Algebras for Tree Decomposable Graphs

Complex problems can be sometimes solved efficiently via recursive decomposition strategies. In this line, the tree decomposition approach equips problems modelled as graphs with tree-like parsing structures. Following Milner’s flowgraph algebra, in a previous paper two of the authors introduced a strong network algebra to represent open graphs (up to isomorphism), so that homomorphic properties of open graphs can be computed via structural recursion. This paper extends this graphical-algebraic foundation to tree decomposable graphs. The correspondence is shown: (i) on the algebraic side by a loose network algebra, which relaxes the restriction reordering and scope extension axioms of the strong one; and (ii) on the graphical side by Milner’s binding bigraphs, and elementary tree decompositions. Conveniently, an interpreted loose algebra gives the evaluation complexity of each graph decomposition. As a key contribution, we apply our results to dynamic programming (DP). The initial statement of the problem is transformed into a term (this is the secondary optimisation problem of DP). Noting that when the scope extension axiom is applied to reduce the scope of the restriction, then also the complexity is reduced (or not changed), only so-called canonical terms (in the loose algebra) are considered. Then, the canonical term is evaluated obtaining a solution which is locally optimal for complexity. Finding a global optimum remains an NP-hard problem

The standard graded property for vertex cover algebras of Quasi-Trees

J. Herzog, T. Hibi, N. V. Trung and X. Zheng characterize the vertex cover algebras which are standard graded. In this paper we give a simple combinatorial criterion for the standard graded property of vertex cover algebras in the case of quasi-trees. We also give an example of how this criterion works and compute the maximal degree of a minimal generator in that case

Minimal generating set for semi-invariants of quivers of dimension two

A minimal (by inclusion) generating set for the algebra of semi-invariants of a quiver of dimension (2,...,2) is established over an infinite field of arbitrary characteristic. The mentioned generating set consists of the determinants of generic matrices and the traces of tree paths of pairwise different multidegrees, where in the case of characteristic different from two we take only admissible paths. As a consequence, we describe relations modulo decomposable semi-invariants.Comment: 30 pages; v2. Examples and applications are added. Some notations and definitions are change

Polar syzygies in characteristic zero: the monomial case

Given a set of forms f={f_1,...,f_m} in R=k[x_1,...,x_n], where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module D(f), whose elements are called differential syzygies of f. There is a distinct submodule P of Z coming from the polynomial relations of f through its transposed Jacobian matrix, the elements of which are called polar syzygies of f. We say that f is polarizable if equality P=Z holds. This paper is concerned with the situation where f are monomials of degree 2, in which case one can naturally associate to them a graph G(f) with loops and translate the problem into a combinatorial one. A main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra k[f] of R and that the converse holds provided the graph G(f) is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of G(f) has diameter at most 2 then f is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.Comment: 33 pages, 15 figure