Near-Linear Time Approximations for Cut Problems via Fair Cuts

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].