An Efficient Algorithm for All-Pairs Bounded Edge Connectivity
Name
453_2023_Article_1203.pdf
Size
522.11 KB
Format
Adobe PDF
Checksum (MD5)
246c6d81e581885d0c4b3cf297920ac2
Author(s)
Akmal, Shyan
Jin, Ce
Date Issued
January 22, 2024
Publisher
Springer US
Citation
Akmal, S., Jin, C. An Efficient Algorithm for All-Pairs Bounded Edge Connectivity. Algorithmica (2024).
Version
Final published version
Abstract
Our work concerns algorithms for a variant of Maximum Flow in unweighted graphs. In the All-Pairs Connectivity (APC) problem, we are given a graph G on n vertices and m edges, and are tasked with computing the maximum number of edge-disjoint paths from s to t (equivalently, the size of a minimum (s, t)-cut) in G, for all pairs of vertices (s, t). Significant algorithmic breakthroughs have recently shown that over undirected graphs, APC can be solved in
$$n^{2+o(1)}$$
n
2
+
o
(
1
)
time, which is essentially optimal. In contrast, the true time complexity of APC over directed graphs remains open: this problem can be solved in
$${\tilde{O}}(m^\omega )$$
O
~
(
m
ω
)
time, where
$$\omega \in [2, 2.373)$$
ω
∈
[
2
,
2.373
)
is the exponent of matrix multiplication, but no matching conditional lower bound is known. Following [Abboud et al. In: 46th International colloquium on automata, languages, and programming, ICALP 2019, July 9-12, 2019, Patras, Greece, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2019], we study a bounded version of
$${{\textsf {APC}}}$$
APC
called the k-Bounded All Pairs Connectivity (k-APC) problem. In this variant of APC, we are given an integer k in addition to the graph G, and are now tasked with reporting the size of a minimum (s, t)-cut only for pairs (s, t) of vertices with min-cut value less than k (if the minimum (s, t)-cut has size at least k, we can just report it is “large” instead of computing the exact value). Our main result is an
$${\tilde{O}}((kn)^\omega )$$
O
~
(
(
k
n
)
ω
)
time algorithm solving k-APC in directed graphs. This is the first algorithm which solves k-APC faster than simply solving the more general APC problem exactly, for all
$$k\ge 3$$
k
≥
3
. This runtime is
$${{\tilde{O}}}(n^\omega )$$
O
~
(
n
ω
)
for all
$$k\le {{\,\textrm{poly}\,}}(\log n)$$
k
≤
poly
(
log
n
)
, which essentially matches the optimal runtime for the
$$k=1$$
k
=
1
case of k-APC, under popular conjectures from fine-grained complexity. Previously, this runtime was only achieved for
$$k\le 2$$
k
≤
2
in general directed graphs [Georgiadis et al. In: 44th international colloquium on automata, languages, and programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2017], and for
$$k\le o(\sqrt{\log n})$$
k
≤
o
(
log
n
)
in the special case of directed acyclic graphs [Abboud et al. In: 46th international colloquium on automata, languages, and programming, ICALP 2019, July 9–12, 2019, Patras, Greece, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2019]. Our result employs the same algebraic framework used in previous work, introduced by [Cheung et al. In: FOCS, 2011]. A direct implementation of this framework involves inverting a large random matrix. Our new algorithm is based off the insight that for solving k-APC, it suffices to invert a low-rank random matrix instead of a generic random matrix. We also obtain a new algorithm for a variant of k-APC, the k-Bounded All-Pairs Vertex Connectivity (k-APVC) problem, where we are now tasked with reporting, for every pair of vertices (s, t), the maximum number of internally vertex-disjoint (rather than edge-disjoint) paths from s to t if this number is less than k, and otherwise reporting that there are at least k internally vertex-disjoint paths from s to t. Our second result is an
$${\tilde{O}}(k^2n^\omega )$$
O
~
(
k
2
n
ω
)
time algorithm solving k-APVC in directed graphs. Previous work showed how to solve an easier version of the k-APVC problem (where answers only need to be returned for pairs of vertices (s, t) which are not edges in the graph) in
$${{\tilde{O}}}((kn)^\omega )$$
O
~
(
(
k
n
)
ω
)
time [Abboud et al. In: 46th International colloquium on automata, languages, and programming, ICALP 2019, July 9–12, 2019, Patras, Greece, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2019]. In comparison, our algorithm solves the full k-APVC problem, and is faster if
$$\omega > 2$$
ω
>
2
.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00453-023-01203-2
