The composition complexity of majority (2205.02374v2)

Abstract: We study the complexity of computing majority as a composition of local functions: [ \text{Maj}_n = h(g_1,\ldots,g_m), ] where each $g_j :{0,1}^{n} \to {0,1}$ is an arbitrary function that queries only $k \ll n$ variables and $h : {0,1}^m \to {0,1}$ is an arbitrary combining function. We prove an optimal lower bound of [ m \ge \Omega\left( \frac{n}{k} \log k \right) ] on the number of functions needed, which is a factor $\Omega(\log k)$ larger than the ideal $m = n/k$. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority. Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions $g_j$, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).

• The paper establishes a rigorous lower bound of f(n, k) = $ \Omega\left(\frac{n}{k} \log k\right) $ for the composition complexity of the majority function.
• It demonstrates how this bound recovers known lower bounds for bounded-width branching programs computing the majority function.
• The work introduces novel information-theoretic techniques for analysis and lays groundwork for future research in small-depth circuit complexity.

The Composition Complexity of Majority

The paper "The composition complexity of majority" authored by Victor Lecomte, Prasanna Ramakrishnan, and Li-Yang Tan presents an in-depth examination of the composite complexity of the majority function, a fundamental problem in theoretical computer science. The paper investigates the complexity associated with expressing the majority function as a composition of local functions, which query only a subset of variables, and an overarching combining function.

Summary of Key Contributions

1. Lower Bound for Composition Complexity: The authors establish a rigorous lower bound on the number of functions required to compute the majority function as a composition of k-local functions. This bound is expressed as $\Omega\left(\frac{n}{k} \log k\right)$, which signifies a composition overhead factor of $\log k$ over the naive partitioning approach. This result indicates that the composition complexity inherently surpasses the ideal case, where each variable is only queried once.
2. Relation to Bounded-Width Branching Programs: The paper extends the lower bound findings to implicate bounded-width branching programs. Specifically, it recovers and provides an alternative proof for the Alon and Maass (1986) lower bound on the length of these programs for majority functions using composition complexity. This correspondence implies that computing the majority function using such a model results in a significant length requirement of $\Omega(n \log n)$.
3. Implications for Small-Depth Circuits: The results presented lay the groundwork for further investigations into small-depth circuit lower bounds, particularly the longstanding open challenge of establishing tight lower bounds for depth-3 circuits. The findings suggest that to exceed current lower bounds for such circuits, new techniques leveraging composition complexity could be fruitful.
4. Information-Theoretic Techniques: The paper introduces novel methods to analyze the information flow within the composition, utilizing mutual information to demonstrate how variable constraints lead to information leakage across the composition network. This approach has the potential to be adapted to even more complex computational models.
5. Multi-Output Functions Influence: A significant element of the work is the demonstration that the majority function's composition complexity can be understood by first considering the Hamming weight function—a multi-output counterpart. This technique hints at a generalizable strategy for tackling other complex boolean functions.

Implications and Future Directions

From a theoretical standpoint, the presented findings advance our knowledge of boolean function complexity, particularly through the lens of composition. Practically, this work provides a framework that could potentially inform the design of efficient algorithms and systems where majority-based decision-making is critical, such as distributed computing networks or parallel processing architectures.

Several intriguing future research pathways emerge from this paper. Extending the applicability of the proposed information-theoretic methods to broader classes of circuits and different function compositions could yield significant insights into circuit complexity. Furthermore, exploring whether these results can be generalized to more complex, real-world majority decision-making scenarios could bridge the gap between theoretical and applied computer science.

Overall, the paper marks a substantial advance in understanding the composition complexity of majority—a foundational element in distributed computing and circuit complexity—and sets the stage for further exploration in this domain.