[Paper Review] Lower Bounds for Multiplication via Network Coding

Peyman Afshani, Casper Benjamin Freksen+2 more|DROPS (Schloss Dagstuhl – Leibniz Center for Informatics)|Feb 28, 2019

Complexity and Algorithms in Graphs14 references4 citations

TL;DR

This paper establishes a conditional $Ω(n\lg n)$ lower bound for Boolean circuits computing multiplication or the shift operation, assuming the undirected $k$-pairs network coding conjecture is true. The result implies that multiplication cannot be computed by constant-degree circuits of size $O(n)$, nearly matching Fürer's upper bound and showing multiplication is strictly harder than addition.

ABSTRACT

Multiplication is one of the most fundamental computational problems, yet its true complexity remains elusive. The best known upper bound, by Fürer, shows that two $n$-bit numbers can be multiplied via a boolean circuit of size $O(n \lg n \cdot 4^{\lg^*n})$, where $\lg^*n$ is the very slowly growing iterated logarithm. In this work, we prove that if a central conjecture in the area of network coding is true, then any constant degree boolean circuit for multiplication must have size $Ω(n \lg n)$, thus almost completely settling the complexity of multiplication circuits. We additionally revisit classic conjectures in circuit complexity, due to Valiant, and show that the network coding conjecture also implies one of Valiant's conjectures.

Motivation & Objective

• To establish a conditional lower bound for the circuit complexity of multiplication, a fundamental computational problem.
• To connect the complexity of multiplication to the central undirected $k$-pairs network coding conjecture.
• To show that even the simpler shift problem requires $Ω(n\lg n)$ circuit size under the same conjecture.
• To demonstrate that the network coding conjecture implies long-standing conjectures in circuit complexity, including Valiant's depth reduction conjecture.
• To provide a new framework for deriving circuit lower bounds using network coding principles.

Proposed method

• Reduces the multiplication problem to the shift problem, where shifting a string by $j$ positions is equivalent to multiplying by $2^j$.
• Constructs a network coding instance with $n/k$ source-sink pairs, modeling the shift operation as a communication task.
• Uses the $\mathcal{F}$-correction game to simulate information flow and derive entropy constraints on edges.
• Applies Shannon's source coding theorem to bound the entropy of messages on edges, linking it to edge capacities.
• Employs multicommodity flow analysis on the undirected version of the network, assuming the network coding conjecture holds.
• Derives a lower bound on the number of edges in a subgraph by combining flow conservation, entropy bounds, and capacity constraints.

Figure 1: The depth $3$ circuit $C$ .

Experimental results

Research questions

• RQ1 Can the complexity of multiplication be bounded below by $\Omega(n\lg n)$ in constant-degree Boolean circuits?
• RQ2 Does the undirected $k$-pairs network coding conjecture imply strong circuit lower bounds for fundamental arithmetic operations?
• RQ3 Is the shift operation, a special case of multiplication, inherently more complex than addition in circuit complexity?
• RQ4 Does the network coding conjecture imply Valiant's conjecture on depth reduction in Boolean circuits?
• RQ5 Can network coding-based reductions be used to derive tight lower bounds for other computational problems?

Key findings

• Assuming the undirected $k$-pairs network coding conjecture, any constant-degree Boolean circuit for multiplication must have size $\Omega(n\lg n)$.
• The same lower bound applies to the simpler shift problem, where an $n$-bit string is shifted by a $\lg n$-bit index.
• The result nearly matches Fürer's upper bound of $O(n\lg n \cdot 4^{\lg^* n})$, showing that multiplication is strictly harder than addition.
• The network coding conjecture implies Valiant's conjecture on depth reduction, linking two major open problems in circuit complexity.
• The proof technique extends to cyclic shifts, showing the same lower bound applies to that variant as well.

Figure 2: Given the $2$ -layer circuit $\Gamma$ spanned by $x_{1},\ldots,x_{n},y_{1},\ldots,y_{n}$ , we construct the communication network graph $G$ .