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Electronic Colloquium on Computational Complexity

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All reports by Author Ian Mertz:

TR21-054 | 14th April 2021
James Cook, Ian Mertz

Encodings and the Tree Evaluation Problem

We show that the Tree Evaluation Problem with alphabet size $k$ and height $h$ can be solved by branching programs of size $k^{O(h/\log h)} + 2^{O(h)}$. This answers a longstanding challenge of Cook et al. (2009) and gives the first general upper bound since the problem's inception.

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TR20-111 | 24th July 2020
Ian Mertz, Toniann Pitassi

Lifting: As Easy As 1,2,3

Revisions: 1

Query-to-communication lifting theorems translate lower bounds on query complexity to lower bounds for the corresponding communication model. In this paper, we give a simplified proof of deterministic lifting (in both the tree-like and dag-like settings). Whereas previous proofs used sophisticated Fourier analytic techniques, our proof uses elementary counting together with ... more >>>

TR20-056 | 17th April 2020
James Cook, Ian Mertz

Catalytic Approaches to the Tree Evaluation Problem

The study of branching programs for the Tree Evaluation Problem, introduced by S. Cook et al. (TOCT 2012), remains one of the most promising approaches to separating L from P. Given a label in $[k]$ at each leaf of a complete binary tree and an explicit function in $[k]^2 \to ... more >>>

TR20-049 | 17th April 2020
Mika Göös, Sajin Koroth, Ian Mertz, Toniann Pitassi

Automating Cutting Planes is NP-Hard

We show that Cutting Planes (CP) proofs are hard to find: Given an unsatisfiable formula $F$,

(1) it is NP-hard to find a CP refutation of $F$ in time polynomial in the length of the shortest such refutation; and

(2) unless Gap-Hitting-Set admits a nontrivial algorithm, one cannot find a ... more >>>

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