Linear MIM-Width of Trees

We provide an $O(n \log n)$ algorithm computing the linear maximum induced matching width of a tree and an optimal layout.Comment: 19 pages, 7 figures, full version of WG19 paper of same nam

On the Linear MIM-width of Trees

Linear MIM-width, and its generalization MIM-width, is a graph width parameter that has become noted for having bounded value on several important graph classes, e.g. interval graphs and permutation graphs. The linear MIM-width of a graph G measures a min-max relation on all maximum induced matchings in bipartite graphs given by a linear layout of the vertices in G, over all possible linear layouts. In this thesis we give an overlook of some of the research that has been done on this parameter, and provide a new result, computing the linear MIM-width of trees in n log n time.Masteroppgåve i informatikkINF399MAMN-PROGMAMN-IN

On the Linear MIM-width of Trees

Linear MIM-width, and its generalization MIM-width, is a graph width parameter that has become noted for having bounded value on several important graph classes, e.g. interval graphs and permutation graphs. The linear MIM-width of a graph G measures a min-max relation on all maximum induced matchings in bipartite graphs given by a linear layout of the vertices in G, over all possible linear layouts. In this thesis we give an overlook of some of the research that has been done on this parameter, and provide a new result, computing the linear MIM-width of trees in n log n time.Masteroppgåve i informatikkINF399MAMN-PROGMAMN-IN

Linear MIM-width of the Square of Trees

Graph parameters measure the amount of structure (or lack thereof) in a graph that makes it amenable to being decomposed in a way that facilitates dynamic programming. Graph decompositions and their associated parameters are important both in practice (as a tool for designing robust algorithms for NP-hard problems) and in theory (relating large classes of problems to the graphs on which they are solvable in polynomial time). Linear MIM-width is a variant of the graph parameter MIM-width, introduced by Vatshelle. MIM-width is a parameter that is constant for many classes of graphs. Most graph classes which have been shown to have constant MIM-width also have constant linear MIM-width. However, computing the (linear) MIM-width of graphs, or showing that it is hard, has proven to be a huge challenge. To date, the only graph class with unbounded linear MIM-width, whose linear MIM-width can be computed in polynomial time, is the trees. In this follow-up, we show that for any tree $T$ with linear MIM-width $k$, the linear MIM-width of its square $T^2$ always lies between $k$ and $2k$, and that these bounds are tight for all $k$.Comment: 11 pages. To appear in NIKT 202

More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width

More applications of the d-neighbor equivalence: acyclicity and connectivity constraints

In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that solves all these problems and whose running time is upper bounded by $2^{O(k)}\cdot n^{O(1)}$, $2^{O(k \log(k))}\cdot n^{O(1)}$, $2^{O(k^2)}\cdot n^{O(1)}$ and $n^{O(k)}$ where $k$ is respectively the clique-width, $\mathbb{Q}$-rank-width, rank-width and maximum induced matching width of a given decomposition. Our meta-algorithm simplifies and unifies the known algorithms for each of the parameters and its running time matches asymptotically also the running times of the best known algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the $d$-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance of this equivalence relation on the algorithmic applications of width measures. We also prove that our framework could be useful for $W[1]$-hard problems parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For these latter problems, we obtain $n^{O(k)}$, $n^{O(k)}$ and $n^{2^{O(k)}}$ time algorithms where $k$ is respectively the clique-width, the $\mathbb{Q}$-rank-width and the rank-width of the input graph

Understanding the complexity of #SAT using knowledge compilation

Two main techniques have been used so far to solve the #P-hard problem #SAT. The first one, used in practice, is based on an extension of DPLL for model counting called exhaustive DPLL. The second approach, more theoretical, exploits the structure of the input to compute the number of satisfying assignments by usually using a dynamic programming scheme on a decomposition of the formula. In this paper, we make a first step toward the separation of these two techniques by exhibiting a family of formulas that can be solved in polynomial time with the first technique but needs an exponential time with the second one. We show this by observing that both techniques implicitely construct a very specific boolean circuit equivalent to the input formula. We then show that every beta-acyclic formula can be represented by a polynomial size circuit corresponding to the first method and exhibit a family of beta-acyclic formulas which cannot be represented by polynomial size circuits corresponding to the second method. This result shed a new light on the complexity of #SAT and related problems on beta-acyclic formulas. As a byproduct, we give new handy tools to design algorithms on beta-acyclic hypergraphs