Approximating acyclicity parameters of sparse hypergraphs

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class

Approximating Acyclicity Parameters of Sparse Hypergraphs

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello (PODS'99, PODS'01) in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx in SODA'06, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. Computing each of these width parameters is known to be an NP-hard problem. Moreover, the (generalized) hypertree width of an n-vertex hypergraph cannot be approximated within a logarithmic factor unless P=NP. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class. This extends the results of Feige, Hajiaghayi, and Lee from STOC'05 on approximating treewidth of H-minor-free graphs.publishedVersio

Efficient parameterized algorithms for data packing

There is a huge gap between the speeds of modern caches and main memories, and therefore cache misses account for a considerable loss of efficiency in programs. The predominant technique to address this issue has been Data Packing: data elements that are frequently accessed within time proximity are packed into the same cache block, thereby minimizing accesses to the main memory. We consider the algorithmic problem of Data Packing on a two-level memory system. Given a reference sequence R of accesses to data elements, the task is to partition the elements into cache blocks such that the number of cache misses on R is minimized. The problem is notoriously difficult: it is NP-hard even when the cache has size 1, and is hard to approximate for any cache size larger than 4. Therefore, all existing techniques for Data Packing are based on heuristics and lack theoretical guarantees. In this work, we present the first positive theoretical results for Data Packing, along with new and stronger negative results. We consider the problem under the lens of the underlying access hypergraphs, which are hypergraphs of affinities between the data elements, where the order of an access hypergraph corresponds to the size of the affinity group. We study the problem parameterized by the treewidth of access hypergraphs, which is a standard notion in graph theory to measure the closeness of a graph to a tree. Our main results are as follows: We show there is a number q* depending on the cache parameters such that (a) if the access hypergraph of order q* has constant treewidth, then there is a linear-time algorithm for Data Packing; (b)the Data Packing problem remains NP-hard even if the access hypergraph of order q*-1 has constant treewidth. Thus, we establish a fine-grained dichotomy depending on a single parameter, namely, the highest order among access hypegraphs that have constant treewidth; and establish the optimal value q* of this parameter. Finally, we present an experimental evaluation of a prototype implementation of our algorithm. Our results demonstrate that, in practice, access hypergraphs of many commonly-used algorithms have small treewidth. We compare our approach with several state-of-the-art heuristic-based algorithms and show that our algorithm leads to significantly fewer cache-misses

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics

On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic

Brambles were introduced as the dual notion to treewidth, one of the most central concepts of the graph minor theory of Robertson and Seymour. Recently, Grohe and Marx showed that there are graphs G, in which every bramble of order larger than the square root of the treewidth is of exponential size in |G|. On the positive side, they show the existence of polynomial-sized brambles of the order of the square root of the treewidth, up to log factors. We provide the first polynomial time algorithm to construct a bramble in general graphs and achieve this bound, up to log-factors. We use this algorithm to construct grid-like minors, a replacement structure for grid-minors recently introduced by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce the notion of a perfect bramble and an algorithm to find one in polynomial time. Perfect brambles are brambles with a particularly simple structure and they also provide us with a subgraph that has bounded degree and still large treewidth; we use them to obtain a meta-theorem on deciding certain parameterized subgraph-closed problems on general graphs in time singly exponential in the parameter. The second part of our work deals with providing a lower bound to Courcelle's famous theorem, stating that every graph property that can be expressed by a sentence in monadic second-order logic (MSO), can be decided by a linear time algorithm on classes of graphs of bounded treewidth. Using our results from the first part of our work we establish a strong lower bound for tractability of MSO on classes of colored graphs

Approximating branchwidth on parametric extensions of planarity

The \textsl{branchwidth} of a graph has been introduced by Roberson and Seymour as a measure of the tree-decomposability of a graph, alternative to treewidth. Branchwidth is polynomially computable on planar graphs by the celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an extension of this algorithm to minor-closed graph classes, further than planar graphs as follows: Let $H_{0}$ be a graph embeddedable in the projective plane and $H_{1}$ be a graph embeddedable in the torus. We prove that every $\{H_{0},H_{1}\}$-minor free graph $G$ contains a subgraph $G'$ where the difference between the branchwidth of $G$ and the branchwidth of $G'$ is bounded by some constant, depending only on $H_{0}$ and $H_{1}$. Moreover, the graph $G'$ admits a tree decomposition where all torsos are planar. This decomposition can be used for deriving an EPTAS for branchwidth: For $\{H_{0},H_{1}\}$-minor free graphs, there is a function $f\colon\mathbb{N}\to\mathbb{N}$ and a $(1+\epsilon)$-approximation algorithm for branchwidth, running in time $\mathcal{O}(n^3+f(\frac{1}{\epsilon})\cdot n),$ for every $\epsilon>0$

Integer programming models for the branchwidth problem

We consider the problem of computing the branchwidth and an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced in 1991 by Robertson and Seymour and were used in the proof of Graph Minors Theorem (GMT), a well known conjecture (Wagner's conjecture) in graph theory. The notions of branchwidth and branch decompositions have been proved to be useful for solving many NP-hard problems that have applications in fields such as graph theory, network design, sensor networks and biology. Branch decompositions have been utilized for problems such as the traveling salesman problem by Cook and Seymour, general minor containment and the branchwidth problem by Hicks by means of the relevant branch decomposition-based algorithms. Branch decomposition-based algorithms are fixed parameter tractable algorithms obtained by combining dynamic programming techniques with branch decompositions. The running time and space of these algorithms strongly depend on the width of the utilized branch decomposition. Thus, finding optimal or close to optimal branch decompositions is very important for the efficiency of the branch decomposition-based algorithms. Motivated by the vastness of the fields of application, we aim to increase the efficiency of the branch decomposition-based algorithms by investigating effective techniques to find optimal branch decompositions. We present three integer programming models for the branchwidth problem. Two similar formulations are based on the relationship of branchwidth problem with a special case of the Steiner tree packing problem. The third formulation is based on the notion of laminar separations. We utilize upper and lower bounds obtained by heuristic algorithms, reduction techniques and cutting planes to increase the efficiency of our models. We use all three models for the branchwidth problem on hypergraphs as well. We compare the performance of three models both on graphs and hypergraphs. Furthermore we use the third model for rank-width problem and also offer a heuristic for finding good rank decompositions. We provide computational results for this problem, which can be a basis of comparison for future formulations

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

FPT algoritms providing constant ratio approximation of hypertree width parameters for hypergraphs of bounded rank

We propose an algorithm whose input are parameters $k$ and $r$ and a hypergraph $H$ of rank at most $r$. The algorithm either returns a tree decomposition of $H$ of generalized hypertree width at most $4k$ or 'NO'. In the latter case, it is guaranteed that the hypertree width of $H$ is greater than $k$. Most importantly, the runtime of the algorithm is \emph{FPT} in $k$ and $r$. The approach extends to fractional hypertree width with a slightly worse approximation ($4k+1$ instead of $4k$). We hope that the results of this paper will give rise to a new research direction whose aim is design of FPT algorithms for computation and approximation of hypertree width parameters for restricted classes of hypergraphs