Reducing CMSO model checking to highly connected graphs

Given a Counting Monadic Second Order (CMSO) sentence psi, the CMSO[psi] problem is defined as follows. The input to CMSO[psi] is a graph G, and the objective is to determine whether G |= psi. Our main theorem states that for every CMSO sentence psi, if CMSO[psi] is solvable in polynomial time on "globally highly connected graphs", then CMSO[psi] is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph G and the task is to find a connected induced subgraph of G such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property

A fixed-parameter tractable algorithm for elimination distance to bounded degree graphs

In the literature on parameterized graph problems, there has been an increased effort in recent years aimed at exploring novel notions of graph edit-distance that are more powerful than the size of a modulator to a specific graph class. In this line of research, Bulian and Dawar [Algorithmica, 75 (2016), pp. 363--382] introduced the notion of elimination distance and showed that deciding whether a given graph has elimination distance at most $k$ to any minor-closed class of graphs is fixed-parameter tractable parameterized by $k$ [Algorithmica, 79 (2017), pp. 139--158]. They showed that graph isomorphism parameterized by the elimination distance to bounded degree graphs is fixed-parameter tractable and asked whether determining the elimination distance to the class of bounded degree graphs is fixed-parameter tractable. Recently, Lindermayr, Siebertz, and Vigny [MFCS 2020, LIPIcs Leibniz Int. Proc. Inform. 170, Wadern Germany, 2020, 65] obtained a fixed-parameter algorithm for this problem in the special case where the input is restricted to $K_5$-minor free graphs. In this paper, we answer the question of Bulian and Dawar in the affirmative for general graphs. In fact, we give a more general result capturing elimination distance to any graph class characterized by a finite set of graphs as forbidden induced subgraphs

Efficient processor management strategies for multicomputer systems

Multicomputers are cost-effective alternatives to the conventional supercomputers. Contemporary processor management schemes tend to underutilize the processors and leave many of the processors in the system idle while jobs are waiting for execution;Instead of designing faster processors or interconnection networks, a substantial performance improvement can be obtained by implementing better processor management strategies. This dissertation studies the performance issues related to the processor management schemes and proposes several ways to enhance the multicomputer systems by means of processor management. The proposed schemes incorporate the concepts of size-reduction, non-contiguous allocation, as well as job migration. Job scheduling using a bypass-queue is also studied. All the proposed schemes are proven effective in improving the system performance via extensive simulations. Each proposed scheme has different implementation cost and constraints. In order to take advantage of these schemes, judicious selection of system parameters is important and is discussed

Randomized contractions meet lean decompositions

We show an algorithm that, given an $n$-vertex graph $G$ and a parameter $k$, in time $2^{O(k \log k)} n^{O(1)}$ finds a tree decomposition of $G$ with the following properties: * every adhesion of the tree decomposition is of size at most $k$, and * every bag of the tree decomposition is $(i,i)$-unbreakable in $G$ for every $1 \leq i \leq k$. Here, a set $X \subseteq V(G)$ is $(a,b)$-unbreakable in $G$ if for every separation $(A,B)$ of order at most $b$ in $G$, we have $|A \cap X| \leq a$ or $|B \cap X| \leq a$. The resulting tree decomposition has arguably best possible adhesion size boundsand unbreakability guarantees. Furthermore, the parametric factor in the running time bound is significantly smaller than in previous similar constructions. These improvements allow us to present parameterized algorithms for Minimum Bisection, Steiner Cut, and Steiner Multicut with improved parameteric factor in the running time bound. The main technical insight is to adapt the notion of lean decompositions of Thomas and the subsequent construction algorithm of Bellenbaum and Diestel to the parameterized setting.Comment: v2: New co-author (Magnus) and improved results on vertex unbreakability of bags, v3: final changes, including new abstrac

Finding Optimal Tree Decompositions

The task of organizing a given graph into a structure called a tree decomposition is relevant in multiple areas of computer science. In particular, many NP-hard problems can be solved in polynomial time if a suitable tree decomposition of a graph describing the problem instance is given as a part of the input. This motivates the task of finding as good tree decompositions as possible, or ideally, optimal tree decompositions. This thesis is about finding optimal tree decompositions of graphs with respect to several notions of optimality. Each of the considered notions measures the quality of a tree decomposition in the context of an application. In particular, we consider a total of seven problems that are formulated as finding optimal tree decompositions: treewidth, minimum fill-in, generalized and fractional hypertreewidth, total table size, phylogenetic character compatibility, and treelength. For each of these problems we consider the BT algorithm of Bouchitté and Todinca as the method of finding optimal tree decompositions. The BT algorithm is well-known on the theoretical side, but to our knowledge the first time it was implemented was only recently for the 2nd Parameterized Algorithms and Computational Experiments Challenge (PACE 2017). The author’s implementation of the BT algorithm took the second place in the minimum fill-in track of PACE 2017. In this thesis we review and extend the BT algorithm and our implementation. In particular, we improve the eciency of the algorithm in terms of both theory and practice. We also implement the algorithm for each of the seven problems considered, introducing a novel adaptation of the algorithm for the maximum compatibility problem of phylogenetic characters. Our implementation outperforms alternative state-of-the-art approaches in terms of numbers of test instances solved on well-known benchmarks on minimum fill-in, generalized hypertreewidth, fractional hypertreewidth, total table size, and the maximum compatibility problem of phylogenetic characters. Furthermore, to our understanding the implementation is the first exact approach for the treelength problem