Extension Complexity of Independent Set Polytopes

Mika Göös, Rahul Jain, Thomas Watson
2018 SIAM journal on computing (Print)  
We exhibit an n-node graph whose independent set polytope requires extended formulations of size exponential in Ω(n/ log n). Previously, no explicit examples of n-dimensional 0/1-polytopes were known with extension complexity larger than exponential in Θ( √ n). Our construction is inspired by a relatively little-known connection between extended formulations and (monotone) circuit depth. Our approach Curiously enough, an analogous √ n-frontier existed in the seemingly unrelated field of
more » ... circuits: Raz and Wigderson [RW92] proved an Ω(m) lower bound for the depth of any monotone circuit computing the matching function on m 2 input bits. This remained the largest monotone depth bound for an explicit function until the recent work of Göös and Pitassi [GP14], who exhibited a function with monotone depth Ω(n/ log n). In short, our idea is to prove an extension complexity analogue of this latter result. The conceptual inspiration for our construction is a relatively little-known connection between Karchmer-Wigderson games [KW88] (which characterize circuit depth) and extended formulations. This "KW/EF connection" (see Section 2 for details) was pointed out by Hrubeš [Hru12] as a nonnegative analogue of a classic rank-based method of Razborov [Raz90] . In this work, we focus only on the monotone setting. For any monotone f : {0, 1} n → {0, 1} we can study the convex hull of its 1-inputs, namely, the polytope The upshot of the KW/EF connection is that extension complexity lower bounds for F follow from a certain type of strengthening of monotone depth lower bounds for f . For example, using this connection, it turns out that Rothvoß's result [Rot14] implies the result of Raz and Wigderson [RW92] in a simple black-box fashion (Section 2.3). Our main technical result is to strengthen the existing monotone depth lower bound from [GP14] into a lower bound for the associated polytope (though we employ substantially different techniques than were used in that paper). The key communication search problem studied in [GP14] is a communication version of the well-known Tseitin problem (see Section 3 for definitions), which has especially deep roots in proof complexity (e.g., [Juk12, §18.7]) and has also been studied in query complexity [LNNW95] . We use information complexity techniques to prove the required Ω(n/ log n) communication lower bound for the relevant variant of the Tseitin problem; information theoretic tools have been used in extension complexity several times [BM13, BP13, BP15]. One relevant work is Huynh and Nordström [HN12] (predecessor to [GP14]), whose information complexity arguments we extend in this work. (Instead of using information complexity, an alternative seemingly promising approach would be to "lift" a strong enough query complexity lower bound for Tseitin into communication complexity. Unfortunately, this approach runs into problems due to limitations in existing query-tocommunication simulation theorems; we discuss this in Section 7.) Theorem 1 follows by reductions from the result for Tseitin (Section 4). Indeed, it was known that the Tseitin problem reduces to the monotone KW game associated with an f : {0, 1} O(n) → {0, 1} that encodes (in a monotone fashion) a certain CSP satisfiability problem. This gives us an extension complexity lower bound for the (explicit) polytope F := conv f −1 (1). As a final step, we give a reduction from F to an independent set polytope. Background Let M be a nonnegative matrix. The nonnegative rank of M , denoted rk + (M ), is the minimum r such that M can be decomposed as a sum i∈[r] R i where each R i is a rank-1 nonnegative matrix. Randomized protocols. Faenza et al. [FFGT14] observed that a nonnegative rank decomposition can be naturally interpreted as a type of randomized protocol that computes the matrix M "in expectation". We phrase this connection precisely as follows: log rk + (M ) + Θ(1) is the minimum communication cost of a private-coin protocol Π whose acceptance probability on each input (x, y)
doi:10.1137/16m109884x fatcat:zzxykn7kojbhjc4f7nxebzs43e