105 research outputs found
Vertex Cover Gets Faster and Harder on Low Degree Graphs
The problem of finding an optimal vertex cover in a graph is a classic
NP-complete problem, and is a special case of the hitting set question. On the
other hand, the hitting set problem, when asked in the context of induced
geometric objects, often turns out to be exactly the vertex cover problem on
restricted classes of graphs. In this work we explore a particular instance of
such a phenomenon. We consider the problem of hitting all axis-parallel slabs
induced by a point set P, and show that it is equivalent to the problem of
finding a vertex cover on a graph whose edge set is the union of two
Hamiltonian Paths. We show the latter problem to be NP-complete, and we also
give an algorithm to find a vertex cover of size at most k, on graphs of
maximum degree four, whose running time is 1.2637^k n^O(1)
Fine-Grained Complexity of Rainbow Coloring and its Variants
Consider a graph G and an edge-coloring c_R:E(G) rightarrow [k]. A rainbow path between u,v in V(G) is a path P from u to v such that for all e,e\u27 in E(P), where e neq e\u27 we have c_R(e) neq c_R(e\u27). In the Rainbow k-Coloring problem we are given a graph G, and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in V(G) there is a rainbow path between u and v in G. Several variants of Rainbow k-Coloring have been studied, two of which are defined as follows. The Subset Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G) times V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all (u,v) in S there is a rainbow path between u and v in G. The problem Steiner Rainbow k-Coloring takes as an input a graph G and a set S subseteq V(G), and the objective is to decide if there exists c_R: E(G) rightarrow [k] such that for all u,v in S there is a rainbow path between u and v in G. In an attempt to resolve open problems posed by Kowalik et al. (ESA 2016), we obtain the following results.
- For every k geq 3, Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|E(G)|)}n^{O(1)}, unless ETH fails.
- For every k geq 3, Steiner Rainbow k-Coloring does not admit an algorithm running in time 2^{o(|S|^2)}n^{O(1)}, unless ETH fails.
- Subset Rainbow k-Coloring admits an algorithm running in time 2^{OO(|S|)}n^{O(1)}. This also implies an algorithm running in time 2^{o(|S|^2)}n^{O(1)} for Steiner Rainbow k-Coloring, which matches the lower bound we obtain
Simultaneous Feedback Vertex Set: A Parameterized Perspective
Given a family of graphs , a graph , and a positive integer
, the -Deletion problem asks whether we can delete at most
vertices from to obtain a graph in . -Deletion
generalizes many classical graph problems such as Vertex Cover, Feedback Vertex
Set, and Odd Cycle Transversal. A graph ,
where the edge set of is partitioned into color classes, is called
an -edge-colored graph. A natural extension of the
-Deletion problem to edge-colored graphs is the
-Simultaneous -Deletion problem. In the latter problem, we
are given an -edge-colored graph and the goal is to find a set
of at most vertices such that each graph , where and , is in . In this work, we
study -Simultaneous -Deletion for being the
family of forests. In other words, we focus on the -Simultaneous
Feedback Vertex Set (-SimFVS) problem. Algorithmically, we show that,
like its classical counterpart, -SimFVS parameterized by is
fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed
constant . In particular, we give an algorithm running in time and a kernel with vertices. The
running time of our algorithm implies that -SimFVS is FPT even when
. We complement this positive result by showing that for
, where is the number of vertices in the input graph,
-SimFVS becomes W[1]-hard. Our positive results answer one of the open
problems posed by Cai and Ye (MFCS 2014)
Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion
Given a graph and a parameter , the Chordal Vertex Deletion (CVD)
problem asks whether there exists a subset of size at most
that hits all induced cycles of size at least 4. The existence of a
polynomial kernel for CVD was a well-known open problem in the field of
Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question
affirmatively by designing a polynomial kernel for CVD of size
, and asked whether one can design a kernel of size
. While we do not completely resolve this question, we design a
significantly smaller kernel of size , inspired by the
-size kernel for Feedback Vertex Set. Furthermore, we introduce the
notion of the independence degree of a vertex, which is our main conceptual
contribution
On the Parameterized Complexity of Contraction to Generalization of Trees
For a family of graphs F, the F-Contraction problem takes as an input a graph G and an integer k, and the goal is to decide if there exists S subseteq E(G) of size at most k such that G/S belongs to F. Here, G/S is the graph obtained from G by contracting all the edges in S. Heggernes et al.[Algorithmica (2014)] were the first to study edge contraction problems in the realm of Parameterized Complexity. They studied cal F-Contraction when F is a simple family of graphs such as trees and paths. In this paper, we study the F-Contraction problem, where F generalizes the family of trees. In particular, we define this generalization in a "parameterized way". Let T_ell be the family of graphs such that each graph in T_ell can be made into a tree by deleting at most ell edges. Thus, the problem we study is T_ell-Contraction. We design an FPT algorithm for T_ell-Contraction running in time O((ncol)^{O(k + ell)} * n^{O(1)}). Furthermore, we show that the problem does not admit a polynomial kernel when parameterized by k. Inspired by the negative result for the kernelization, we design a lossy kernel for T_ell-Contraction of size O([k(k + 2ell)] ^{(lceil {frac{alpha}{alpha-1}rceil + 1)}})
A Finite Algorithm for the Realizabilty of a Delaunay Triangulation
The \emph{Delaunay graph} of a point set is the
plane graph with the vertex-set and the edge-set that contains
if there exists a disc whose intersection with is exactly .
Accordingly, a triangulated graph is \emph{Delaunay realizable} if there
exists a triangulation of the Delaunay graph of some , called a \emph{Delaunay triangulation} of , that is
isomorphic to . The objective of \textsc{Delaunay Realization} is to compute
a point set that realizes a given graph (if such
a exists). Known algorithms do not solve \textsc{Delaunay Realization} as
they are non-constructive. Obtaining a constructive algorithm for
\textsc{Delaunay Realization} was mentioned as an open problem by Hiroshima et
al.~\cite{hiroshima2000}. We design an -time constructive
algorithm for \textsc{Delaunay Realization}. In fact, our algorithm outputs
sets of points with {\em integer} coordinates
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