Parameterized bounded-depth Frege is not optimal

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's

On the Algebraic Proof Complexity of Tensor Isomorphism

The Tensor Isomorphism problem (TI) has recently emerged as having connections to multiple areas of research within complexity and beyond, but the current best upper bound is essentially the brute force algorithm. Being an algebraic problem, TI (or rather, proving that two tensors are non-isomorphic) lends itself very naturally to algebraic and semi-algebraic proof systems, such as the Polynomial Calculus (PC) and Sum of Squares (SoS). For its combinatorial cousin Graph Isomorphism, essentially optimal lower bounds are known for approaches based on PC and SoS (Berkholz & Grohe, SODA'17). Our main results are an Ω(n) lower bound on PC degree or SoS degree for Tensor Isomorphism, and a nontrivial upper bound for testing isomorphism of tensors of bounded rank. We also show that PC cannot perform basic linear algebra in sub-linear degree, such as comparing the rank of two matrices (which is essentially the same as 2-TI), or deriving BA = I from AB = I. As linear algebra is a key tool for understanding tensors, we introduce a strictly stronger proof system, PC+Inv, which allows as derivation rules all substitution instances of the implication AB = I → BA = I. We conjecture that even PC+Inv cannot solve TI in polynomial time either, but leave open getting lower bounds on PC+Inv for any system of equations, let alone those for TI. We also highlight many other open questions about proof complexity approaches to TI

Adequação Estratégica das Áreas de Trade Marketing das Empresas de Bens de Consumo Atuando no Brasil

A atuação diferenciada nos pontos de venda torna-se cada vez mais relevante para a efetividade das cadeias de suprimentos da indústria de bens de consumo. Como resposta a este desafio surge, nas empresas, a área de trade marketing (TM), cujo objetivo é unificar as estratégias de marketing da indústria e do varejo no que diz respeito à sua efetiva implantação dentro do ponto de venda. O objetivo deste trabalho é identificar quais os atributos que dificultam e quais aqueles que contribuem para que a área de trade marketing esteja adequada à estratégia de negócio das empresas de bens de consumo. O estudo foi realizado por meio de uma pesquisa exploratória descritiva, cujo instrumento de coleta foi um questionário semiestruturado, respondido por 36 executivos de 20 empresas de diferentes segmentos de bens de consumo não duráveis. O resultado obtido com a pesquisa possibilitou a validação da lista de atributos definida a partir do ferramental de análise estratégica Genérico Integrativo (GI) de Macedo-Soares (2001), além da construção de listas adicionais de referência que podem ajudar para uma construção mais coerente e sólida da área de Trade Marketing dentro das organizações do segmento de bens de consumo