Flow-augmentation I: Directed graphs

TL;DR

This paper introduces a randomized flow-augmentation algorithm for directed graphs that helps solve several fixed-parameter tractability problems, including Chain SAT and weighted directed cut problems, by probabilistically modifying the graph.

Contribution

It presents a novel randomized and deterministic flow-augmentation method for directed graphs, enabling FPT algorithms for problems previously considered open.

Findings

Proves fixed-parameter tractability of Chain SAT.

Establishes FPT results for weighted directed cut problems.

Confirms a conjecture relating List H-Coloring and vertex-deletion problems.

Abstract

We show a flow-augmentation algorithm in directed graphs: There exists a randomized polynomial-time algorithm that, given a directed graph $G$, two vertices $s,t \in V(G)$, 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 $2^{-\mathrm{poly}(k)}$ the set $Z$ becomes a minimum $st$-cut in the resulting graph. We also provide a deterministic counterpart of this procedure. 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} or Weighted Directed Feedback Vertex Set. 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.