The seminal work of Benczu}r and Karger demonstrated cut sparsifiers of near-linear size, with several applications throughout theoretical computer science. Subsequent extensions have yielded sparsifiers for hypergraph cuts and more recently linear codes over Abelian groups. A decade ago, Kogan and Krauthgamer asked about the sparsifiability of arbitrary constraint satisfaction ... more >>>

We present the first algorithms for polynomial identity testing (PIT) of read-$4$ arithmetic formulas in the non-multilinear setting. Specifically, we give a polynomial-time PIT algorithm in the whitebox model and a quasi-polynomial-time algorithm in the blackbox model. Since our techniques are based on proving hardness of representation results, we extend ... more >>>

We construct an explicit distribution $\mathbf{D}$ over $\{0,1\}^N$ that exhibits an essentially optimal separation between adaptive and non-adaptive cell-probe sampling. The distribution can be sampled exactly when each output bit is allowed two adaptive probes to an arbitrarily long sequence of independent uniform symbols from $[N]$. In contrast, any non-adaptive ... more >>>