An Exponential Lower Bound on OBDD Refutations for Pigeonhole Formulas

Abstract

Haken proved that every resolution refutation of the pigeonhole formula has at least exponential size. Groote and Zantema proved that a particular OBDD computation of the pigeonhole formula has an exponential size. Here we show that any arbitrary OBDD refutation of the pigeonhole formula has an exponential size, too: we prove that the size of one of the intermediate OBDDs is Ω(1.025 n ).

Fork-forests in bi-colored complete bipartite graphs

“…Finding f (G, c) allows to determine the corresponding parameter in matrices and to prove the conjecture of Tveretina et al in [6]. In particular, our result f (n) = (1 − 1 √ 2 )n is an improvement of the previously known bound f (n) ≥ 1 2 (1 − 1 √ 2 )n from [6]. This in turn improves the lower bound on resolution for ordered binary decision diagrams.…”

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Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2-edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G = K n,n are colored with black and white such that the number of black edges differs from the number of white edges by at most 1, then there are at least n(1 − 1/ √ 2) vertex-disjoint forks with centers in the same partite set of G. Here, a fork is a graph formed by two adjacent edges of different colors.

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Ordered Binary Decision Diagrams, Pigeonhole Formulas and Beyond1

“…Lemma 1 was presented for the first time in [16], but with a smaller coefficient c = 1 2 − 1 4 √ 2 ≈ 0.146. This lemma is of interest from a point of view of Ramsey Theory that typically asks questions of the form: How many elements of some structure must there be to guarantee that a particular property will hold?…”

“…Lemma 1 presents another combinatorial property of a matrix containing entries equally colored white and black. In comparison with [16] we present another proof that gives us a better c = • One can choose m rows, and in every of these rows a white and a black entry, such that all these 2m entries are in different columns.…”

“…It was proven for the first time in [16] that OBDD proof systems with the two rules Axiom and Join, corresponding to the Apply method, have an exponential lower bound on refutations of the pigeonhole formula. However, the lower bound Ω(1.14 n ) presented in this paper is stricter in comparison with Ω(1.025 n ) in [16]. We also demonstrate a family of CNFs that requires exponential increase for all OBDD refutations based on Apply method, i.e.…”