Stratified Static Analysis Based on Variable Dependencies

In static analysis by abstract interpretation, one often uses widening operators in order to enforce convergence within finite time to an inductive invariant. Certain widening operators, including the classical one over finite polyhedra, exhibit an unintuitive behavior: analyzing the program over a subset of its variables may lead a more precise result than analyzing the original program! In this article, we present simple workarounds for such behavior

Uncapacitated Flow-based Extended Formulations

An extended formulation of a polytope is a linear description of this polytope using extra variables besides the variables in which the polytope is defined. The interest of extended formulations is due to the fact that many interesting polytopes have extended formulations with a lot fewer inequalities than any linear description in the original space. This motivates the development of methods for, on the one hand, constructing extended formulations and, on the other hand, proving lower bounds on the sizes of extended formulations. Network flows are a central paradigm in discrete optimization, and are widely used to design extended formulations. We prove exponential lower bounds on the sizes of uncapacitated flow-based extended formulations of several polytopes, such as the (bipartite and non-bipartite) perfect matching polytope and TSP polytope. We also give new examples of flow-based extended formulations, e.g., for 0/1-polytopes defined from regular languages. Finally, we state a few open problems

On the extension complexity of combinatorial polytopes

In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large number of graphs, including those used in quantum information and suspensions of cubic planar graphs.Comment: 15 pages, 3 figures, 2 table

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the {\L}ojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in \O_n using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of {\L}ojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the {\L}ojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.Comment: 25 pages. Updated with minor change

Comparative morphology of configurations with reduced part count derived from the octahedral-tetrahedral truss

Morphology (the study of structure and form) of the octahedral-tetrahedral (octet) truss is described. Both the geometry and symmetry of the octet truss are considered. Morphological techniques based on symmetry operations are presented which enable the derivation of reduced-part-count truss configurations from the octet truss by removing struts and nodes. These techniques are unique because their Morphological origination and they allow for the systematic generation and analysis of a large variety of structures. Methods for easily determining the part count and redundancy of infinite truss configurations are presented. Nine examples of truss configurations obtained by applying the derivation techniques are considered. These configurations are structurally stable while at the same time exhibiting significant reductions in part count. Some practical and analytical considerations, such as structural performance, regarding the example reduced-part-count truss geometries are briefly discussed

Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP$_{{\mathbb R}}$ under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201