Abstract
Research on the similarity of a graph to being a tree—called the treewidth of the graph—has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). The parameter treedepth has a similar story. It was popularized through the graph sparsity project and is theoretically well understood—but the first practical algorithm was discovered only recently by Trimble (IPEC 2020) and is based on the same paradigm. We give an alternative and unifying view on such algorithms from the perspective of the corresponding configuration graphs in certain two-player games. This results in a single algorithm that can compute a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We complement this algorithm with a novel randomized data structure that accelerates the enumeration of subproblems in positive-instance driven algorithms.
Highlights
Graph decompositions are an important tool in modern algorithmic graph theory that provide a structured representation of a graph
We extend a known data structure used by such algorithms, called block sieve, with a randomized component based on the well-known color coding technique
In contrast, are (i) much more compact, as they are lazily constructed, (ii) use randomization to provide a guarantee that non-compatible elements are pruned with high probability, and (iii) allow a trade-off between the time spend in the data structure and the amount of elements that will be pruned. We extend this theoretical analysis by an implementation of the data structure in our treedepth solver positive-instance driven (PID)? and an experimental evaluation (Sections 6)
Summary
Graph decompositions are an important tool in modern algorithmic graph theory that provide a structured representation of a graph. A graph decomposition comes along with a width measure that indicates how well a graph can be decomposed. Many problems that presumably cannot be solved in polynomial time on general graphs can be solved efficiently on graphs that admit a certain decomposition of small width [1]. The most prominent example of a width measure is treewidth, which (on an intuitive level) measures the similarity of a graph to a tree. This parameter is a cornerstone of parameterized algorithms [2] and its success has led to its integration into many different fields. Treewidth has been studied in the context of machine learning [3,4,5], modelchecking [6,7], SAT-solving [8,9,10], QBF-solving [11,12], CSP-solving [13,14], or ILP S [15,16,17,18,19]
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