Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for $k$-CNFs:
every k-CNF is a sub-exponential size disjunction of $k$-CNFs with a linear
number of clauses. This lemma has subsequently played a key role in the study
of the exact complexity of the satisfiability problem. A natural question is
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We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$ and an error parameter $\varepsilon > 0$, our algorithm runs in space $\tilde{O}(k\log (N\cdot ... more >>>

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 >>>

In this work, we study the space complexity of sketching the intersection profile of a distribution $D$ on $2^{[n]}$. Specifically, we seek a succinct data structure that, for any query set $S \subseteq [n]$, approximates the quantity $\Pr_{T \sim D}[T \cap S \neq \emptyset]$ to within a small constant additive ... more >>>

We extend the recent work of Reis and Rothvoss on sparsifying sums of $\ell_1$ norms to the more general task of sparsifying (Minkowski) sums of centrally symmetric, convex sets. As our main result, we prove that for any $\varepsilon > 0$ and centrally symmetric, convex sets $C_1, \ldots, C_m\subseteq\mathbb R^n$ ... more >>>