11,302 research outputs found
Finding a Small Number of Colourful Components
A partition (V_1,...,V_k) of the vertex set of a graph G with a (not necessarily proper) colouring c is colourful if no two vertices in any V_i have the same colour and every set V_i induces a connected graph. The Colourful Partition problem, introduced by Adamaszek and Popa, is to decide whether a coloured graph (G,c) has a colourful partition of size at most k. This problem is related to the Colourful Components problem, introduced by He, Liu and Zhao, which is to decide whether a graph can be modified into a graph whose connected components form a colourful partition by deleting at most p edges.
Despite the similarities in their definitions, we show that Colourful Partition and Colourful Components may have different complexities for restricted instances. We tighten known NP-hardness results for both problems by closing a number of complexity gaps. In addition, we prove new hardness and tractability results for Colourful Partition. In particular, we prove that deciding whether a coloured graph (G,c) has a colourful partition of size 2 is NP-complete for coloured planar bipartite graphs of maximum degree 3 and path-width 3, but polynomial-time solvable for coloured graphs of treewidth 2.
Rather than performing an ad hoc study, we use our classical complexity results to guide us in undertaking a thorough parameterized study of Colourful Partition. We show that this leads to suitable parameters for obtaining FPT results and moreover prove that Colourful Components and Colourful Partition may have different parameterized complexities, depending on the chosen parameter
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Families of graph-different Hamilton paths
Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D)
be the maximum of the cardinality of a set of Hamiltonian paths in the complete
graph K_n such that the union of any two paths from the family contains a not
necessarily induced cycle of some length from D. We determine or bound the
asymptotics of M(n;D) in various special cases. This problem is closely related
to that of the permutation capacity of graphs and constitutes a further
extension of the problem area around Shannon capacity. We also discuss how to
generalize our cycle-difference problems and present an example where cycles
are replaced by 4-cliques. These problems are in a natural duality to those of
graph intersection, initiated by Erd\"os, Simonovits and S\'os. The lack of
kernel structure as a natural candidate for optimum makes our problems quite
challenging
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Approximately Counting Embeddings into Random Graphs
Let H be a graph, and let C_H(G) be the number of (subgraph isomorphic)
copies of H contained in a graph G. We investigate the fundamental problem of
estimating C_H(G). Previous results cover only a few specific instances of this
general problem, for example, the case when H has degree at most one
(monomer-dimer problem). In this paper, we present the first general subcase of
the subgraph isomorphism counting problem which is almost always efficiently
approximable. The results rely on a new graph decomposition technique.
Informally, the decomposition is a labeling of the vertices such that every
edge is between vertices with different labels and for every vertex all
neighbors with a higher label have identical labels. The labeling implicitly
generates a sequence of bipartite graphs which permits us to break the problem
of counting embeddings of large subgraphs into that of counting embeddings of
small subgraphs. Using this method, we present a simple randomized algorithm
for the counting problem. For all decomposable graphs H and all graphs G, the
algorithm is an unbiased estimator. Furthermore, for all graphs H having a
decomposition where each of the bipartite graphs generated is small and almost
all graphs G, the algorithm is a fully polynomial randomized approximation
scheme.
We show that the graph classes of H for which we obtain a fully polynomial
randomized approximation scheme for almost all G includes graphs of degree at
most two, bounded-degree forests, bounded-length grid graphs, subdivision of
bounded-degree graphs, and major subclasses of outerplanar graphs,
series-parallel graphs and planar graphs, whereas unbounded-length grid graphs
are excluded.Comment: Earlier version appeared in Random 2008. Fixed an typo in Definition
3.
Quantum Commuting Circuits and Complexity of Ising Partition Functions
Instantaneous quantum polynomial-time (IQP) computation is a class of quantum
computation consisting only of commuting two-qubit gates and is not universal
in the sense of standard quantum computation. Nevertheless, it has been shown
that if there is a classical algorithm that can simulate IQP efficiently, the
polynomial hierarchy (PH) collapses at the third level, which is highly
implausible. However, the origin of the classical intractability is still less
understood. Here we establish a relationship between IQP and computational
complexity of the partition functions of Ising models. We apply the established
relationship in two opposite directions. One direction is to find subclasses of
IQP that are classically efficiently simulatable in the strong sense, by using
exact solvability of certain types of Ising models. Another direction is
applying quantum computational complexity of IQP to investigate (im)possibility
of efficient classical approximations of Ising models with imaginary coupling
constants. Specifically, we show that there is no fully polynomial randomized
approximation scheme (FPRAS) for Ising models with almost all imaginary
coupling constants even on a planar graph of a bounded degree, unless the PH
collapses at the third level. Furthermore, we also show a multiplicative
approximation of such a class of Ising partition functions is at least as hard
as a multiplicative approximation for the output distribution of an arbitrary
quantum circuit.Comment: 36 pages, 5 figure
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