On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress

Künnemann, Marvin

doi:10.48550/arxiv.1806.09189

Cited by 1 publication

(1 citation statement)

References 38 publications

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“…This result could have applications in computation-delegation settings and may be of interest in other contexts. In particular, since our nondeterministic witness can be constructed deterministically efficiently, namely, in polynomial but super-linear time, it provides a potentially interesting certifying algorithm [MMNS11, ABMR11] (see [Kün18] for a recent paper with a further discussion of the connections to fine-grained complexity). Our final non-reducibility result is as follows.…”

Section: Our Resultsmentioning

confidence: 99%

New Algorithms and Lower Bounds for All-Pairs Max-Flow in Undirected Graphs

Abboud¹,

Krauthgamer²,

Trabelsi³

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the time-complexity of the All-Pairs Max-Flow problem: Given a graph with n nodes and m edges, compute for all pairs of nodes the maximum-flow value between them. If Max-Flow (the version with a given source-sink pair s, t) can be solved in time T (m), then an O(n 2 ) · T (m) is a trivial upper bound. But can we do better?For directed graphs, recent results in fine-grained complexity suggest that this time bound is essentially optimal. In contrast, for undirected graphs with edge capacities, a seminal algorithm of Gomory and Hu (1961) runs in much faster time O(n)·T (m). Under the plausible assumption that Max-Flow can be solved in near-linear time m 1+o(1) , this half-century old algorithm yields an nm 1+o(1) bound. Several other algorithms have been designed through the years, including O(mn) time for unit-capacity edges (unconditionally), but none of them break the O(mn) barrier. Meanwhile, no super-linear lower bound was shown for undirected graphs.We design the first hardness reductions for All-Pairs Max-Flow in undirected graphs, giving an essentially optimal lower bound for the node-capacities setting. For edge capacities, our efforts to prove similar lower bounds have failed, but we have discovered a surprising new algorithm that breaks the O(mn) barrier for graphs with unit-capacity edges! Assuming T (m) = m 1+o(1) , our algorithm runs in time m 3/2+o(1) and outputs a cut-equivalent tree (similarly to the Gomory-Hu algorithm). Even with current Max-Flow algorithms we improve state-of-the-art as long as m = O(n 5/3−ε ). Finally, we explain the lack of lower bounds by proving a non-reducibility result. This result is based on a new quasi-linear timeÕ(m) non-deterministic algorithm for constructing a cut-equivalent tree and may be of independent interest. * A full version appears at arXiv:1901.01412 † 1 Throughout, we focus on computing the value of the flow (rather than an actual flow), which is equal to the value of the minimum st-cut by the famous max-flow/min-cut theorem [FF56].2 SETH asserts that for every fixed ε > 0 there is an integer k ≥ 3, such that kSAT on n variables and m clauses cannot be solved in time 2 (1−ε)n m O(1) .3 Notice that a minimum st-cut in T consists of a single edge that has minimum capacity along the unique st-path in T , and removing this edge disconnects T to two connected components. A flow-equivalent tree has the weaker property that for every pair of nodes s, t, the maximum st-flow value in T equals that in G. The key difference is that flow-equivalence maintains only the values of the flows (and thus also of the corresponding cuts).

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Section: Our Resultsmentioning

confidence: 99%