Common Information and Unique Disjointness

Abstract

We provide a new framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in [1]. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we can compute the (almost) exact common information of UDISJ partial matrix. We also establish robustness of this estimation under various perturbations of the UDISJ partial matrix, where rows and columns are randomly or adversarially removed or where entries are randomly or adversarially altered. This robustness translates, via a variant of Yannakakis' Factorization Theorem, to lower bounds on the average case and adversarial approximate extension complexity. We present the first family of polytopes, the hard pair introduced in [2] related to the CLIQUE problem, with high average case and adversarial approximate extension complexity. We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

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“…This suggests that perhaps an idea analogous to our and-amplification-based communication lower bound proof could be used to "bootstrap" the result of [6] to get the tight lower bound for maximum-clique. In Section 5 we confirm that this is indeed the case, thus obtaining a proof of the tight bound that is simpler than the ones given in [8,7] and avoids their use of information-theoretic methods (instead relying only on the corruption-based argument of [6]). …”

“…Combining this result with Corollary 1.2 yields an alternative proof of Theorem 1.3 and shows the class equality SBP = USBP. Furthermore, this result implies that the tight extended formulation lower bound for maximum-clique (discussed in Section 1.5 and Section 5) can be derived from Razborov's corruption lemma [35] in a black-box way, without needing any of the machinery in [6,8,7]: Specifically, any ε-UDISJ matrix having nonnegative rank r can be interpreted as witnessing R priv α, (1 − ε)α (UDISJ) ≤ log r + O(1) (for some α > 0). This protocol can be and-amplified to witness USBP(UDISJ) ≤ O((log r)/ε), and thus SBP(UDISJ) ≤ O((log r)/ε + log n).…”

“…The lower bound in (i) was improved to 2 Ω(εn) in [8], and a simpler alternative proof of this stronger bound was given in [7]. Combining such nonnegative rank lower bounds with (ii) yields lower bounds on the approximate extended formulation complexity for maximum-clique.…”

“…In [8], this result was improved using information-theoretic methods to show a tight lower bound for maximum-clique: for every positive constant δ < 1, such n 1−δ -approximate extended formulations have complexity 2 Ω(n δ ) . In [7], a simplified proof of the latter result was given, also using information-theoretic methods.…”

“…Both of the subsequent papers [8,7] employed information-theoretic methods. We now show that an idea analogous to our and-amplification-based communication lower bound proof (Theorem 1.4 and Theorem 1.5) can be used to "bootstrap" (i) to get the stronger 2 Ω(εn) lower bound, thus bypassing the information-theoretic methods of [8,7].…”