35 research outputs found
Courcelle\u27s Theorem: Overview and Applications
Courcelle\u27s Theorem states that any graph property expressible in monadic second order logic can be decidedin O(f(k)n) for graphs of treewidth k. This paper gives a broad overview of how this theorem is proved and outlines tools available to help express graph properties in monadic second order logic
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
Waiter–Client and Client–Waiter games
In this thesis, we consider two types of positional games; - and - games. Each round in a biased (:) game begins with Waiter offering a+b free elements of the board to Client. Client claims elements among these and the remaining elements are claimed by Waiter. Waiter wins in a Waiter-Client game if he can force Client to fully claim a , otherwise Client wins. In a Client-Waiter game, Client wins if he can claim a winning set himself, else Waiter wins.
We estimate the of four different (:) Waiter-Client and Client-Waiter games. This is the unique value of Waiter's bias at which the player with a winning strategy changes. We find its asymptotic value for both versions of the complete-minor and non-planarity games and give bounds for both versions of the non--colourability and -SAT games. Our results show that these games exhibit a heuristic called the .
We also find sharp probability thresholds for the appearance of a graph in the random graph (,) on which Waiter and Client win the (:) Waiter-Client and Client-Waiter Hamiltonicity games respectively
The complexity of approximating bounded-degree Boolean #CSP
AbstractThe degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of {0}, {1} and binary implication. Otherwise, there is no FPRAS unless NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs
Faster Detours in Undirected Graphs
The -Detour problem is a basic path-finding problem: given a graph on
vertices, with specified nodes and , and a positive integer , the
goal is to determine if has an -path of length exactly , where is the length of a shortest path from to
. The -Detour problem is NP-hard when is part of the input, so
researchers have sought efficient parameterized algorithms for this task,
running in time, for as slow-growing as possible.
We present faster algorithms for -Detour in undirected graphs, running in
randomized and deterministic
time. The previous fastest algorithms for this problem took randomized and deterministic time
[Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact
that detecting a path of a given length in an undirected graph is easier if we
are promised that the path belongs to what we call a "bipartitioned" subgraph,
where the nodes are split into two parts and the path must satisfy constraints
on those parts. Previously, this idea was used to obtain the fastest known
algorithm for finding paths of length in undirected graphs
[Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017].
Our work has direct implications for the -Longest Detour problem: in this
problem, we are given the same input as in -Detour, but are now tasked with
determining if has an -path of length at least
Our results for k-Detour imply that we can solve -Longest Detour in randomized and deterministic time.
The previous fastest algorithms for this problem took
randomized and deterministic time [Fomin et al.,
STACS 2022]
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