Time and Parallelizability Results for Parity Games with Bounded Tree and DAG Width

Parity games are a much researched class of games in NP intersect CoNP that are not known to be in P. Consequently, researchers have considered specialised algorithms for the case where certain graph parameters are small. In this paper, we study parity games on graphs with bounded treewidth, and graphs with bounded DAG width. We show that parity games with bounded DAG width can be solved in O(n^(k+3) k^(k + 2) (d + 1)^(3k + 2)) time, where n, k, and d are the size, treewidth, and number of priorities in the parity game. This is an improvement over the previous best algorithm, given by Berwanger et al., which runs in n^O(k^2) time. We also show that, if a tree decomposition is provided, then parity games with bounded treewidth can be solved in O(n k^(k + 5) (d + 1)^(3k + 5)) time. This improves over previous best algorithm, given by Obdrzalek, which runs in O(n d^(2(k+1)^2)) time. Our techniques can also be adapted to show that the problem of solving parity games with bounded treewidth lies in the complexity class NC^2, which is the class of problems that can be efficiently parallelized. This is in stark contrast to the general parity game problem, which is known to be P-hard, and thus unlikely to be contained in NC

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number $k$ of nodes and the other player controls the $n-k$ other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct priorities and $m$ is the number of edges. For all games with $k = o(n)$ this improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree

Graph-theoretic Properties of Control Flow Graphs and Applications

This thesis deals with determining appropriate width parameters of control flow graphs so that certain computationally hard problems of practical interest become efficiently solvable. A well-known result of Thorup states that the treewidth of control flow graphs arising from structured (goto-free) programs is at most six. However, since a control flow graph is inherently directed, it is very likely that using a digraph width measure would give better algorithms for problems where directional properties of edges are important. One such problem, parity game, is closely related to the μ-calculus model checking problem in software verification and is known to be tractable on graphs of bounded DAG-width, Kelly-width or entanglement. Motivated by this, we show that the DAG-width of control flow graphs arising from structured programs is at most three and give a linear-time algorithm to compute the corresponding DAG decomposition. Using similar techniques, we show that Kelly-width of control flow graphs is also bounded by three. Additionally, we also show that control flow graphs can have unbounded entanglement. In light of these results, we revisit the complexity of the μ-calculus model checking problem on these special graph classes and show that we can obtain better running times for control flow graphs

Parity Games of Bounded Tree- and Clique-Width

Abstract. In this paper it is shown that deciding the winner of a parity game is in LogCFL, if the underlying graph has bounded tree-width, and in LogDCFL, if the tree-width is at most 2. It is also proven that parity games of bounded clique-width can be solved in LogCFL via a log-space reduction to the bounded tree-width case, assuming that a k-expression for the parity game is part of the input.

Parity Games of Bounded Tree-Depth

The exact complexity of solving parity games is a major open problem. Several authors have searched for efficient algorithms over specific classes of graphs. In particular, Obdr\v{z}\'{a}lek showed that for graphs of bounded tree-width or clique-width, the problem is in $\mathrm{P}$, which was later improved by Ganardi, who showed that it is even in $\mathrm{LOGCFL}$ (with an additional assumption for clique-width case). Here we extend this line of research by showing that for graphs of bounded tree-depth the problem of solving parity games is in logspace uniform $\text{AC}^0$. We achieve this by first considering a parameter that we obtain from a modification of clique-width, which we call shallow clique-width. We subsequently provide a suitable reduction.Comment: This is the full version of the paper that has been accepted at CSL 2023 and is going to be published in Leibniz International Proceedings in Informatics (LIPIcs

Constrained Clustering Problems and Parity Games

Clustering is a fundamental tool in data mining. It partitions points into groups (clusters) and may be used to make decisions for each point based on its group. We study several clustering objectives. We begin with studying the Euclidean k-center problem. The k-center problem is a classical combinatorial optimization problem which asks to select k centers and assign each input point in a set P to one of the centers, such that the maximum distance of any input point to its assigned center is minimized. The Euclidean k-center problem assumes that the input set P is a subset of a Euclidean space and that each location in the Euclidean space can be chosen as a center. We focus on the special case with k = 1, the smallest enclosing ball problem: given a set of points in m-dimensional Euclidean space, find the smallest sphere enclosing all the points. We combine known results about convex optimization with structural properties of the smallest enclosing ball to create a new algorithm. We show that on instances with rational coefficients our new algorithm computes the exact center of the optimal solutions and has a worst-case run time that is polynomial in the size of the input. We use the new algorithm to show that we can solve the Euclidean k-center problem in polynomial time for constant k and dimension m. The general unconstrained clustering problems are mostly very well studied. The k-center problem for example allows for elegant 2-approximation algorithms(Gonzalez 1985, Hochbaum,Shmoys 1986). However, the situation becomes significantly more difficult when constraints are added to the problem. We first look at the fair clustering. The fairness constraint is motivated by the fact that the general process of computing a clustering may harm protected (minority) classes if the clustering algorithm does not adequately represent them in desirable clusters -- especially if the data is already biased. At NIPS 2017, Chierichetti et al. proposed a model for fair clustering requiring the representation in each cluster to (approximately) preserve the global fraction of each protected class. Restricting to two protected classes, they developed both a 4-approximation algorithm for the fair k-center problem and an O(t)-approximation algorithm for the fair k-median problem, where t is a parameter for the fairness model. For multiple protected classes, the best known result is a 14-approximation algorithm for fair k-center (Rösner, Schmidt 2018). We extend and improve the known results. Firstly, we give a 5-approximation algorithm for the fair k-center problem with multiple protected classes. Secondly, we propose a relaxed fairness notion under which we can give bicriteria constant-factor approximation algorithms for the fair version of all of the classical clustering objectives (k-center, k-supplier, k-median, k-means and facility location). The latter approximation algorithms are achieved by a framework that takes an arbitrary existing unfair (integral) solution and a fair (fractional) LP solution and combines them into an essentially fair clustering with a weakly supervised rounding scheme. In this way, a fair clustering can be established belatedly, in a situation where for example the centers are already fixed. The second clustering constraint we study is privacy: Here, we are asked to only open a center when at least l points will be assigned to it. We raise the question whether a general method can be derived to turn an approximation algorithm for a clustering problem with some constraints into an approximation algorithm that additionally respects privacy. We show how to combine privacy with several other constraints and obtain approximation algorithms for the k-center problem with several combinations of constraints. In this dissertation we also study parity games, a two player game played on a directed graph. We study the case in which one of the two players controls only a small number k of nodes and the other player controls the n-k other nodes of the game. Our main result is a fixed-parameter-tractable algorithm that solves bipartite parity games in time k^{O(sqrt{k})} O(n^3), and general parity games in time (p+k)^{O(sqrt{k})} O(pnm), where p is the number of distinct priorities and m is the number of edges. For all games with k = o(n) this improves the previously fastest algorithm by Jurdziński, Paterson, and Zwick (2008). We also obtain novel kernelization results and an improved deterministic algorithm for parity games on graphs with small average node-degree

Programming Languages and Systems

This open access book constitutes the proceedings of the 29th European Symposium on Programming, ESOP 2020, which was planned to take place in Dublin, Ireland, in April 2020, as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The actual ETAPS 2020 meeting was postponed due to the Corona pandemic. The papers deal with fundamental issues in the specification, design, analysis, and implementation of programming languages and systems