Range Avoidance in Boolean Circuits via Turan-type Bounds

2025-03-21

AI Summary

This paper studies the Range Avoidance problem (RAV) for Boolean circuits: given an $n$-input, $m$-output circuit $C$, find an $m$-bit string $y$ not in its image. While RAV is hard for general circuits when $m > n$, no deterministic polynomial-time algorithm was known for monotone $mathsf{NC}^0_3$ circuits—despite extensive study. The authors first formulate monotone $mathsf{NC}^0_3$-RAV as a hypergraph Turán-type extremal problem and establish a novel Turán theorem for loose $chi$-cycles. Leveraging this combinatorial insight, they design the first deterministic polynomial-time algorithm for monotone $mathsf{NC}^0_3$-RAV, improving the solvability threshold from the prior $m geq n^2 / log n$ to the optimal $m > n$. This substantially advances the frontier of deterministic tractability for RAV and yields new tools for explicit constructions of hard functions, linear codes, and other pseudorandom objects.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Stochastic Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Given a circuit $C : {0,1}^n o {0,1}^m$ from a circuit class $F$, with $m>n$, finding a $y in {0,1}^m$ such that $forall x in {0,1}^n$, $C(x) e y$, is the range avoidance problem (denoted by $F$-$avoid$). Deterministic polynomial time algorithms (even with access to $NP$ oracles) solving this problem is known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for $NC^0_2$-$avoid$ when $m>n$, and for $NC^0_3$-$avoid$ when $m ge frac{n^2}{log n}$, where $NC^0_k$ is the class of circuits with bounded fan-in which have constant depth and the output depends on at most $k$ of the input bits. On the other hand, it is also known that $NC^0_3$-$avoid$ when $m = n+Oleft(n^{2/3} ight)$ is at least as hard as explicit construction of rigid matrices. In this paper, we propose a new approach to solving range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind - for a fixed $k$-uniform hypergraph $H'$, what is the maximum number of edges that can exist in a $k$-uniform hypergraph $H$ which does not have a sub-hypergraph isomorphic to $H'$? We use our approach to show (using known Turan-type bounds) that there is a constant $c$ such that $mon$-$NC^0_3$-$avoid$ can be solved in deterministic polynomial time when $m>cn^2$. To improve the stretch constraint to linear, we show a new Turan-type theorem for a hypergraph structure (which we call the the loose $chi$-cycles) and use it to show that $mon$-$NC^0_3$-$avoid$ can be solved in deterministic polynomial time when $m>n$, thus improving the known bounds of $NC^0_3$-avoid for the case of monotone circuits.

Problem

• Solving range avoidance in Boolean circuits via hypergraphs
• Improving deterministic polynomial time algorithms for NC^0_3-avoid
• Establishing new Turan-type bounds for hypergraph structures

Innovation

• Hypergraph-based approach for range avoidance
• Turan-type bounds applied to circuit problems
• New Turan-type theorem for loose chi-cycles