31 research outputs found

    On the communication complexity of sparse set disjointness and exists-equal problems

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    In this paper we study the two player randomized communication complexity of the sparse set disjointness and the exists-equal problems and give matching lower and upper bounds (up to constant factors) for any number of rounds for both of these problems. In the sparse set disjointness problem, each player receives a k-subset of [m] and the goal is to determine whether the sets intersect. For this problem, we give a protocol that communicates a total of O(k\log^{(r)}k) bits over r rounds and errs with very small probability. Here we can take r=\log^{*}k to obtain a O(k) total communication \log^{*}k-round protocol with exponentially small error probability, improving on the O(k)-bits O(\log k)-round constant error probability protocol of Hastad and Wigderson from 1997. In the exist-equal problem, the players receive vectors x,y\in [t]^n and the goal is to determine whether there exists a coordinate i such that x_i=y_i. Namely, the exists-equal problem is the OR of n equality problems. Observe that exists-equal is an instance of sparse set disjointness with k=n, hence the protocol above applies here as well, giving an O(n\log^{(r)}n) upper bound. Our main technical contribution in this paper is a matching lower bound: we show that when t=\Omega(n), any r-round randomized protocol for the exists-equal problem with error probability at most 1/3 should have a message of size \Omega(n\log^{(r)}n). Our lower bound holds even for super-constant r <= \log^*n, showing that any O(n) bits exists-equal protocol should have \log^*n - O(1) rounds

    A Lower Bound for Sampling Disjoint Sets

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    Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

    Cell-probe Lower Bounds for Dynamic Problems via a New Communication Model

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    In this paper, we develop a new communication model to prove a data structure lower bound for the dynamic interval union problem. The problem is to maintain a multiset of intervals I\mathcal{I} over [0,n][0, n] with integer coordinates, supporting the following operations: - insert(a, b): add an interval [a,b][a, b] to I\mathcal{I}, provided that aa and bb are integers in [0,n][0, n]; - delete(a, b): delete a (previously inserted) interval [a,b][a, b] from I\mathcal{I}; - query(): return the total length of the union of all intervals in I\mathcal{I}. It is related to the two-dimensional case of Klee's measure problem. We prove that there is a distribution over sequences of operations with O(n)O(n) insertions and deletions, and O(n0.01)O(n^{0.01}) queries, for which any data structure with any constant error probability requires Ω(nlogn)\Omega(n\log n) time in expectation. Interestingly, we use the sparse set disjointness protocol of H\aa{}stad and Wigderson [ToC'07] to speed up a reduction from a new kind of nondeterministic communication games, for which we prove lower bounds. For applications, we prove lower bounds for several dynamic graph problems by reducing them from dynamic interval union

    Lower bounds on the size of semidefinite programming relaxations

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    We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on nn-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nc2^{n^c}, for some constant c>0c > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1)O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT
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