Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction Problem over a fixed template is solvable in polynomial time if the algebra of polymorphisms associated to the template lies in a Taylor variety, and is NP-complete otherwise. This paper provides two new characterizations of finitely generated Taylor varieties. The first characterization is using absorbing subalgebras and the second one cyclic terms. These new conditions allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the authors) and the characterization of locally finite Taylor varieties using weak near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and self-contained way

Distributed Decision Through Self-Synchronizing Sensor Networks in the Presence of Propagation Delays and Asymmetric Channels

In this paper we propose and analyze a distributed algorithm for achieving globally optimal decisions, either estimation or detection, through a self-synchronization mechanism among linearly coupled integrators initialized with local measurements. We model the interaction among the nodes as a directed graph with weights (possibly) dependent on the radio channels and we pose special attention to the effect of the propagation delay occurring in the exchange of data among sensors, as a function of the network geometry. We derive necessary and sufficient conditions for the proposed system to reach a consensus on globally optimal decision statistics. One of the major results proved in this work is that a consensus is reached with exponential convergence speed for any bounded delay condition if and only if the directed graph is quasi-strongly connected. We provide a closed form expression for the global consensus, showing that the effect of delays is, in general, the introduction of a bias in the final decision. Finally, we exploit our closed form expression to devise a double-step consensus mechanism able to provide an unbiased estimate with minimum extra complexity, without the need to know or estimate the channel parameters.Comment: To be published on IEEE Transactions on Signal Processin

Symmetric Determinantal Representation of Formulas and Weakly Skew Circuits

We deploy algebraic complexity theoretic techniques for constructing symmetric determinantal representations of for00504925mulas and weakly skew circuits. Our representations produce matrices of much smaller dimensions than those given in the convex geometry literature when applied to polynomials having a concise representation (as a sum of monomials, or more generally as an arithmetic formula or a weakly skew circuit). These representations are valid in any field of characteristic different from 2. In characteristic 2 we are led to an almost complete solution to a question of B\"urgisser on the VNP-completeness of the partial permanent. In particular, we show that the partial permanent cannot be VNP-complete in a finite field of characteristic 2 unless the polynomial hierarchy collapses.Comment: To appear in the AMS Contemporary Mathematics volume on Randomization, Relaxation, and Complexity in Polynomial Equation Solving, edited by Gurvits, Pebay, Rojas and Thompso

Forest matrices around the Laplacian matrix

We study the matrices Q_k of in-forests of a weighted digraph G and their connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representations for the generalized inverses, the powers, and some eigenvectors of L. The normalized in-forest matrices are row stochastic; the normalized matrix of maximum in-forests is the eigenprojection of the Laplacian matrix, which provides an immediate proof of the Markov chain tree theorem. A source of these results is the fact that matrices Q_k are the matrix coefficients in the polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's matrices for -L. Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection; Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic Graph Theor