On the Power of Many One-Bit Provers

We study the class of languages, denoted by \MIP[k, 1-\epsilon, s], which have $k$-prover games where each prover just sends a \emph{single} bit, with completeness $1-\epsilon$ and soundness error $s$. For the case that $k=1$ (i.e., for the case of interactive proofs), Goldreich, Vadhan and Wigderson ({\em Computational Complexity'02}) demonstrate that \SZK exactly characterizes languages having 1-bit proof systems with"non-trivial" soundness (i.e., $1/2 < s \leq 1-2\epsilon$). We demonstrate that for the case that $k\geq 2$, 1-bit $k$-prover games exhibit a significantly richer structure: + (Folklore) When $s \leq \frac{1}{2^k} - \epsilon$, \MIP[k, 1-\epsilon, s] = \BPP; + When $\frac{1}{2^k} + \epsilon \leq s < \frac{2}{2^k}-\epsilon$, \MIP[k, 1-\epsilon, s] = \SZK; + When $s \ge \frac{2}{2^k} + \epsilon$, \AM \subseteq \MIP[k, 1-\epsilon, s]; + For $s \le 0.62 k/2^k$ and sufficiently large $k$, \MIP[k, 1-\epsilon, s] \subseteq \EXP; + For $s \ge 2k/2^{k}$, \MIP[k, 1, 1-\epsilon, s] = \NEXP. As such, 1-bit $k$-prover games yield a natural "quantitative" approach to relating complexity classes such as \BPP,\SZK,\AM, \EXP, and \NEXP. We leave open the question of whether a more fine-grained hierarchy (between \AM and \NEXP) can be established for the case when $s \geq \frac{2}{2^k} + \epsilon$

Cubical coloring -- fractional covering by cuts and semidefinite programming

We introduce a new graph invariant that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that circular cliques play for the circular chromatic number. The fact that the defined parameter attains on these graphs the `correct' value suggests that the definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engstr\"om, F\"arnqvist, Jonsson, and Thapper. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246--265]).Comment: 17 page

Fast Distributed Approximation for Max-Cut

Finding a maximum cut is a fundamental task in many computational settings. Surprisingly, it has been insufficiently studied in the classic distributed settings, where vertices communicate by synchronously sending messages to their neighbors according to the underlying graph, known as the $\mathcal{LOCAL}$ or $\mathcal{CONGEST}$ models. We amend this by obtaining almost optimal algorithms for Max-Cut on a wide class of graphs in these models. In particular, for any $\epsilon > 0$, we develop randomized approximation algorithms achieving a ratio of $(1-\epsilon)$ to the optimum for Max-Cut on bipartite graphs in the $\mathcal{CONGEST}$ model, and on general graphs in the $\mathcal{LOCAL}$ model. We further present efficient deterministic algorithms, including a $1/3$-approximation for Max-Dicut in our models, thus improving the best known (randomized) ratio of $1/4$. Our algorithms make non-trivial use of the greedy approach of Buchbinder et al. (SIAM Journal on Computing, 2015) for maximizing an unconstrained (non-monotone) submodular function, which may be of independent interest

Streaming Lower Bounds for Approximating MAX-CUT

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 max cut value. On the other extreme, if one allows $\tilde{O}(n)$ space, then a near-optimal solution to the max cut value can be obtained by storing an $\tilde{O}(n)$-size sparsifier that essentially preserves the max cut. An intriguing question is if poly-logarithmic space suffices to obtain a non-trivial approximation to the max-cut value (that is, beating the factor $2$). It was recently shown that the problem of estimating the size of a maximum matching in a graph admits a non-trivial approximation in poly-logarithmic space. Our main result is that any streaming algorithm that breaks the $2$-approximation barrier requires $\tilde{\Omega}(\sqrt{n})$ space even if the edges of the input graph are presented in random order. Our result is obtained by exhibiting a distribution over graphs which are either bipartite or $\frac{1}{2}$-far from being bipartite, and establishing that $\tilde{\Omega}(\sqrt{n})$ space is necessary to differentiate between these two cases. Thus as a direct corollary we obtain that $\tilde{\Omega}(\sqrt{n})$ space is also necessary to test if a graph is bipartite or $\frac{1}{2}$-far from being bipartite. We also show that for any $\epsilon > 0$, any streaming algorithm that obtains a $(1 + \epsilon)$-approximation to the max cut value when edges arrive in adversarial order requires $n^{1 - O(\epsilon)}$ space, implying that $\Omega(n)$ space is necessary to obtain an arbitrarily good approximation to the max cut value

Intersection and mixing times for reversible chains

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