Cuts and connectivity in graphs and hypergraphs

In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following: - We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation. - We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2+epsilon)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier. - We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span. - We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-1/448 approximation algorithm for the global bicut problem

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

High-Quality Hypergraph Partitioning

This dissertation focuses on computing high-quality solutions for the NP-hard balanced hypergraph partitioning problem: Given a hypergraph and an integer $k$, partition its vertex set into $k$ disjoint blocks of bounded size, while minimizing an objective function over the hyperedges. Here, we consider the two most commonly used objectives: the cut-net metric and the connectivity metric. Since the problem is computationally intractable, heuristics are used in practice - the most prominent being the three-phase multi-level paradigm: During coarsening, the hypergraph is successively contracted to obtain a hierarchy of smaller instances. After applying an initial partitioning algorithm to the smallest hypergraph, contraction is undone and, at each level, refinement algorithms try to improve the current solution. With this work, we give a brief overview of the field and present several algorithmic improvements to the multi-level paradigm. Instead of using a logarithmic number of levels like traditional algorithms, we present two coarsening algorithms that create a hierarchy of (nearly) $n$ levels, where $n$ is the number of vertices. This makes consecutive levels as similar as possible and provides many opportunities for refinement algorithms to improve the partition. This approach is made feasible in practice by tailoring all algorithms and data structures to the $n$-level paradigm, and developing lazy-evaluation techniques, caching mechanisms and early stopping criteria to speed up the partitioning process. Furthermore, we propose a sparsification algorithm based on locality-sensitive hashing that improves the running time for hypergraphs with large hyperedges, and show that incorporating global information about the community structure into the coarsening process improves quality. Moreover, we present a portfolio-based initial partitioning approach, and propose three refinement algorithms. Two are based on the Fiduccia-Mattheyses (FM) heuristic, but perform a highly localized search at each level. While one is designed for two-way partitioning, the other is the first FM-style algorithm that can be efficiently employed in the multi-level setting to directly improve $k$-way partitions. The third algorithm uses max-flow computations on pairs of blocks to refine $k$-way partitions. Finally, we present the first memetic multi-level hypergraph partitioning algorithm for an extensive exploration of the global solution space. All contributions are made available through our open-source framework KaHyPar. In a comprehensive experimental study, we compare KaHyPar with hMETIS, PaToH, Mondriaan, Zoltan-AlgD, and HYPE on a wide range of hypergraphs from several application areas. Our results indicate that KaHyPar, already without the memetic component, computes better solutions than all competing algorithms for both the cut-net and the connectivity metric, while being faster than Zoltan-AlgD and equally fast as hMETIS. Moreover, KaHyPar compares favorably with the current best graph partitioning system KaFFPa - both in terms of solution quality and running time

Tight MIP Formulations for Bounded Up/Down Times and Interval-Dependent Start-Ups

Switching machines on and off is an important aspect of unit commitment problems and production planning problems, among others. Here we study tight mixed integer program- ming formulations for two aspects of such problems: bounded length on- and off-intervals, and interval-dependent start-ups. For the problem with both these aspects we develop a tight (convex hull) formulation involving additional variables. For the bounded interval problem we present a tight net- work dual formulation based on new integer variables that allows us to simultaneously treat lower and upper bounds on the interval lengths. This in turn leads to more general results, including simpler proofs of known tight formulations for problems with just lower bounds. For the interval-dependent start-up problem we develop a path formulation that allows us to describe the convex hull of solutions in the space of machine-on and interval-dependent start-up variable