Solving hard cut problems via flow-augmentation

We present a new technique for designing fixed-parameter 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 ⊆ E(G) of size at most k such that •in G – Z, s and t are in distinct connected components,•every edge of Z connects two distinct connected components of G – Z, and•if we define the set Zs, t ⊆ Z as those edges e ∊ Z for which there exists an (s, t)-path Pe with E(Pe) ∩ Z = e, then Zs, t separates s from t. We prove that in the above scenario one can in randomized time k(1) (|V(G)| + |E(G)|) add a number of edges to the graph so that with probably at least 2–(k log k) no added edge connects two components of G – Z, and Zs, t becomes a minimum cut between s and t. This additional property becomes a handy lever in applications. For example, consider the question of an (s, t)-cut of cardinality at most k and of minimum possible weight (assuming edge weights in G). While the problem is NP-hard in general, it easily reduces to the maximum flow / minimum cut problem if we additionally assume that k is the minimum possible cardinality of an (s, t)-cut in G. Hence, we immediately obtain that the aforementioned problem admits an 2(k log k) n(1)-time randomized fixed-parameter algorithm. We apply our method to obtain a randomized fixed-parameter 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 (see below), and was unamenable to other techniques for parameterized algorithms in graph cut problems, such as Randomized Contractions, Treewidth Reduction or Shadow Removal. In fact, we go one step further. To demonstrate the power of the approach, we consider more generally the Boolean Min CSP(Γ)-problems, a.k.a. Min SAT(Γ), parameterized by the solution cost. This is a framework of optimization problems that includes problems such as Almost 2-SAT and the notorious i-Chain SAT problem. We are able to show that every problem Min SAT(Γ) is either (1) FPT, (2) W[1]-hard, or (3) able to express the soft constraint (u → 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. In other words, flow-augmentation is powerful enough to let us solve every fixed-parameter tractable problem in the class, except those that explicitly encompass directed graph cuts.