A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection

Given an undirected graph with edge costs and node weights, the minimum bisection problem asks for a partition of the nodes into two parts of equal weight such that the sum of edge costs between the parts is minimized. We give a polynomial time bicriteria approximation scheme for bisection on planar graphs. Specifically, let $W$ be the total weight of all nodes in a planar graph $G$. For any constant $\varepsilon > 0$, our algorithm outputs a bipartition of the nodes such that each part weighs at most $W/2 + \varepsilon$ and the total cost of edges crossing the partition is at most $(1+\varepsilon)$ times the total cost of the optimal bisection. The previously best known approximation for planar minimum bisection, even with unit node weights, was $O(\log n)$. Our algorithm actually solves a more general problem where the input may include a target weight for the smaller side of the bipartition.Comment: To appear in STOC 201

Balanced Partitions of Trees and Applications

We study the k-BALANCED PARTITIONING problem in which the vertices of a graph are to be partitioned into k sets of size at most ceil(n/k) while minimising the cut size, which is the number of edges connecting vertices in different sets. The problem is well studied for general graphs, for which it cannot be approximated within any factor in polynomial time. However, little is known about restricted graph classes. We show that for trees k-BALANCED PARTITIONING remains surprisingly hard. In particular, approximating the cut size is APX-hard even if the maximum degree of the tree is constant. If instead the diameter of the tree is bounded by a constant, we show that it is NP-hard to approximate the cut size within n^c, for any constant c<1. In the face of the hardness results, we show that allowing near-balanced solutions, in which there are at most (1+eps)ceil(n/k) vertices in any of the k sets, admits a PTAS for trees. Remarkably, the computed cut size is no larger than that of an optimal balanced solution. In the final section of our paper, we harness results on embedding graph metrics into tree metrics to extend our PTAS for trees to general graphs. In addition to being conceptually simpler and easier to analyse, our scheme improves the best factor known on the cut size of near-balanced solutions from O(log^{1.5}(n)/eps^2) [Andreev and Räcke TCS 2006] to 0(log n), for weighted graphs. This also settles a question posed by Andreev and Räcke of whether an algorithm with approximation guarantees on the cut size independent from eps exists.ISSN:1868-896

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

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

08431 Abstracts Collection -- Moderately Exponential Time Algorithms

From $19/10/2008$ to $24/10/2008$, the Dagstuhl Seminar 08431 ``Moderately Exponential Time Algorithms \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition