46 research outputs found
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
Local algorithms in (weakly) coloured graphs
A local algorithm is a distributed algorithm that completes after a constant
number of synchronous communication rounds. We present local approximation
algorithms for the minimum dominating set problem and the maximum matching
problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured
graph, both problems admit a local algorithm with the approximation factor
, where is the maximum degree of the graph. We also give
a matching lower bound proving that there is no local algorithm with a better
approximation factor for either of these problems. Furthermore, we show that
the stronger assumption of a 2-colouring does not help in the case of the
dominating set problem, but there is a local approximation scheme for the
maximum matching problem in 2-coloured graphs.Comment: 14 pages, 3 figure
Probabilistic Spectral Sparsification In Sublinear Time
In this paper, we introduce a variant of spectral sparsification, called
probabilistic -spectral sparsification. Roughly speaking,
it preserves the cut value of any cut with an
multiplicative error and a additive error. We show how
to produce a probabilistic -spectral sparsifier with
edges in time
time for unweighted undirected graph. This gives fastest known sub-linear time
algorithms for different cut problems on unweighted undirected graph such as
- An time -approximation
algorithm for the sparsest cut problem and the balanced separator problem.
- A time approximation minimum s-t cut algorithm
with an additive error
A Local Computation Approximation Scheme to Maximum Matching
We present a polylogarithmic local computation matching algorithm which
guarantees a (1-\eps)-approximation to the maximum matching in graphs of
bounded degree.Comment: Appears in Approx 201
Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy
We define the matching measure of a lattice L as the spectral measure of the
tree of self-avoiding walks in L. We connect this invariant to the
monomer-dimer partition function of a sequence of finite graphs converging to
L.
This allows us to express the monomer-dimer free energy of L in terms of the
measure. Exploiting an analytic advantage of the matching measure over the
Mayer series then leads to new, rigorous bounds on the monomer-dimer free
energies of various Euclidean lattices. While our estimates use only the
computational data given in previous papers, they improve the known bounds
significantly.Comment: 18 pages, 3 figure
Distributed Maximum Matching in Bounded Degree Graphs
We present deterministic distributed algorithms for computing approximate
maximum cardinality matchings and approximate maximum weight matchings. Our
algorithm for the unweighted case computes a matching whose size is at least
(1-\eps) times the optimal in \Delta^{O(1/\eps)} +
O\left(\frac{1}{\eps^2}\right) \cdot\log^*(n) rounds where is the number
of vertices in the graph and is the maximum degree. Our algorithm for
the edge-weighted case computes a matching whose weight is at least (1-\eps)
times the optimal in
\log(\min\{1/\wmin,n/\eps\})^{O(1/\eps)}\cdot(\Delta^{O(1/\eps)}+\log^*(n))
rounds for edge-weights in [\wmin,1].
The best previous algorithms for both the unweighted case and the weighted
case are by Lotker, Patt-Shamir, and Pettie~(SPAA 2008). For the unweighted
case they give a randomized (1-\eps)-approximation algorithm that runs in
O((\log(n)) /\eps^3) rounds. For the weighted case they give a randomized
(1/2-\eps)-approximation algorithm that runs in O(\log(\eps^{-1}) \cdot
\log(n)) rounds. Hence, our results improve on the previous ones when the
parameters , \eps and \wmin are constants (where we reduce the
number of runs from to ), and more generally when
, 1/\eps and 1/\wmin are sufficiently slowly increasing functions
of . Moreover, our algorithms are deterministic rather than randomized.Comment: arXiv admin note: substantial text overlap with arXiv:1402.379
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Optimal Dynamic Distributed MIS
Finding a maximal independent set (MIS) in a graph is a cornerstone task in
distributed computing. The local nature of an MIS allows for fast solutions in
a static distributed setting, which are logarithmic in the number of nodes or
in their degrees. The result trivially applies for the dynamic distributed
model, in which edges or nodes may be inserted or deleted. In this paper, we
take a different approach which exploits locality to the extreme, and show how
to update an MIS in a dynamic distributed setting, either \emph{synchronous} or
\emph{asynchronous}, with only \emph{a single adjustment} and in a single
round, in expectation. These strong guarantees hold for the \emph{complete
fully dynamic} setting: Insertions and deletions, of edges as well as nodes,
gracefully and abruptly. This strongly separates the static and dynamic
distributed models, as super-constant lower bounds exist for computing an MIS
in the former.
Our results are obtained by a novel analysis of the surprisingly simple
solution of carefully simulating the greedy \emph{sequential} MIS algorithm
with a random ordering of the nodes. As such, our algorithm has a direct
application as a -approximation algorithm for correlation clustering. This
adds to the important toolbox of distributed graph decompositions, which are
widely used as crucial building blocks in distributed computing.
Finally, our algorithm enjoys a useful \emph{history-independence} property,
meaning the output is independent of the history of topology changes that
constructed that graph. This means the output cannot be chosen, or even biased,
by the adversary in case its goal is to prevent us from optimizing some
objective function.Comment: 19 pages including appendix and reference
Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems
Friedland's Lower Matching Conjecture asserts that if is a --regular
bipartite graph on vertices, and denotes the number of
matchings of size , then where . When
, this conjecture reduces to a theorem of Schrijver which says that a
--regular bipartite graph on vertices has at least
perfect matchings. L. Gurvits
proved an asymptotic version of the Lower Matching Conjecture, namely he proved
that
In this paper, we prove the Lower Matching Conjecture. In fact, we will prove
a slightly stronger statement which gives an extra factor
compared to the conjecture if is separated away from and , and is
tight up to a constant factor if is separated away from . We will also
give a new proof of Gurvits's and Schrijver's theorems, and we extend these
theorems to --biregular bipartite graphs