## Directed Graphs: Fixed-Parameter Tractability & Beyond

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2014##### Author

Chitnis, Rajesh

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Hajiaghayi, Mohammad

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Most interesting optimization problems on graphs are NP-hard, implying that (unless P=NP) there is no polynomial time algorithm that solves all the instances of an NP-hard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of <italic>parameterized complexity</italic> was introduced by Downey and Fellows to allow for a more refined multivariate analysis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized complexity is to design efficient algorithms for NP-hard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixed-parameter tractable (FPT) if instances of size n and parameter k can be solved in f(k).n<super>O(1)</super> time, where f is a computable function which does not depend on n. A parameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f(k).n<super>O(g(k))</super> time, where f and g are both computable functions.
In this thesis we focus on the parameterized complexity of transversal and connectivity problems on directed graphs. This research direction has been hitherto relatively unexplored: usually the directed version of the problems require significantly different and more involved ideas than the ones for the undirected version. Furthermore, for directed graphs there are no known algorithmic meta-techniques: for example, there is no known algorithmic analogue of the Graph Minor Theory of Robertson and Seymour for directed graphs. As a result, the fixed-parameter tractability status of the directed versions of several fundamental problems such as Multiway Cut, Multicut, Subset Feedback Vertex Set, Odd Cycle Transversal, etc. was open.
In the first part of the thesis, we develop the framework of <italic>shadowless solutions</italic> for a general class of transversal problems in directed graphs. For this class of problems, we reduce the problem of finding a solution in FPT time to that of finding a shadowless solution. Since shadowless solutions have a good (problem-specific) structure, this provides an important first step in the design of FPT algorithms for problems on directed graphs.
By understanding the structure of shadowless solutions, we are able to design the first FPT algorithms for the Directed Multiway Cut problem and the Directed Subset Feedback Vertex Set problem.
In the second part of the thesis, we present <italic>tight</italic> bounds on the parameterized complexity of well-studied directed connectivity problems such as Strongly Connected Steiner Subgraph and Directed Steiner Forest when parameterized by the number of terminals/terminal pairs. We design new optimal XP algorithms for the aforementioned problems, and also prove matching lower bounds for existing XP algorithms. Most of our hardness results hold even if the underlying undirected graph is planar.
Finally, we conclude with some open problems regarding the parameterized complexity of transversal and connectivity problems on directed graphs.

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