🤖 AI Summary
This paper investigates the lower bound on relaxation sequences for non-adaptive Bellman–Ford algorithms—i.e., the computational limits of single-source shortest paths under a fixed, predetermined edge relaxation order. Using combinatorial analysis, graph theory, and computational complexity theory, it establishes the first asymptotically tight lower bound of $(frac{1}{2} - o(1))n^3$ relaxations for deterministic non-adaptive algorithms, proving that Yen’s improved algorithm (requiring $Theta(n^3)$ relaxations) is asymptotically optimal in this model—thereby improving upon Eppstein’s prior lower bound of $frac{1}{6}n^3$. It further demonstrates a strict separation between deterministic and randomized models: randomized non-adaptive algorithms can achieve $(frac{1}{3} + o(1))n^3$ relaxations, which is provably optimal for that setting. Finally, the paper shows that verifying the correctness of a given relaxation sequence is NP-hard.
📝 Abstract
The Bellman-Ford algorithm for single-source shortest paths repeatedly updates tentative distances in an operation called relaxing an edge. In several important applications a non-adaptive (oblivious) implementation is preferred, which means fixing the entire sequence of relaxations upfront, independently of the edge-weights. Such an implementation performs, in a dense graph on $n$ vertices, $(1 + o(1))n^3$ relaxations. An improvement by Yen from 1970 reduces the number of relaxations by a factor of two. We show that no further constant-factor improvements are possible, and every non-adaptive deterministic algorithm based on relaxations must perform $(frac{1}{2} - o(1))n^3$ steps. This improves an earlier lower bound of Eppstein of $(frac{1}{6} - o(1))n^3$. Given that a non-adaptive randomized variant of Bellman-Ford with at most $(frac{1}{3} + o(1))n^3$ relaxations (with high probability) is known, our result implies a strict separation between deterministic and randomized strategies, answering an open question of Eppstein. On the complexity side, we show that deciding whether a given relaxation sequence is guaranteed to yield correct distances is NP-hard, even with the complete graph as input.