10 research outputs found
Fixed-parameter tractability of multicut parameterized by the size of the cutset
Given an undirected graph , a collection of
pairs of vertices, and an integer , the Edge Multicut problem ask if there
is a set of at most edges such that the removal of disconnects
every from the corresponding . Vertex Multicut is the analogous
problem where is a set of at most vertices. Our main result is that
both problems can be solved in time , i.e.,
fixed-parameter tractable parameterized by the size of the cutset in the
solution. By contrast, it is unlikely that an algorithm with running time of
the form exists for the directed version of the problem, as
we show it to be W[1]-hard parameterized by the size of the cutset
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
Directed Graphs: Fixed-Parameter Tractability & Beyond
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 parameterized complexity 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).nO(1) 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).nO(g(k)) 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 shadowless solutions 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 tight 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
Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable
Given a graph and an integer , the Feedback Vertex Set (FVS) problem
asks if there is a vertex set of size at most that hits all cycles in
the graph. The fixed-parameter tractability status of FVS in directed graphs
was a long-standing open problem until Chen et al. (STOC '08) showed that it is
FPT by giving a time algorithm. In the subset versions of
this problems, we are given an additional subset of vertices (resp., edges)
and we want to hit all cycles passing through a vertex of (resp. an edge of
). Recently, the Subset Feedback Vertex Set in undirected graphs was shown
to be FPT by Cygan et al. (ICALP '11) and independently by Kakimura et al.
(SODA '12). We generalize the result of Chen et al. (STOC '08) by showing that
Subset Feedback Vertex Set in directed graphs can be solved in time
. By our result, we complete the picture for feedback
vertex set problems and their subset versions in undirected and directed
graphs. Besides proving the fixed-parameter tractability of Directed Subset
Feedback Vertex Set, we reformulate the random sampling of important separators
technique in an abstract way that can be used for a general family of
transversal problems. Moreover, we modify the probability distribution used in
the technique to achieve better running time; in particular, this gives an
improvement from to in the parameter dependence of
the Directed Multiway Cut algorithm of Chitnis et al. (SODA '12).Comment: To appear in ACM Transactions on Algorithms. A preliminary version
appeared in ICALP '12. We would like to thank Marcin Pilipczuk for pointing
out a missing case in the conference version which has been considered in
this version. Also, we give an single exponential FPT algorithm improving on
the double exponential algorithm from the conference versio
Fixed-Parameter Tractability of Directed Multicut with Three Terminal Pairs Parameterized by the Size of the Cutset: Twin-width Meets Flow-Augmentation
We show fixed-parameter tractability of the Directed Multicut problem withthree terminal pairs (with a randomized algorithm). This problem, given adirected graph , pairs of vertices (called terminals) ,, and , and an integer , asks to find a set of at most non-terminal vertices in that intersect all -paths, all-paths, and all -paths. The parameterized complexity of thiscase has been open since Chitnis, Cygan, Hajiaghayi, and Marx provedfixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, andPilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairscase at SODA 2016. On the technical side, we use two recent developments in parameterizedalgorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem withfew variables and constraints over a large ordered domain.We observe that thisproblem can be in turn encoded as an FO model-checking task over a structureconsisting of a few 0-1 matrices. We look at this problem through the lenses oftwin-width, a recently introduced structural parameter [Bonnet, Kim,Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] thesaid FO model-checking task can be done in FPT time if the said matrices havebounded grid rank. To complete the proof, we show an irrelevant vertex rule: Ifany of the matrices in the said encoding has a large grid minor, a vertexcorresponding to the ``middle'' box in the grid minor can be proclaimedirrelevant -- not contained in the sought solution -- and thus reduced.<br
Fixed-parameter tractability of Directed Multicut with three terminal pairs parameterized by the size of the cutset: twin-width meets flow-augmentation
We show fixed-parameter tractability of the Directed Multicut problem with
three terminal pairs (with a randomized algorithm). This problem, given a
directed graph , pairs of vertices (called terminals) ,
, and , and an integer , asks to find a set of at most
non-terminal vertices in that intersect all -paths, all
-paths, and all -paths. The parameterized complexity of this
case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved
fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and
Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs
case at SODA 2016.
On the technical side, we use two recent developments in parameterized
algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,
Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with
few variables and constraints over a large ordered domain.We observe that this
problem can be in turn encoded as an FO model-checking task over a structure
consisting of a few 0-1 matrices. We look at this problem through the lenses of
twin-width, a recently introduced structural parameter [Bonnet, Kim,
Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,
Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the
said FO model-checking task can be done in FPT time if the said matrices have
bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If
any of the matrices in the said encoding has a large grid minor, a vertex
corresponding to the ``middle'' box in the grid minor can be proclaimed
irrelevant -- not contained in the sought solution -- and thus reduced
Parameterized Algorithms for Generalizations of Directed Feedback Vertex Set
The Directed Feedback Vertex Set (DFVS) problem takes as input a directed
graph~ and seeks a smallest vertex set~ that hits all cycles in . This
is one of Karp's 21 -complete problems. Resolving the
parameterized complexity status of DFVS was a long-standing open problem until
Chen et al. [STOC 2008, J. ACM 2008] showed its fixed-parameter tractability
via a -time algorithm, where .
Here we show fixed-parameter tractability of two generalizations of DFVS:
- Find a smallest vertex set such that every strong component of
has size at most~: we give an algorithm solving this problem in time
. This generalizes an algorithm by Xiao
[JCSS 2017] for the undirected version of the problem.
- Find a smallest vertex set such that every non-trivial strong component
of is 1-out-regular: we give an algorithm solving this problem in time
.
We also solve the corresponding arc versions of these problems by
fixed-parameter algorithms