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Electronic Colloquium on Computational Complexity

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TR25-034 | 20th March 2025
Neha Kuntewar, Jayalal Sarma

Range Avoidance in Boolean Circuits via Turan-type Bounds

Given a circuit $C : \{0,1\}^n \to \{0,1\}^m$ from a circuit class $F$, with $m > n$, finding a $y \in \{0,1\}^m$ such that $\forall x \in \{0,1\}^n$, $C(x) \ne y$, is the range avoidance problem (denoted by $F$-AVOID). It is known that deterministic polynomial time algorithms (even with access ... more >>>


TR25-033 | 18th March 2025
Bruno Pasqualotto Cavalar, Igor Oliveira

Boolean Circuit Complexity and Two-Dimensional Cover Problems

We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the ... more >>>


TR25-032 | 21st March 2025
Jonas Conneryd, Susanna F. de Rezende, Jakob Nordström, Shuo Pang, Kilian Risse

Graph Colouring Is Hard on Average for Polynomial Calculus and Nullstellensatz

We prove that polynomial calculus (and hence also Nullstellensatz) over any field requires linear degree to refute that sparse random regular graphs, as well as sparse Erd?s-Rényi random graphs, are 3-colourable. Using the known relation between size and degree for polynomial calculus proofs, this implies strongly exponential lower bounds on ... more >>>



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