An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions

In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real space. The incrementality comes from the first part of the algorithm, where a complex cylindrical tree is constructed by refining a previous complex cylindrical tree with a polynomial constraint. We have implemented our algorithm in Maple. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature

Computing Shrub-Depth Decompositions

Shrub-depth is a width measure of graphs which, roughly speaking, corresponds to the smallest depth of a tree into which a graph can be encoded. It can be thought of as a low-depth variant of clique-width (or rank-width), similarly as treedepth is a low-depth variant of treewidth. We present an fpt algorithm for computing decompositions of graphs of bounded shrub-depth. To the best of our knowledge, this is the first algorithm which computes the decomposition directly, without use of rank-width decompositions and FO or MSO logic

A Backtracking-Based Algorithm for Computing Hypertree-Decompositions

Hypertree decompositions of hypergraphs are a generalization of tree decompositions of graphs. The corresponding hypertree-width is a measure for the cyclicity and therefore tractability of the encoded computation problem. Many NP-hard decision and computation problems are known to be tractable on instances whose structure corresponds to hypergraphs of bounded hypertree-width. Intuitively, the smaller the hypertree-width, the faster the computation problem can be solved. In this paper, we present the new backtracking-based algorithm det-k-decomp for computing hypertree decompositions of small width. Our benchmark evaluations have shown that det-k-decomp significantly outperforms opt-k-decomp, the only exact hypertree decomposition algorithm so far. Even compared to the best heuristic algorithm, we obtained competitive results as long as the hypergraphs are not too large.Comment: 19 pages, 6 figures, 3 table

Size-Constrained Tree Decompositions

Tree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this report, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k ≥ 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs

Minimum Size Tree-Decompositions

International audienceTree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree- decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic program- ming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree- decompositions. It is then interesting to try to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k >= 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k >= 4 and polynomial for k <= 2. On going work also suggests it is polynomial for k = 3

On computing tree and path decompositions with metric constraints on the bags

We here investigate on the complexity of computing the \emph{tree-length} and the \emph{tree-breadth} of any graph $G$, that are respectively the best possible upper-bounds on the diameter and the radius of the bags in a tree decomposition of $G$. \emph{Path-length} and \emph{path-breadth} are similarly defined and studied for path decompositions. So far, it was already known that tree-length is NP-hard to compute. We here prove it is also the case for tree-breadth, path-length and path-breadth. Furthermore, we provide a more detailed analysis on the complexity of computing the tree-breadth. In particular, we show that graphs with tree-breadth one are in some sense the hardest instances for the problem of computing the tree-breadth. We give new properties of graphs with tree-breadth one. Then we use these properties in order to recognize in polynomial-time all graphs with tree-breadth one that are planar or bipartite graphs. On the way, we relate tree-breadth with the notion of \emph{$k$-good} tree decompositions (for $k=1$), that have been introduced in former work for routing. As a byproduct of the above relation, we prove that deciding on the existence of a $k$-good tree decomposition is NP-complete (even if $k=1$). All this answers open questions from the literature.Comment: 50 pages, 39 figure

An Experimental Study of the Treewidth of Real-World Graph Data

Treewidth is a parameter that measures how tree-like a relational instance is, and whether it can reasonably be decomposed into a tree. Many computation tasks are known to be tractable on databases of small treewidth, but computing the treewidth of a given instance is intractable. This article is the first large-scale experimental study of treewidth and tree decompositions of real-world database instances (25 datasets from 8 different domains, with sizes ranging from a few thousand to a few million vertices). The goal is to determine which data, if any, can benefit of the wealth of algorithms for databases of small treewidth. For each dataset, we obtain upper and lower bound estimations of their treewidth, and study the properties of their tree decompositions. We show in particular that, even when treewidth is high, using partial tree decompositions can result in data structures that can assist algorithms

A Unifying Framework for Characterizing and Computing Width Measures

Algorithms for computing or approximating optimal decompositions for decompositional parameters such as treewidth or clique-width have so far traditionally been tailored to specific width parameters. Moreover, for mim-width, no efficient algorithms for computing good decompositions were known, even under highly restrictive parameterizations. In this work we identify ?-branchwidth as a class of generic decompositional parameters that can capture mim-width, treewidth, clique-width as well as other measures. We show that while there is an infinite number of ?-branchwidth parameters, only a handful of these are asymptotically distinct. We then develop fixed-parameter and kernelization algorithms (under several structural parameterizations) that can approximate every possible ?-branchwidth, providing a unifying parameterized framework that can efficiently obtain near-optimal tree-decompositions, k-expressions, as well as optimal mim-width decompositions

Minimum Size Tree-decompositions

International audienceTree-decompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs