2,935 research outputs found
Rainbow Coloring Hardness via Low Sensitivity Polymorphisms
A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs.
Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover
On the fine-grained complexity of rainbow coloring
The Rainbow k-Coloring problem asks whether the edges of a given graph can be
colored in colors so that every pair of vertices is connected by a rainbow
path, i.e., a path with all edges of different colors. Our main result states
that for any , there is no algorithm for Rainbow k-Coloring running in
time , unless ETH fails.
Motivated by this negative result we consider two parameterized variants of
the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et
al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set of
pairs of vertices and we ask if there is a coloring in which all the pairs in
are connected by rainbow paths. We show that Subset Rainbow k-Coloring is
FPT when parameterized by . We also study Maximum Rainbow k-Coloring
problem, where we are additionally given an integer and we ask if there is
a coloring in which at least anti-edges are connected by rainbow paths. We
show that the problem is FPT when parameterized by and has a kernel of size
for every (thus proving that the problem is FPT), extending the
result of Ananth et al. [FSTTCS 2011]
Computing Minimum Rainbow and Strong Rainbow Colorings of Block Graphs
A path in an edge-colored graph is rainbow if no two edges of it are
colored the same. The graph is rainbow-connected if there is a rainbow path
between every pair of vertices. If there is a rainbow shortest path between
every pair of vertices, the graph is strongly rainbow-connected. The
minimum number of colors needed to make rainbow-connected is known as the
rainbow connection number of , and is denoted by . Similarly,
the minimum number of colors needed to make strongly rainbow-connected is
known as the strong rainbow connection number of , and is denoted by
. We prove that for every , deciding whether
is NP-complete for split graphs, which form a subclass
of chordal graphs. Furthermore, there exists no polynomial-time algorithm for
approximating the strong rainbow connection number of an -vertex split graph
with a factor of for any unless P = NP. We
then turn our attention to block graphs, which also form a subclass of chordal
graphs. We determine the strong rainbow connection number of block graphs, and
show it can be computed in linear time. Finally, we provide a polynomial-time
characterization of bridgeless block graphs with rainbow connection number at
most 4.Comment: 13 pages, 3 figure
Algorithms and Bounds for Very Strong Rainbow Coloring
A well-studied coloring problem is to assign colors to the edges of a graph
so that, for every pair of vertices, all edges of at least one shortest
path between them receive different colors. The minimum number of colors
necessary in such a coloring is the strong rainbow connection number
(\src(G)) of the graph. When proving upper bounds on \src(G), it is natural
to prove that a coloring exists where, for \emph{every} shortest path between
every pair of vertices in the graph, all edges of the path receive different
colors. Therefore, we introduce and formally define this more restricted edge
coloring number, which we call \emph{very strong rainbow connection number}
(\vsrc(G)).
In this paper, we give upper bounds on \vsrc(G) for several graph classes,
some of which are tight. These immediately imply new upper bounds on \src(G)
for these classes, showing that the study of \vsrc(G) enables meaningful
progress on bounding \src(G). Then we study the complexity of the problem to
compute \vsrc(G), particularly for graphs of bounded treewidth, and show this
is an interesting problem in its own right. We prove that \vsrc(G) can be
computed in polynomial time on cactus graphs; in contrast, this question is
still open for \src(G). We also observe that deciding whether \vsrc(G) = k
is fixed-parameter tractable in and the treewidth of . Finally, on
general graphs, we prove that there is no polynomial-time algorithm to decide
whether \vsrc(G) \leq 3 nor to approximate \vsrc(G) within a factor
, unless PNP
Weak embeddings of posets to the Boolean lattice
The goal of this paper is to prove that several variants of deciding whether
a poset can be (weakly) embedded into a small Boolean lattice, or to a few
consecutive levels of a Boolean lattice, are NP-complete, answering a question
of Griggs and of Patkos. As an equivalent reformulation of one of these
problems, we also derive that it is NP-complete to decide whether a given graph
can be embedded to the two middle levels of some hypercube
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