A new counting polynomial, called the “Omega” Ω(G, x) polynomial, is proposed on the ground of quasi-orthogonal cut “qoc” edge strips in a bipartite lattice. Within a qoc not all cut edges are necessarily orthogonal, meaning not all are pairwise codistant. Two topological indices: CI (Cluj-Ilmenau), eventually equal to the well-known PI index, in planar, bipartite graphs and IΩ are defined on t...

This paper concerns the uniform random generation and the approximate counting of combinatorial structures admitting an ambiguous description. We propose a general framework to study the complexity of these problems and present some applications to specific classes of languages. In particular, we give a uniform random generation algorithm for finitely ambiguous contextfree languages of the same...

We give a quantum-inspired Opnq algorithm computing the Tutte polynomial of a lattice path matroid, where n is the size of the ground set of the matroid. Furthermore, this can be improved to Opnq arithmetic operations if we evaluate the Tutte polynomial on a given input, fixing the values of the variables. The best existing algorithm, found in 2004, was Opnq, and the problem has only been known...

The parallel complexity class NC has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. [CMTV98] considered arithmetizations of two of these classes, #NC and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in FLo...

We present a novel extension to the permutation group enumeration technique which is well known to have polynomial time algorithms. We use this extended technique to count all the perfect matchings in a bipartite graph. We further show how this technique can be used for solving a class of #P-complete counting problems by NC-algorithms, where the solution space of the associated search problems ...

We study a problem of classiication of linear homogeneous second-order ODE's with polynomial coeecients based on qualitative properties of singularities. The corresponding combinatorial problem of counting the number of classes is then solved in terms of the initial number of singularities. Un probl eme combinatoire en classiication des EDO lin eaires du second ordre R esum e Nous etudions un p...

We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [26, 33, 27], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [23]. To interpolate between the hard case of counting matchings and the eas...

The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to derive the formulas for counting the sizes of Markov equivalence classes. We ...

For every polynomial q, we present worst-case to average-case (almost-linear-time) reductions for a class of problems in P that are widely conjectured not to be solvable in time q. These classes contain, for example, the problems of counting the number of k-cliques in a graph, for any fixed k ≥ 3. In general, we consider the class of problems that consist of counting the number of local neighbo...

A variety of counting problems on 3-regular planar graphs are considered in this paper. We give a sufficient condition which guarantees that the coefficients of a homogeneous polynomial can be uniquely determined by its values on a recurrence sequence. This result enables us to use the polynomial interpolation technique in high dimension to prove the #P-completeness of problems on graphs with s...