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Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses

Nick Fischer

Abstract

The APSP Hypothesis states that the All-Pairs Shortest Paths (APSP) problem requires time $n^{3-o(1)}$ on graphs with polynomially bounded integer edge weights. Two increasingly stronger assumptions are the Strong APSP Hypothesis and the Directed Unweighted APSP Hypothesis, which state that the fastest-known APSP algorithms on graphs with small weights and unweighted graphs, respectively, are best-possible. In this paper, we design an efficient universe reduction for APSP, which proves that these three hypotheses are, in fact, equivalent, conditioned on $ω= 2$ and a plausible additive combinatorics assumption. Along the way, we resolve the fine-grained complexity of many long-standing graph and matrix problems with "intermediate" complexity such as Node-Weighted APSP, All-Pairs Bottleneck Paths, Monotone Min-Plus Product in certain settings, and many others, by designing matching APSP-based lower bounds.

Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses

Abstract

The APSP Hypothesis states that the All-Pairs Shortest Paths (APSP) problem requires time on graphs with polynomially bounded integer edge weights. Two increasingly stronger assumptions are the Strong APSP Hypothesis and the Directed Unweighted APSP Hypothesis, which state that the fastest-known APSP algorithms on graphs with small weights and unweighted graphs, respectively, are best-possible. In this paper, we design an efficient universe reduction for APSP, which proves that these three hypotheses are, in fact, equivalent, conditioned on and a plausible additive combinatorics assumption. Along the way, we resolve the fine-grained complexity of many long-standing graph and matrix problems with "intermediate" complexity such as Node-Weighted APSP, All-Pairs Bottleneck Paths, Monotone Min-Plus Product in certain settings, and many others, by designing matching APSP-based lower bounds.

Paper Structure

This paper contains 56 sections, 53 theorems, 39 equations, 2 figures.

Table of Contents

  1. Introduction
  2. All-Pairs Shortest Paths
  3. Min-Plus Product
  4. Universe Reductions for APSP
  5. Small-Universe Reduction: From Small to No Weights
  6. Large-Universe Reduction: From Large to Small Weights
  7. Step 1: Doubling Reduction
  8. Step 2: Sum-Order-Preserving Hashing
  9. Critical Remarks
  10. APSP-Based Lower Bounds for Intermediate Problems
  11. Lower Bound 1: Node-Weighted APSP
  12. Lower Bound 2: All-Pairs Bottleneck Paths
  13. Lower Bound 3: Bounded-Difference and Monotone Min-Plus Product
  14. And More ...
  15. Proof Ideas
  16. ...and 41 more sections

Key Result

theorem 1.7

APSP in directed unweighted graphs cannot be solved in time $O(n^{2.5-\epsilon})$ (for any constant $\epsilon > 0$), unless the Strong APSP Hypothesis fails. In particular, conditioned on $\omega = 2$, the Strong APSP and Directed Unweighted APSP Hypotheses are equivalent.

Figures (2)

  • Figure 1: Illustrates our fine-grained reductions and resulting lower bounds assuming that $\omega = 2$. Each arrow symbolizes a (tight) fine-grained reduction. Unlabeled arrows correspond to trivial reductions. The dashed arrows are conditioned on the additive combinatorics \ref{['hypo:hashing']}.
  • Figure 2: Illustrates the four steps in the reduction from Low-Rank Exact Triangle to Uniform Low-Doubling Exact Triangle. (The dashed arrow symbolizes a recursive dependence; see \ref{['sec:exact-tri-low-rank:sec:low-rank-to-unif-regular']}.)

Theorems & Definitions (77)

  • theorem 1.7: Strong APSP Implies Directed Unweighted APSP
  • theorem 1.8: Zwick's Algorithm is Optimal
  • theorem 1.9: Shoshan--Zwick Algorithm is Optimal
  • theorem 1.10: Uniform Low-Doubling APSP
  • theorem 1.11: Sum-Order-Preserving Hashing with Quasi-Polynomial Bounds AmirkhanyanBC18Sanders12
  • theorem 1.12: APSP Conditionally Implies Strong APSP
  • theorem 1.13: Node-Weighted APSP
  • theorem 1.14: All-Pairs Bottleneck Paths
  • theorem 1.15: Row-Bounded-Difference Row-Monotone Min-Plus Product
  • definition 1.15: Select-Plus Rank
  • ...and 67 more