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# Eun Jung Kim (김은정), Directed flow-augmentation

## Tuesday, August 9, 2022 @ 4:30 PM - 5:30 PM KST

We show a flow-augmentation algorithm in directed graphs: There exists a polynomial-time algorithm that, given a directed graph G, two integers $s,t\in V(G)$, and an integer $k$, adds (randomly) to $G$ a number of arcs such that for every minimal st-cut $Z$ in $G$ of size at most $k$, with probability $2^{−\operatorname{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph.

The directed flow-augmentation tool allows us to prove fixed-parameter tractability of a number of problems parameterized by the cardinality of the deletion set, whose parameterized complexity status was repeatedly posed as open problems:

(1) Chain SAT, defined by Chitnis, Egri, and Marx [ESA’13, Algorithmica’17],

(2) a number of weighted variants of classic directed cut problems, such as Weighted st-Cut, Weighted Directed Feedback Vertex Set, or Weighted Almost 2-SAT.

By proving that Chain SAT is FPT, we confirm a conjecture of Chitnis, Egri, and Marx that, for any graph H, if the List H-Coloring problem is polynomial-time solvable, then the corresponding vertex-deletion problem is fixed-parameter tractable.

Joint work with Stefan Kratsch, Marcin Pilipczuk, Magnus Wahlström.