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

Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

Solving Set Constraint Satisfaction Problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems