We consider the problem of estimating the value of MAX-CUT in a graph in the streaming model of computation. At one extreme, there is a trivial 2-approximation for this problem that uses only O(log n) space, namely, count the number of edges and output half of this value as the estimate for the size of the MAX-CUT. On the other extreme, for any fixed ε > 0, if one allowsÕ(n) space, a (1 + ε)-approximate solution to the MAX-CUT value can be obtained by storing anÕ(n)-size sparsifier that essentially preserves MAX-CUT value.Our main result is that any (randomized) single pass streaming algorithm that breaks the 2-approximation barrier requires Ω(n)-space, thus resolving the space complexity of any non-trivial approximations of the MAX-CUT value to within polylogarithmic factors in the single pass streaming model. We achieve the result by presenting a tight analysis of the Implicit Hidden Partition Problem introduced by Kapralov et al. [SODA'17] for an arbitrarily large number of players. In this problem a number of players receive random matchings of Ω(n) size together with random bits on the edges, and their task is to determine whether the bits correspond to parities of some hidden bipartition, or are just uniformly random.Unlike all previous Fourier analytic communication lower bounds, our analysis does not directly use bounds on the ℓ 2 norm of Fourier coefficients of a typical message at any given weight level that follow from hypercontractivity. Instead, we use the fact that graphs received by players are sparse (matchings) to obtain strong upper bounds on the ℓ 1 norm of the Fourier coefficients of the messages of individual players using their special structure, and then argue, using the convolution theorem, that similar strong bounds on the ℓ 1 norm are essentially preserved (up to an exponential loss in the number of players) once messages of different players are combined. We feel that our main technique is likely of independent interest.

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In this paper, we provide new proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime $$\Delta <1$$ Δ < 1 . As an application, we provide a short, fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2. This latter result is at the base of a number of recent results, in particular the rigorous proof of continuity/discontinuity of the phase transition of the random-cluster model, the localization/delocalization behaviour of the six-vertex height function when $$a=b=1$$ a = b = 1 and $$c\ge 1$$ c ≥ 1 , and the rotational invariance of the six-vertex model and the Fortuin–Kasteleyn percolation.

The pentagonal numbers are the integers given by p 5 (n) = n(3n − 1)/2 (n = 0, 1, 2, . . .). Let (b, c, d) be one of the triples (1, 1, 2), (1, 2, 3), (1, 2, 6) and (2,3,4). We show that each n = 0, 1, 2, . . . can be written as w+bx+cy+dz with w, x, y, z pentagonal numbers, which was first conjectured by Z.-W. Sun in 2016. In particular, any nonnegative integer is a sum of five pentagonal numbers two of which are equal; this refines a classical result of Cauchy claimed by Fermat.