(1 + Ω(1))-Αpproximation to MAX-CUT Requires Linear Space

Abstract

We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. We show that there exists a constant * > 0 such that any randomized streaming algorithm that computes a (1 + * )-approximation to MAX-CUT requires Ω(n) space on an n vertex graph. By contrast, there are algorithms that produce a (1 + )-approximation in space O(n/ 2 ) for every > 0. Our result is the first linear space lower bound for the task of approximating the max cut value and partially answers an open question from the literature [2]. The prior state of the art ruled out (2 − )-approximation inÕ( √ n) space or (1 + )-approximation in n 1−O( ) space, for any > 0.Previous lower bounds for the MAX-CUT problem relied, in essence, on a lower bound on the communication complexity of the following task: Several players are each given some edges of a graph and they wish to determine if the union of these edges is -close to forming a bipartite graph, using one-way communication. The previous works proved a lower bound of Ω( √ n) for this task when = 1/2, and n 1−O( ) for every > 0, even when one of the players is given a candidate bipartition of the graph and the graph is promised to be bipartite with respect to this partition or -far from bipartite. This added information was essential in enabling the previous analyses but also yields a weak bound since, with this extra information, there is an n 1−O( ) communication protocol for this problem. In this work, we give an Ω(n) lower bound on the communication complexity of the original problem (without the extra information) for = Ω(1) in the three-player setting. Obtaining this Ω(n) lower bound on the communication complexity is the main technical result in this paper. We achieve it by a

An optimal space lower bound for approximating MAX-CUT

“…Implicit Hidden Partition (IHP) Problem of [KKSV17]. We define a parametrized class of problems IHP(n, α, T ) for positive integers n and T and real α ∈ (0, 1/2) as follows: IHP(n, α, T ) is a T -player problem with public inputs M t ∈ {0, 1} αn×n being incidence matrices of matchings (so their rows sum to 2 and columns sum to at most 1), and the private inputs are w t ∈ {0, 1} αn .…”

“…Our main technical contribution is a nearly optimal lower bound on the communication complexity of the Implicit Hidden Partition problem introduced in [KKSV17]. The implicit hidden partition problem is a multiple-player communication game where many players are given labellings of sparse subsets of edges and must determine if they are consistent with a bipartition of vertices.…”

“…The main technical tool underlying our analysis is a novel and general way of using the convolution theorem in Fourier analysis to analyze information conveyed by the combined messages of multiple players (which corresponds to the intersection of their individual messages). While this idea has recently been used for proving lower bounds on the streaming complexity of MAX-CUT [KKSV17] and the sketching complexity of subgraph counting [KKP18], in both of these works the convolution theorem is applied in a rather restricted setting. Specifically, the structure of the Fourier transforms of the messages of the players is such that convolution simply amounts to multiplication in Fourier domain, i.e.…”

“…In our setting convolving the Fourier transforms of the players' messages leads to contributions across different levels of the weight spectrum, and analyzing such processes requires a new technique. Our main insight is the idea of controlling the ℓ 1 norm of the Fourier transform of the intersection of the players' messages as opposed to the ℓ 2 norm, bounds on which follow more naturally as a consequence of hypercontractivity (note that ℓ 2 bounds on various levels of the Fourier spectrum that follow from the hypercontractive inequality underlie the analysis of the Boolean Hidden Matching problem of [GKK + 08], as well as recent works on streaming and sketching lower bounds through Fourier analysis [KKS15,KK14,KKSV17,KKP18]). Conceptually, the idea of controlling the ℓ 1 norm stems from the fact that since individual players receive parity information of some hidden vector X across edges of a sparse graph (a matching), strong upper bounds on the ℓ 1 norm of the Fourier transform of the corresponding player's message follow (due to sparsity of the graph), and these ℓ 1 bounds remain approximately preserved when the players' functions are multiplied (i.e.…”

Testing Graph Clusterability: Algorithms and Lower Bounds

“…The reduction to MAX-CUT follows using rather standard techniques (e.g. is very similar to [KKSV17]; see Section 4.2.2). The reduction to PartitionTesting (1, Ω(1), ǫ, 1) is more delicate and novel: the difficulty is that we need to ensure that the introduction of random noise Z e on the edge labels produces graphs that have the expansion property (in contrast, the MAX-CUT reduction produces graphs with a linear fraction of isolated nodes).…”