4,462 research outputs found
Oriented coloring on recursively defined digraphs
Coloring is one of the most famous problems in graph theory. The coloring
problem on undirected graphs has been well studied, whereas there are very few
results for coloring problems on directed graphs. An oriented k-coloring of an
oriented graph G=(V,A) is a partition of the vertex set V into k independent
sets such that all the arcs linking two of these subsets have the same
direction. The oriented chromatic number of an oriented graph G is the smallest
k such that G allows an oriented k-coloring. Deciding whether an acyclic
digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the
chromatic number of an oriented graph is an NP-hard problem. This motivates to
consider the problem on oriented co-graphs. After giving several
characterizations for this graph class, we show a linear time algorithm which
computes an optimal oriented coloring for an oriented co-graph. We further
prove how the oriented chromatic number can be computed for the disjoint union
and order composition from the oriented chromatic number of the involved
oriented co-graphs. It turns out that within oriented co-graphs the oriented
chromatic number is equal to the length of a longest oriented path plus one. We
also show that the graph isomorphism problem on oriented co-graphs can be
solved in linear time.Comment: 14 page
Log-space Algorithms for Paths and Matchings in k-trees
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width 2 [JT07]. However, for
graphs of tree-width larger than 2, no bound better than NL is known. In this
paper, we improve these bounds for k-trees, where k is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed k-trees, and for computation of shortest and longest
paths in directed acyclic k-trees.
Besides the path problems mentioned above, we also consider the problem of
deciding whether a k-tree has a perfect macthing (decision version), and if so,
finding a perfect match- ing (search version), and prove that these two
problems are L-complete. These problems are known to be in P and in RNC for
general graphs, and in SPL for planar bipartite graphs [DKR08].
Our results settle the complexity of these problems for the class of k-trees.
The results are also applicable for bounded tree-width graphs, when a
tree-decomposition is given as input. The technique central to our algorithms
is a careful implementation of divide-and-conquer approach in log-space, along
with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201
Notes on the Schreier graphs of the Grigorchuk group
The paper is concerned with the space of the marked Schreier graphs of the
Grigorchuk group and the action of the group on this space. In particular, we
describe an invariant set of the Schreier graphs corresponding to the action on
the boundary of the binary rooted tree and dynamics of the group action
restricted to this invariant set.Comment: 33 pages, 4 figure
Grad and Classes with Bounded Expansion II. Algorithmic Aspects
Classes of graphs with bounded expansion are a generalization of both proper
minor closed classes and degree bounded classes. Such classes are based on a
new invariant, the greatest reduced average density (grad) of G with rank r,
∇r(G). These classes are also characterized by the existence of several
partition results such as the existence of low tree-width and low tree-depth
colorings. These results lead to several new linear time algorithms, such as an
algorithm for counting all the isomorphs of a fixed graph in an input graph or
an algorithm for checking whether there exists a subset of vertices of a priori
bounded size such that the subgraph induced by this subset satisfies some
arbirtrary but fixed first order sentence. We also show that for fixed p,
computing the distances between two vertices up to distance p may be performed
in constant time per query after a linear time preprocessing. We also show,
extending several earlier results, that a class of graphs has sublinear
separators if it has sub-exponential expansion. This result result is best
possible in general
Combinatorics of compactified universal Jacobians
We use orientations on stable graphs to express the combinatorial structure
of the compactified universal Jacobians in degrees g-1 and g over the moduli
space of stable curves, \Mgb, and construct for them graded stratifications
compatible with the one of \Mgb. In particular, for a stable curve we exhibit
graded stratifications of the compactified Jacobians in terms of totally
cyclic, respectively rooted, orientations on subgraphs of its dual graph.Comment: Final version, to appear in Advances in Mathematics. 41 page
Any-k: Anytime Top-k Tree Pattern Retrieval in Labeled Graphs
Many problems in areas as diverse as recommendation systems, social network
analysis, semantic search, and distributed root cause analysis can be modeled
as pattern search on labeled graphs (also called "heterogeneous information
networks" or HINs). Given a large graph and a query pattern with node and edge
label constraints, a fundamental challenge is to nd the top-k matches ac-
cording to a ranking function over edge and node weights. For users, it is di
cult to select value k . We therefore propose the novel notion of an any-k
ranking algorithm: for a given time budget, re- turn as many of the top-ranked
results as possible. Then, given additional time, produce the next lower-ranked
results quickly as well. It can be stopped anytime, but may have to continues
until all results are returned. This paper focuses on acyclic patterns over
arbitrary labeled graphs. We are interested in practical algorithms that
effectively exploit (1) properties of heterogeneous networks, in particular
selective constraints on labels, and (2) that the users often explore only a
fraction of the top-ranked results. Our solution, KARPET, carefully integrates
aggressive pruning that leverages the acyclic nature of the query, and
incremental guided search. It enables us to prove strong non-trivial time and
space guarantees, which is generally considered very hard for this type of
graph search problem. Through experimental studies we show that KARPET achieves
running times in the order of milliseconds for tree patterns on large networks
with millions of nodes and edges.Comment: To appear in WWW 201
The Complexity of Bisimulation and Simulation on Finite Systems
In this paper the computational complexity of the (bi)simulation problem over
restricted graph classes is studied. For trees given as pointer structures or
terms the (bi)simulation problem is complete for logarithmic space or NC,
respectively. This solves an open problem from Balc\'azar, Gabarr\'o, and
S\'antha. Furthermore, if only one of the input graphs is required to be a
tree, the bisimulation (simulation) problem is contained in AC (LogCFL). In
contrast, it is also shown that the simulation problem is P-complete already
for graphs of bounded path-width
Linear Time Subgraph Counting, Graph Degeneracy, and the Chasm at Size Six
We consider the problem of counting all k-vertex subgraphs in an input graph, for any constant k. This problem (denoted SUB-CNT_k) has been studied extensively in both theory and practice. In a classic result, Chiba and Nishizeki (SICOMP 85) gave linear time algorithms for clique and 4-cycle counting for bounded degeneracy graphs. This is a rich class of sparse graphs that contains, for example, all minor-free families and preferential attachment graphs. The techniques from this result have inspired a number of recent practical algorithms for SUB-CNT_k. Towards a better understanding of the limits of these techniques, we ask: for what values of k can SUB_CNT_k be solved in linear time?
We discover a chasm at k=6. Specifically, we prove that for k < 6, SUB_CNT_k can be solved in linear time. Assuming a standard conjecture in fine-grained complexity, we prove that for all k ? 6, SUB-CNT_k cannot be solved even in near-linear time
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