24 research outputs found
Maximum flow is approximable by deterministic constant-time algorithm in sparse networks
We show a deterministic constant-time parallel algorithm for finding an
almost maximum flow in multisource-multitarget networks with bounded degrees
and bounded edge capacities. As a consequence, we show that the value of the
maximum flow over the number of nodes is a testable parameter on these
networks.Comment: 8 page
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
What does the local structure of a planar graph tell us about its global structure?
The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G?
In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)
Random local algorithms
Consider the problem when we want to construct some structure on a bounded
degree graph, e.g. an almost maximum matching, and we want to decide about each
edge depending only on its constant radius neighbourhood. We show that the
information about the local statistics of the graph does not help here. Namely,
if there exists a random local algorithm which can use any local statistics
about the graph, and produces an almost optimal structure, then the same can be
achieved by a random local algorithm using no statistics.Comment: 9 page
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
An Efficient Partitioning Oracle for Bounded-Treewidth Graphs
Partitioning oracles were introduced by Hassidim et al. (FOCS 2009) as a
generic tool for constant-time algorithms. For any epsilon > 0, a partitioning
oracle provides query access to a fixed partition of the input bounded-degree
minor-free graph, in which every component has size poly(1/epsilon), and the
number of edges removed is at most epsilon*n, where n is the number of vertices
in the graph.
However, the oracle of Hassidimet al. makes an exponential number of queries
to the input graph to answer every query about the partition. In this paper, we
construct an efficient partitioning oracle for graphs with constant treewidth.
The oracle makes only O(poly(1/epsilon)) queries to the input graph to answer
each query about the partition.
Examples of bounded-treewidth graph classes include k-outerplanar graphs for
fixed k, series-parallel graphs, cactus graphs, and pseudoforests. Our oracle
yields poly(1/epsilon)-time property testing algorithms for membership in these
classes of graphs. Another application of the oracle is a poly(1/epsilon)-time
algorithm that approximates the maximum matching size, the minimum vertex cover
size, and the minimum dominating set size up to an additive epsilon*n in graphs
with bounded treewidth. Finally, the oracle can be used to test in
poly(1/epsilon) time whether the input bounded-treewidth graph is k-colorable
or perfect.Comment: Full version of a paper to appear in RANDOM 201