Tight Bounds for Maximal Identifiability of Failure Nodes in Boolean Network Tomography

We study maximal identifiability, a measure recently introduced in Boolean Network Tomography to characterize networks' capability to localize failure nodes in end-to-end path measurements. We prove tight upper and lower bounds on the maximal identifiability of failure nodes for specific classes of network topologies, such as trees and $d$-dimensional grids, in both directed and undirected cases. We prove that directed $d$-dimensional grids with support $n$ have maximal identifiability $d$ using $2d(n-1)+2$ monitors; and in the undirected case we show that $2d$ monitors suffice to get identifiability of $d-1$. We then study identifiability under embeddings: we establish relations between maximal identifiability, embeddability and graph dimension when network topologies are model as DAGs. Our results suggest the design of networks over $N$ nodes with maximal identifiability $\Omega(\log N)$ using $O(\log N)$ monitors and a heuristic to boost maximal identifiability on a given network by simulating $d$-dimensional grids. We provide positive evidence of this heuristic through data extracted by exact computation of maximal identifiability on examples of small real networks

On the proof complexity of Paris-harrington and off-diagonal ramsey tautologies

We study the proof complexity of Paris-Harrington’s Large Ramsey Theorem for bi-colorings of graphs and of off-diagonal Ramsey’s Theorem. For Paris-Harrington, we prove a non-trivial conditional lower bound in Resolution and a non-trivial upper bound in bounded-depth Frege. The lower bound is conditional on a (very reasonable) hardness assumption for a weak (quasi-polynomial) Pigeonhole principle in RES(2). We show that under such an assumption, there is no refutation of the Paris-Harrington formulas of size quasipolynomial in the number of propositional variables. The proof technique for the lower bound extends the idea of using a combinatorial principle to blow up a counterexample for another combinatorial principle beyond the threshold of inconsistency. A strong link with the proof complexity of an unbalanced off-diagonal Ramsey principle is established. This is obtained by adapting some constructions due to Erdos and Mills. ˝ We prove a non-trivial Resolution lower bound for a family of such off-diagonal Ramsey principles

Degree complexity for a modified pigeonhole principle

We consider a modification of the pigeonhole principle, M P H P, introduced by Goerdt in [7]. M P H P is defined over n pigeons and log n holes, and more than one pigeon can go into a hole (according to some rules). Using a technique of Razborov [9] and simplified by Impagliazzo, Pudlak and Sgall [8], we prove that any Polynomial Calculus refutation of a set of polynomials encoding the M P H P, requires degree Omega(log n). We also prove a simple Lemma giving a simulation of Resolution by Polynomial Calculus. Using this lemma, and a Resolution upper bound by Goerdt [7], we obtain that the degree lower bound is tight. Our lower bound establishes the optimality of the tree-like Resolution simulation by the Polynomial Calculus given in [6]

On vanishing sums of roots of unity in polynomial calculus and sum-of-squares

Vanishing sums of roots of unity can be seen as a natural generalization of knapsack from Boolean variables to variables taking values over the roots of unity. We show that these sums are hard to prove for polynomial calculus and for sum-of-squares, both in terms of degree and size.The first author was supported by the MICIN grants PID2019-109137GB-C22 and IJC2018-035334-I, and partially by the grant PID2019-109137GB-C21.Peer ReviewedPostprint (published version

Monotone Proofs of the Pigeon Hole Principle

Lecture Notes in Computer Science. Geneva, Switzerland, July 9-15

A predicative and decidable characterization of the polynomial classes of languages

Characterizations of PTIME, PSPACE, the polynomial hierarchy and its elements are given, which are decidable (membership can be decided by syntactic inspection to the constructions), predicative (according to points of view by Leivant and others), and are obtained by means of increasing restrictions to course-of-values recursion on trees (represented in a dialect of Lisp). (C) 2001 Elsevier Science B.V. All rights reserved

Space proof complexity for random 3-CNFs

We investigate the space complexity of refuting 3-CNFs in Resolution and algebraic systems. We prove that every Polynomial Calculus with Resolution refutation of a random 3-CNF φ in n variables requires, with high probability, distinct monomials to be kept simultaneously in memory. The same construction also proves that every Resolution refutation of φ requires, with high probability, clauses each of width to be kept at the same time in memory. This gives a lower bound for the total space needed in Resolution to refute φ. These results are best possible (up to a constant factor) and answer questions about space complexity of 3-CNFs