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

Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal-dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem, respectively. These results match the currently known best results for purely edge-weighted graphs.Comment: To appear in SWAT 201

Approximation algorithms for node-weighted prize-collecting Steiner tree problems on planar graphs

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a ($2.88 + \epsilon$)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain

The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks

It is challenging to design large and low-cost communication networks. In this paper, we formulate this challenge as the prize-collecting Steiner Tree Problem (PCSTP). The objective is to minimize the costs of transmission routes and the disconnected monetary or informational profits. Initially, we note that the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to improve suboptimal solutions to PCSTP. Based on these techniques, we propose two fast heuristic algorithms: the first one is a quasilinear time heuristic algorithm that is faster and consumes less memory than other algorithms; and the second one is an improvement of a stateof-the-art polynomial time heuristic algorithm that can find high-quality solutions at a speed that is only inferior to the first one. We demonstrate the competitiveness of our heuristic algorithms by comparing them with the state-of-the-art ones on the largest existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we generate new instances that are even larger (1 000 000 vertices and 10 000 000 edges) to further demonstrate their advantages in large networks. The state-ofthe-art algorithms are too slow to find high-quality solutions for instances of this size, whereas our new heuristic algorithms can do this in around 6 to 45s on a personal computer. Ultimately, we apply our post-processing techniques to update the bestknown solution for a notoriously difficult benchmark instance to show that they can improve near-optimal solutions to PCSTP. In conclusion, we demonstrate the usefulness of our heuristic algorithms and post-processing techniques for designing large and low-cost communication networks

On the performance of a cavity method based algorithm for the Prize-Collecting Steiner Tree Problem on graphs

We study the behavior of an algorithm derived from the cavity method for the Prize-Collecting Steiner Tree (PCST) problem on graphs. The algorithm is based on the zero temperature limit of the cavity equations and as such is formally simple (a fixed point equation resolved by iteration) and distributed (parallelizable). We provide a detailed comparison with state-of-the-art algorithms on a wide range of existing benchmarks networks and random graphs. Specifically, we consider an enhanced derivative of the Goemans-Williamson heuristics and the DHEA solver, a Branch and Cut Linear/Integer Programming based approach. The comparison shows that the cavity algorithm outperforms the two algorithms in most large instances both in running time and quality of the solution. Finally we prove a few optimality properties of the solutions provided by our algorithm, including optimality under the two post-processing procedures defined in the Goemans-Williamson derivative and global optimality in some limit cases