Affiliation:
1. Indian Institute of Technology Kanpur, India
2. Limestone, Tower Research Capital LLC, United States
Abstract
Let
G
be a directed multi-graph on
n
vertices and
m
edges with a designated source vertex
s
and a designated sink vertex
t
. We study the (
s,t
)-cuts of capacity minimum+1 and as an important application of them, we give a solution to the dual-edge sensitivity for (
s,t
)-mincuts—reporting an (
s,t
)-mincut upon failure or insertion of any pair of edges.
Picard and Queyranne [Mathematical Programming Studies, 13(1): 8–16 (1980)] showed that there exists a directed acyclic graph (DAG) that compactly stores all minimum (
s,t
)-cuts of
G
. This structure also acts as an oracle for the single-edge sensitivity of minimum (
s,t
)-cut. For undirected multi-graphs, Dinitz and Nutov [STOC, 509–518 (1995)] showed that there exists an 𝒪(
n
) size 2-level Cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (
s,t
)-cuts, no such compact structure exists till date. We present the following structural and algorithmic results on minimum+1 (
s,t
)-cuts.
(1)
Structure:
There is an 𝒪(
m
) size 2-level DAG structure that stores all minimum+1
(s,t)
-cuts of
G
such that each minimum+1 (
s,t
)-cut appears as 3-transversal cut—it intersects any path in this structure at most thrice. We also show that there is an 𝒪(
mn
) size structure for storing and characterizing all minimum+1
(s,t)
-cuts in terms of 1-transversal cuts.
(2)
Data structure:
There exists an 𝒪(
n
2
) size data structure that, given a pair of vertices {u,v} that are not separated by an (
s,t
)-mincut, can determine in 𝒪(1) time if there exists a minimum+1 (
s,t
)-cut, say (
A,B
), such that
s,u ∊ A
and
v,t∊ B
; the corresponding cut can be reported in 𝒪(|
B
|) time.
(3)
Sensitivity oracle:
There exists an 𝒪(
n
2
) size data structure that solves the dual-edge sensitivity problem for
(s,t)
-mincuts. It takes 𝒪(1) time to report the capacity of a resulting
(s,t)
-mincut
(A,B)
and 𝒪(|
B
|) time to report the cut.
(4)
Lower bounds:
For the data structure problems addressed in results (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems—all-pairs directed reachability problem, the dual-edge sensitivity problem for (
s,t
)-mincuts, and the problem of reporting the capacity of ({
x,y
}, {
u,v
})-mincut for any four vertices
x,y,u,v
in
G
. Assuming the Directed Reachability Hypothesis by Patrascu [SIAM J. Computing, 827–847 (2011)] and Goldstein et al. [WADS, 421–436 (2017)], this leads to
\(\tilde{\Omega }(n^2)\)
lower bounds on the space for the latter two problems.
Funder
Tapas Mishra Memorial Chair at Indian Institute of Technology Kanpur
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)