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Electronic Colloquium on Computational Complexity

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All reports by Author Benjamin Rossman:

TR20-186 | 9th December 2020
Benjamin Rossman

Shrinkage of Decision Lists and DNF Formulas

Revisions: 1

We establish nearly tight bounds on the expected shrinkage of decision lists and DNF formulas under the $p$-random restriction $\mathbf R_p$ for all values of $p \in [0,1]$. For a function $f$ with domain $\{0,1\}^n$, let $\mathrm{DL}(f)$ denote the minimum size of a decision list that computes $f$. We show ... more >>>

TR20-061 | 28th April 2020
Deepanshu Kush, Benjamin Rossman

Tree-depth and the Formula Complexity of Subgraph Isomorphism

For a fixed "pattern" graph $G$, the $\textit{colored}$ $G\textit{-subgraph isomorphism problem}$ (denoted $\mathrm{SUB}(G)$) asks, given an $n$-vertex graph $H$ and a coloring $V(H) \to V(G)$, whether $H$ contains a properly colored copy of $G$. The complexity of this problem is tied to parameterized versions of $\mathit{P}$ ${=}?$ $\mathit{NP}$ and $\mathit{L}$ ... more >>>

TR17-022 | 13th February 2017
Benjamin Rossman, Srikanth Srinivasan

Separation of AC$^0[\oplus]$ Formulas and Circuits

This paper gives the first separation between the power of {\em formulas} and {\em circuits} of equal depth in the $\mathrm{AC}^0[\oplus]$ basis (unbounded fan-in AND, OR, NOT and MOD$_2$ gates). We show, for all $d(n) \le O(\frac{\log n}{\log\log n})$, that there exist {\em polynomial-size depth-$d$ circuits} that are not equivalent ... more >>>

TR16-206 | 24th December 2016
Benjamin Rossman

An Improved Homomorphism Preservation Theorem From Lower Bounds in Circuit Complexity

Previous work of the author [39] showed that the Homomorphism Preservation Theorem of classical model theory remains valid when its statement is restricted to finite structures. In this paper, we give a new proof of this result via a reduction to lower bounds in circuit complexity, specifically on the AC$^0$ ... more >>>

TR15-143 | 31st August 2015
Benjamin Rossman

The Average Sensitivity of Bounded-Depth Formulas

We show that unbounded fan-in boolean formulas of depth $d+1$ and size $s$ have average sensitivity $O(\frac{1}{d}\log s)^d$. In particular, this gives a tight $2^{\Omega(d(n^{1/d}-1))}$ lower bound on the size of depth $d+1$ formulas computing the PARITY function. These results strengthen the corresponding $2^{\Omega(n^{1/d})}$ and $O(\log s)^d$ bounds for circuits ... more >>>

TR13-169 | 2nd December 2013
Benjamin Rossman

Formulas vs. Circuits for Small Distance Connectivity

We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance $k(n)$ Connectivity, which asks whether two specified nodes in a graph of size $n$ are connected by a path of length at most $k(n)$. This problem is solvable (by ... more >>>

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