1,518 research outputs found
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.
Induced Minor Free Graphs: Isomorphism and Clique-width
Given two graphs and , we say that contains as an induced
minor if a graph isomorphic to can be obtained from by a sequence of
vertex deletions and edge contractions. We study the complexity of Graph
Isomorphism on graphs that exclude a fixed graph as an induced minor. More
precisely, we determine for every graph that Graph Isomorphism is
polynomial-time solvable on -induced-minor-free graphs or that it is
GI-complete. Additionally, we classify those graphs for which
-induced-minor-free graphs have bounded clique-width. These two results
complement similar dichotomies for graphs that exclude a fixed graph as an
induced subgraph, minor, or subgraph.Comment: 16 pages, 5 figures. An extended abstract of this paper previously
appeared in the proceedings of the 41st International Workshop on
Graph-Theoretic Concepts in Computer Science (WG 2015
Capturing Topology in Graph Pattern Matching
Graph pattern matching is often defined in terms of subgraph isomorphism, an
NP-complete problem. To lower its complexity, various extensions of graph
simulation have been considered instead. These extensions allow pattern
matching to be conducted in cubic-time. However, they fall short of capturing
the topology of data graphs, i.e., graphs may have a structure drastically
different from pattern graphs they match, and the matches found are often too
large to understand and analyze. To rectify these problems, this paper proposes
a notion of strong simulation, a revision of graph simulation, for graph
pattern matching. (1) We identify a set of criteria for preserving the topology
of graphs matched. We show that strong simulation preserves the topology of
data graphs and finds a bounded number of matches. (2) We show that strong
simulation retains the same complexity as earlier extensions of simulation, by
providing a cubic-time algorithm for computing strong simulation. (3) We
present the locality property of strong simulation, which allows us to
effectively conduct pattern matching on distributed graphs. (4) We
experimentally verify the effectiveness and efficiency of these algorithms,
using real-life data and synthetic data.Comment: VLDB201
Complexity of Token Swapping and its Variants
In the Token Swapping problem we are given a graph with a token placed on
each vertex. Each token has exactly one destination vertex, and we try to move
all the tokens to their destinations, using the minimum number of swaps, i.e.,
operations of exchanging the tokens on two adjacent vertices. As the main
result of this paper, we show that Token Swapping is -hard parameterized
by the length of a shortest sequence of swaps. In fact, we prove that, for
any computable function , it cannot be solved in time where is the number of vertices of the input graph, unless the ETH
fails. This lower bound almost matches the trivial -time algorithm.
We also consider two generalizations of the Token Swapping, namely Colored
Token Swapping (where the tokens have different colors and tokens of the same
color are indistinguishable), and Subset Token Swapping (where each token has a
set of possible destinations). To complement the hardness result, we prove that
even the most general variant, Subset Token Swapping, is FPT in nowhere-dense
graph classes.
Finally, we consider the complexities of all three problems in very
restricted classes of graphs: graphs of bounded treewidth and diameter, stars,
cliques, and paths, trying to identify the borderlines between polynomial and
NP-hard cases.Comment: 23 pages, 7 Figure
Fixed-parameter tractability, definability, and model checking
In this article, we study parameterized complexity theory from the
perspective of logic, or more specifically, descriptive complexity theory.
We propose to consider parameterized model-checking problems for various
fragments of first-order logic as generic parameterized problems and show how
this approach can be useful in studying both fixed-parameter tractability and
intractability. For example, we establish the equivalence between the
model-checking for existential first-order logic, the homomorphism problem for
relational structures, and the substructure isomorphism problem. Our main
tractability result shows that model-checking for first-order formulas is
fixed-parameter tractable when restricted to a class of input structures with
an excluded minor. On the intractability side, for every t >= 0 we prove an
equivalence between model-checking for first-order formulas with t quantifier
alternations and the parameterized halting problem for alternating Turing
machines with t alternations. We discuss the close connection between this
alternation hierarchy and Downey and Fellows' W-hierarchy.
On a more abstract level, we consider two forms of definability, called Fagin
definability and slicewise definability, that are appropriate for describing
parameterized problems. We give a characterization of the class FPT of all
fixed-parameter tractable problems in terms of slicewise definability in finite
variable least fixed-point logic, which is reminiscent of the Immerman-Vardi
Theorem characterizing the class PTIME in terms of definability in least
fixed-point logic.Comment: To appear in SIAM Journal on Computin
- …