Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910082427518976 |
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| author | Fischer, Nick |
| author_facet | Fischer, Nick |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27736 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses Fischer, Nick Data Structures and Algorithms 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. |
| title | Universe Reduction for APSP: Equivalence of Three Fine-Grained Hypotheses |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2603.27736 |