Prize-Collecting Steiner Networks via Iterative Rounding

Abstract. In this paper we design an iterative rounding approach for the classic prize-collecting Steiner forest problem and more generally the prize-collecting survivable Steiner network design problem. We show as an structural result that in each iteration of our algorithm there is an LP variable in a basic feasible solution which is at least one-third-integral resulting a 3-approximation algorithm for this problem. In addition, we show this factor 3 in our structural result is indeed tight for prize-collecting Steiner forest and thus prize-collecting survivable Steiner network design. This especially answers negatively the previous belief that one might be able to obtain an approximation factor better than 3 for these problems using a natural iterative rounding approach. Our structural result is extending the celebrated iterative rounding approach of Jain [13] by using several new ideas some from more complicated linear algebra. The approach of this paper can be also applied to get a constant factor (bicriteria-)approximation algorithm for degree constrained prize-collecting network design problems. We emphasize that though in theory we can prove existence of only an LP variable of at least one-third-integral, in practice very often in each iteration there exists a variable of integral or almost integral which results in a much better approximation factor than provable factor 3 in this paper (see patent application [11]). This is indeed the advantage of our algorithm in this paper over previous approximation algorithms for prize-collecting Steiner forest with the same or slightly better provable approximation factors.

Approximation Complexity of Optimization Problems : Structural Foundations and Steiner Tree Problems

In this thesis we study the approximation complexity of the Steiner Tree Problem and related problems as well as foundations in structural complexity theory. The Steiner Tree Problem is one of the most fundamental problems in combinatorial optimization. It asks for a shortest connection of a given set of points in an edge-weighted graph. This problem and its numerous variants have applications ranging from electrical engineering, VLSI design and transportation networks to internet routing. It is closely connected to the famous Traveling Salesman Problem and serves as a benchmark problem for approximation algorithms. We give a survey on the Steiner tree Problem, obtaining lower bounds for approximability of the (1,2)-Steiner Tree Problem by combining hardness results of Berman and Karpinski with reduction methods of Bern and Plassmann. We present approximation algorithms for the Steiner Forest Problem in graphs and bounded hypergraphs, the Prize Collecting Steiner Tree Problem and related problems where prizes are given for pairs of terminals. These results are based on the Primal-Dual method and the Local Ratio framework of Bar-Yehuda. We study the Steiner Network Problem and obtain combinatorial approximation algorithms with reasonable running time for two special cases, namely the Uniform Uncapacitated Case and the Prize Collecting Uniform Uncapacitated Case. For the general case, Jain's algorithms obtains an approximation ratio of 2, based on the Ellipsoid Method. We obtain polynomial time approximation schemes for the Dense Prize Collecting Steiner Tree Problem, Dense k-Steiner Problem and the Dense Class Steiner Tree Problem based on the methods of Karpinski and Zelikovsky for approximating the Dense Steiner Tree Problem. Motivated by the question which parameters make the Steiner Tree problem hard to solve, we make an excurs into Fixed Parameter Complexity, focussing on structural aspects of the W-Hierarchy. We prove a Speedup Theorem for the classes FPT and SP and versions if Levin's Lower Bound Theorem for the class SP as well as for Randomized Space Complexity. Starting from the approximation schemes for the dense Steiner Tree problems, we deal with the efficiency of polynomial time approximation schemes in general. We separate the class EPTAS from PTAS under some reasonable complexity theoretic assumption. The same separation was achieved by Cesaty and Trevisan under some assumtion from Fixed Parameter Complexity. We construct an oracle under which our assumtion holds but that of Cesati and Trevisan does not, which implies that using relativizing proof techniques one cannot show that our assumption implies theirs

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 Θ [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

Iterative Rounding Approximation Algorithms in Network Design

Iterative rounding has been an increasingly popular approach to solving network design optimization problems ever since Jain introduced the concept in his revolutionary 2-approximation for the Survivable Network Design Problem (SNDP). This paper looks at several important iterative rounding approximation algorithms and makes improvements to some of their proofs. We generalize a matrix restatement of Nagarajan et al.'s token argument, which we can use to simplify the proofs of Jain's 2-approximation for SNDP and Fleischer et al.'s 2-approximation for the Element Connectivity (ELC) problem. Lau et al. show how one can construct a (2,2B + 3)-approximation for the degree bounded ELC problem, and this thesis provides the proof. We provide some structural results for basic feasible solutions of the Prize-Collecting Steiner Tree problem, and introduce a new problem that arises, which we call the Prize-Collecting Generalized Steiner Tree problem

The cavity approach for Steiner trees packing problems

The Belief Propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple "interacting" trees have to be sought within a given graph. Depending on the interaction among trees we distinguish the vertex-disjoint Steiner trees problem, where trees cannot share nodes, from the edge-disjoint Steiner trees problem, where edges cannot be shared by trees but nodes can be members of multiple trees. Several practical problems of huge interest in network design can be mapped into these two variants, for instance, the physical design of Very Large Scale Integration (VLSI) chips. The formalism described here relies on two components edge-variables that allows us to formulate a massage-passing algorithm for the V-DStP and two algorithms for the E-DStP differing in the scaling of the computational time with respect to some relevant parameters. We will show that one of the two formalisms used for the edge-disjoint variant allow us to map the max-sum update equations into a weighted maximum matching problem over proper bipartite graphs. We developed a heuristic procedure based on the max-sum equations that shows excellent performance in synthetic networks (in particular outperforming standard multi-step greedy procedures by large margins) and on large benchmark instances of VLSI for which the optimal solution is known, on which the algorithm found the optimum in two cases and the gap to optimality was never larger than 4 %

Locating leak detecting sensors in a water distribution network by solving prize-collecting Steiner arborescence problems

We consider the problem of optimizing a novel acoustic leakage detection system for urban water distribution networks. The system is composed of a number of detectors and transponders to be placed in a choice of hydrants such as to provide a desired coverage under given budget restrictions. The problem is modeled as a particular Prize-Collecting Steiner Arborescence Problem. We present a branch-and-cut-and-bound approach taking advantage of the special structure at hand which performs well when compared to other approaches. Furthermore, using a suitable stopping criterion, we obtain approximations of provably excellent quality (in most cases actually optimal solutions). The test bed includes the real water distribution network from the Lausanne region, as well as carefully randomly generated realistic instance