Abstract
We introduce the notion of {\em fair cuts} as an approach to leverage
approximate $(s,t)$-mincut (equivalently $(s,t)$-maxflow) algorithms in
undirected graphs to obtain near-linear time approximation algorithms for
several cut problems. Informally, for any $\alpha\geq 1$, an $\alpha$-fair
$(s,t)$-cut is an $(s,t)$-cut such that there exists an $(s,t)$-flow that uses
$1/\alpha$ fraction of the capacity of \emph{every} edge in the cut. (So, any
$\alpha$-fair cut is also an $\alpha$-approximate mincut, but not vice-versa.)
We give an algorithm for $(1+\epsilon)$-fair $(s,t)$-cut in
$\tilde{O}(m)$-time, thereby matching the best runtime for
$(1+\epsilon)$-approximate $(s,t)$-mincut [Peng, SODA '16]. We then demonstrate
the power of this approach by showing that this result almost immediately leads
to several applications:
- the first nearly-linear time $(1+\epsilon)$-approximation algorithm that
computes all-pairs maxflow values (by constructing an approximate Gomory-Hu
tree). Prior to our work, such a result was not known even for the special case
of Steiner mincut [Dinitz and Vainstein, STOC '94; Cole and Hariharan, STOC
'03];
- the first almost-linear-work subpolynomial-depth parallel algorithms for
computing $(1+\epsilon)$-approximations for all-pairs maxflow values (again via
an approximate Gomory-Hu tree) in unweighted graphs;
- the first near-linear time expander decomposition algorithm that works even
when the expansion parameter is polynomially small; this subsumes previous
incomparable algorithms [Nanongkai and Saranurak, FOCS '17; Wulff-Nilsen, FOCS
'17; Saranurak and Wang, SODA '19].
