-Steiner tree (see later elaboration). Their framework can be used for the robust facility location problem, but gives only a logarithmic approximation. We present the first constant-factor approximation algorithms for the robust k-Steiner tree (with exponential number of scenarios) and robust uncapacitated facility location problems. Our algorithms are combinatorial and are based on guessing the optimum cost and clustering to aggregate nearby vertices. For the robust k-Steiner forest problem on trees and with uniform multiplicative increase factor for Stage 2 (also known as inflation), we present a constant approximation. We show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question of (Dhamdhere et al. in Proc. of 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05), pp. 367---378, 2005) and (Golovin et al. in Proc. of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS), 2006).

Fairness in Routing and Load Balancing

We consider the issue of network routing subject to explicit fairness conditions. The optimization of fairness criteria interacts in a complex fashion with the optimization of network utilization and throughput; in this work, we undertake an investigation of this relationship through the framework of approximation algorithms. In a range of settings including both high-speed networks and Internet applications, max–min fairness has emerged as a widely accepted formulation of the notion of fairness. Informally, we say that an allocation of bandwidth is max–min fair if there is no way to give more bandwidth to any connection without decreasing the allocation to a connection of lesser or equal bandwidth. Given a collection of transmission routes, this criterion imposes a certain equilibrium condition on the bandwidth allocation, and some simple flow control mechanisms converge quickly to this equilibrium state. Indeed, the vast majority of previous work on max–min fairness has focused on this issue of associating rates with connections that are specified by a fixed set of paths. Very little work has been devoted to understanding the relationship between the way in which one selects paths for routing, and the amount of throughput one obtains from the resulting max–min fair allocation on these paths. In this work, we consider the problem of selecting paths for routing in order to provide a bandwidth allocation that is as fair as possible (in the max–min sense). We obtain the first approximation algorithms for this basic optimization problem, for single-source unsplittable routings in an arbitrary directed graph. Special cases of our model include several fundamental load balancing problems, endowing them with a natural fairness criterion to which our approach can be applied.

Reactive local search techniques for the maximum k-conjunctive constraint satisfaction problem ( MAX-k-CCSP)

In this paper the performance of the Hamming-based Reactive Tabu Search algorithm (H-RTS) previously proposed for the Maximum Satisfiability problem is studied for the different Maximum k-Conjunctive Constraint Satisfaction problem. In addition, the use of non-oblivious functions recently proposed in the framework of approximation algorithms is investigated. In particular, two relevant special cases of the Maximum k-Conjunctive Constraint Satisfaction problem are considered: Maximum Directed Cut and Maximum Independent Set in cubic graphs. The preliminary diversification-bias analysis of the basic components shows a remarkable difference between the two problems, and the derived predictions are then validated by extensive experiments with the complete H-RTS algorithm. The performance of H-RTS is compared with that of Simulated Annealing and simple Repeated Local Search.

Approximate Set Covering in Uniform Hypergraphs

The weighted set covering problem, restricted to the class ofr-uniform hypergraphs, is considered. We propose a new approach, based on a recent result of Aharoni, Holzman, and Krivelevich about the ratio of integer and fractional covering numbers ink-colorabler-uniform hypergraphs. This approach, applied to hypergraphs of maximal degree bounded by Δ, yields an algorithm with approximation ratior(1−c/Δ1/(r−1)). Next, we combine this approach with an adaptation of the local ratio theorem of Bar-Yehuda and Even for hypergraphs and present a general framework of approximation algorithms, based on subhypergraph exclusion. An application of this scheme is described, providing an algorithm with approximation ratior(1−c/n(r−1)/r) for hypergraphs onnvertices. We discuss also the limitations of this approach.

Reactive search, a history-sensitive heuristic for MAX-SAT

The Reactive Search (RS) method proposes the integration of a simple history-sensitive (machine learning) scheme into local search for the on-line determination of free parameters. In this paper a new RS algorithm is proposed for the approximated solution of the Maximum Satisfiability problem: a component based on local search with temporary prohibitions (Tabu Search) is complemented with a reactive scheme that determines the appropriate value of the prohibition parameter by monitoring the Hamming distance along the search trajectory. The proposed algorithm (H-RTS) can therefore be characterized as a dynamic version of Tabu Search. In addition, the non-oblivious functions recently introduced in the framework of approximation algorithms are used to discover a better local optimum in the initial part of the search The algorithm is developed in two phases. First the bias-diversification properties of individual candidate components are analyzed by extensive empirical evaluation, then a reactive scheme is added to the winning component, based on Tabu Search. The final tests on a benchmark of random MAX-3-SAT and MAX-4-SAT problems demonstrate the superiority of H-RTS with respect to alternative heuristics.