Solving hard cut problems via flow-augmentation

We present a new technique for designing FPT algorithms for graph cut problems in undirected graphs, which we call flow augmentation. Our technique is applicable to problems that can be phrased as a search for an (edge) $(s,t)$-cut of cardinality at most $k$ in an undirected graph $G$ with designated terminals $s$ and $t$. More precisely, we consider problems where an (unknown) solution is a set $Z \subseteq E(G)$ of size at most $k$ such that (1) in $G-Z$, $s$ and $t$ are in distinct connected components, (2) every edge of $Z$ connects two distinct connected components of $G-Z$, and (3) if we define the set $Z_{s,t} \subseteq Z$ as these edges $e \in Z$ for which there exists an $(s,t)$-path $P_e$ with $E(P_e) \cap Z = \{e\}$, then $Z_{s,t}$ separates $s$ from $t$. We prove that in this scenario one can in randomized time $k^{O(1)} (|V(G)|+|E(G)|)$ add a number of edges to the graph so that with $2^{-O(k \log k)}$ probability no added edge connects two components of $G-Z$ and $Z_{s,t}$ becomes a minimum cut between $s$ and $t$. We apply our method to obtain a randomized FPT algorithm for a notorious "hard nut" graph cut problem we call Coupled Min-Cut. This problem emerges out of the study of FPT algorithms for Min CSP problems, and was unamenable to other techniques for parameterized algorithms in graph cut problems, such as Randomized Contractions, Treewidth Reduction or Shadow Removal. To demonstrate the power of the approach, we consider more generally Min SAT($\Gamma$), parameterized by the solution cost. We show that every problem Min SAT($\Gamma$) is either (1) FPT, (2) W[1]-hard, or (3) able to express the soft constraint $(u \to v)$, and thereby also the min-cut problem in directed graphs. All the W[1]-hard cases were known or immediate, and the main new result is an FPT algorithm for a generalization of Coupled Min-Cut.