Faster all-pairs shortest paths via circuit complexity (1312.6680v2)

Abstract: We present a new randomized method for computing the min-plus product (a.k.a., tropical product) of two $n \times n$ matrices, yielding a faster algorithm for solving the all-pairs shortest path problem (APSP) in dense $n$-node directed graphs with arbitrary edge weights. On the real RAM, where additions and comparisons of reals are unit cost (but all other operations have typical logarithmic cost), the algorithm runs in time [\frac{n^3}{2^{\Omega(\log n)^{1/2}}}] and is correct with high probability. On the word RAM, the algorithm runs in $n^3/2^{\Omega(\log n)^{1/2}} + n^{2+o(1)}\log M$ time for edge weights in $([0,M] \cap {\mathbb Z})\cup{\infty}$. Prior algorithms used either $n^3/(\log^c n)$ time for various $c \leq 2$, or $O(M^{\alpha}n^{\beta})$ time for various $\alpha > 0$ and $\beta > 2$. The new algorithm applies a tool from circuit complexity, namely the Razborov-Smolensky polynomials for approximately representing ${\sf AC}^0[p]$ circuits, to efficiently reduce a matrix product over the $(\min,+)$ algebra to a relatively small number of rectangular matrix products over ${\mathbb F}_2$, each of which are computable using a particularly efficient method due to Coppersmith. We also give a deterministic version of the algorithm running in $n^3/2^{\log^{\delta} n}$ time for some $\delta > 0$, which utilizes the Yao-Beigel-Tarui translation of ${\sf AC}^0[m]$ circuits into "nice" depth-two circuits.

• The paper improves algorithmic efficiency in APSP by applying circuit complexity to reduce the min-plus matrix product, achieving a time complexity of n^3/2^{Ω((log n)^{1/2})}.
• It leverages Razborov-Smolensky polynomial techniques to transform min-plus operations into XOR computations, optimizing performance on dense graphs.
• The algorithm adapts to both real RAM and word RAM models, offering significant theoretical and practical advancements for fundamental graph problems.

Overview of "Faster All-Pairs Shortest Paths via Circuit Complexity"

The paper by Ryan Williams introduces a novel randomized algorithm for the all-pairs shortest paths problem (APSP), leveraging techniques from circuit complexity. This problem, fundamental to computer science, involves finding the shortest paths between all pairs of nodes in a weighted directed graph. The algorithm presented distinctly improves upon prior approaches by integrating linear algebra techniques with circuit complexity theory, specifically using Razborov-Smolensky polynomials. The resultant method solves APSP more efficiently than previous algorithms, with a time complexity of $n^3/2^{\Omega(\log n)^{1/2}}$ on dense $n$-node graphs.

Key Contributions

1. Algorithmic Efficiency: Williams applies a circuit complexity approach to accelerate the min-plus matrix product, a key computational component of APSP. The algorithm runs faster than the previously best-known time of $n^3/\log^c n$ for various constants $c \leq 2$.
2. Use of Circuit Complexity: The algorithm uses Razborov-Smolensky techniques to represent $\AC^0[p]$ circuits efficiently. This represents a pioneering application of circuit complexity to matrix multiplication in a manner that improves practical computational time.
3. Reduction from Min-Plus to XOR Operations: Williams reduces the problem to several rectangular matrix products over the field $\F_2$. This reduction exploits specific polynomial representations of $\AC^0$ operations, which traditionally involve challenging overhead costs.
4. Implementation on Different Computational Models: The algorithm accounts for both the real RAM and word RAM computational models. While the real RAM model removes some computational overhead, the word RAM model includes additive factors for edge weight considerations, maintaining practicality across different computational settings.

Implications and Future Directions

The paper suggests that merging circuit complexity with traditional algorithmic approaches can lead to significant computational improvements in fundamental graph problems. This method could inspire similar interventions in other algorithmic problems where complexity bounds are tied to fundamental computational limits.

• Theoretical Implications: This work challenges preconceived notions about APSP's complexity, which could potentially shift the theoretical boundaries of subcubic equivalence for a class of graph problems closely related to APSP.
• Practical Implications: The algorithm's emphasis on using bit-level manipulations opens avenues for implementation on architectures stressing efficient bitwise operations. Additionally, the novel integration of algebra with circuit complexity could inspire practical algorithms in other domains involving matrix operations.

Conclusion

Ryan Williams' paper extends the boundaries of computational theory applied to APSP and sets a stage for exploring circuit complexity as a tool for other computational problems. By employing a theoretical approach rooted in circuit complexity, the paper mitigates the computational expense historically associated with matrix multiplication in APSP, offering a substantive improvement over logarithmic factor reductions. Future research may explore extending these techniques to achieve truly subcubic time complexities and apply them in other contexts where similar computational trade-offs are of interest.