Decision trees, monotone functions, and semimatroids

We define decision trees for monotone functions on a simplicial complex. We define homology decidability of monotone functions, and show that various monotone functions related to semimatroids are homology decidable. Homology decidability is a generalization of semi-nonevasiveness, a notion due to Jonsson. The motivating example is the complex of bipartite graphs, whose Betti numbers are unknown in general. We show that these monotone functions have optimum decision trees, from which we can compute relative Betti numbers of related pairs of simplicial complexes. Moreover, these relative Betti numbers are coefficients of evaluations of the Tutte polynomial, and every semimatroid collapses onto its broken circuit complex.Comment: 16 page

Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201

Connected Components Labeling on DRAGs

In this paper we introduce a new Connected Components Labeling (CCL) algorithm which exploits a novel approach to model decision problems as Directed Acyclic Graphs with a root, which will be called Directed Rooted Acyclic Graphs (DRAGs). This structure supports the use of sets of equivalent actions, as required by CCL, and optimally leverages these equivalences to reduce the number of nodes (decision points). The advantage of this representation is that a DRAG, differently from decision trees usually exploited by the state-of-the-art algorithms, will contain only the minimum number of nodes required to reach the leaf corresponding to a set of condition values. This combines the benefits of using binary decision trees with a reduction of the machine code size. Experiments show a consistent improvement of the execution time when the model is applied to CCL

The complexity of dominating set reconfiguration

Suppose that we are given two dominating sets $D_s$ and $D_t$ of a graph $G$ whose cardinalities are at most a given threshold $k$. Then, we are asked whether there exists a sequence of dominating sets of $G$ between $D_s$ and $D_t$ such that each dominating set in the sequence is of cardinality at most $k$ and can be obtained from the previous one by either adding or deleting exactly one vertex. This problem is known to be PSPACE-complete in general. In this paper, we study the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem remains PSPACE-complete even for planar graphs, bounded bandwidth graphs, split graphs, and bipartite graphs. We then give a general scheme to construct linear-time algorithms and show that the problem can be solved in linear time for cographs, trees, and interval graphs. Furthermore, for these tractable cases, we can obtain a desired sequence such that the number of additions and deletions is bounded by $O(n)$, where $n$ is the number of vertices in the input graph

Abstract and Concrete Decision Graphs for Choosing Extensions of Argumentation Frameworks

Most argumentation semantics allow for multiple extensions, which raises the question of how to choose among extensions. We propose to study this question as a decision problem. Inspired by decision trees commonly used in economics, we introduce the notion of a decision graph for deciding between the multiple extensions of a given AF in a given semantics. We distinguish between abstract decision graphs and concrete instantiations thereof. Inspired by the principle-based approach to argumentation, we formulate two principles that mappings from argumentation frameworks to decision graphs should satisfy, the principles of decision-graph directionality and that of directional decision-making. We then propose a concrete instantiation of decision graphs, which satisfies one of these principles. Finally, we discuss the potential for further research based on this novel methodology

Abstract and Concrete Decision Graphs for Choosing Extensions of Argumentation Frameworks - Technical Report

Most argumentation semantics allow for multiple extensions, which raises the question of how to choose among extensions. We propose to study this question as a decision problem. Inspired by decision trees commonly used in economics, we introduce the notion of a decision graph for deciding between the multiple extensions of a given AF in a given semantics. We distinguish between abstract decision graphs and concrete instantiations thereof. Inspired by the principle-based approach to argumentation, we formulate two principles that mappings from argumentation frameworks to decision graphs should satisfy, the principle of decision-graph directionality and the one of directional decision-making. We then propose a concrete instantiation of decision graphs, which satisfies one of these principles. Finally, we discuss the potential for further research based on this novel methodology

О вычислительной эффективности одного алгоритма для нахождения остовного леса графа с минимальным (максимальным) весом

In work effective realization of «greedy» algorithm for finding minimum (maximum) spanning woods (trees) of an undirected weighed graph is considered. Is given the rating of the expected computing time of algorithm is 0 (M), where M — number of edges in a graph. Is shown, that the offered algorithm is better than a Prim’s algorithm for graphs with number of edges less, than N2/6, where N — number of vertices in a graph. The experimental research of algorithm on the graphs, containing from 499500 up to 71994000 edges, has shown its high computing efficiency and his can be recommended for the decision of practical problems on rarefied graphs or networks of the big dimension