# Size-Treewidth Tradeoffs for Circuits Computing the Element Distinctness Function

## Abstract

In this work we study the relationship between size and treewidth of circuits computing variants of the element distinctness function. First, we show that for each n, any circuit of treewidth t computing the element distinctness function δ n : {0, 1}n → {0, 1} must have size at least $$\Omega (\frac {n^{2}}{2^{O(t)} \log n})$$. This result provides a non-trivial generalization of a super-linear lower bound for the size of Boolean formulas (treewidth 1) due to Nečiporuk. Subsequently, we turn our attention to read-once circuits, which are circuits where each variable labels at most one input vertex. For each n, we show that any read-once circuit of treewidth t and size s computing a variant τ n : {0, 1}n → {0, 1} of the element distinctness function must satisfy the inequality $$t\cdot \log s \geq \Omega (\frac {n}{\log n})$$. Using this inequality in conjunction with known results in structural graph theory, we show that for each fixed graph H, read-once circuits computing τ n which exclude H as a minor must have size at least Ω(n 2/log4 n). For certain well studied functions, such as the triangle-freeness function, this last lower bound can be improved to Ω(n 2/log 2n).

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1. 1.

Recently, this lower bound was improved to (3 + 1/86) no(n) .

2. 2.

We say that a circuit $$\mathcal {C}$$ is H-minor-free if its underlying undirected graph excludes H as a minor.

3. 3.

An in-branching tree is a directed tree where all edges are oriented toward the root. An out-branching tree is a directed tree where all edges are oriented toward the leaves.

4. 4.

The structure (N, F)of the decomposition remains the same. Only function assigning bags to nodes in N isupdated.

5. 5.

For instance, the identity of ∧ is 1, while the identity of ∨ is 0. The requirement of an identity can easily be removed by replacing Condition 1b with a slightly more complicated initialization condition.

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## Acknowledgements

The author would like to thank Pavel Pudlák for interesting discussions on circuit lower-bounds, Pavel Hrubeš for pointing me out to reference , Michal Koucký and Bruno Loff for useful feedback during a seminar presentation of this work, and anonymous referees for valuable comments.

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Correspondence to Mateus de Oliveira Oliveira.