39 research outputs found
Partitioning problems in dense hypergraphs
AbstractWe study the general partitioning problem and the discrepancy problem in dense hypergraphs. Using the regularity lemma (SzemerĂ©di, Problemes Combinatories et Theorie des Graphes (1978), pp. 399â402) and its algorithmic version proved in Czygrinow and Rödl (SIAM J. Comput., to appear), we give polynomial-time approximation schemes for the general partitioning problem and for the discrepancy problem
Exact bounds for distributed graph colouring
We prove exact bounds on the time complexity of distributed graph colouring.
If we are given a directed path that is properly coloured with colours, by
prior work it is known that we can find a proper 3-colouring in communication rounds. We close the gap between upper and
lower bounds: we show that for infinitely many the time complexity is
precisely communication rounds.Comment: 16 pages, 3 figure
Node Labels in Local Decision
The role of unique node identifiers in network computing is well understood
as far as symmetry breaking is concerned. However, the unique identifiers also
leak information about the computing environment - in particular, they provide
some nodes with information related to the size of the network. It was recently
proved that in the context of local decision, there are some decision problems
such that (1) they cannot be solved without unique identifiers, and (2) unique
node identifiers leak a sufficient amount of information such that the problem
becomes solvable (PODC 2013).
In this work we give study what is the minimal amount of information that we
need to leak from the environment to the nodes in order to solve local decision
problems. Our key results are related to scalar oracles that, for any given
, provide a multiset of labels; then the adversary assigns the
labels to the nodes in the network. This is a direct generalisation of the
usual assumption of unique node identifiers. We give a complete
characterisation of the weakest oracle that leaks at least as much information
as the unique identifiers.
Our main result is the following dichotomy: we classify scalar oracles as
large and small, depending on their asymptotic behaviour, and show that (1) any
large oracle is at least as powerful as the unique identifiers in the context
of local decision problems, while (2) for any small oracle there are local
decision problems that still benefit from unique identifiers.Comment: Conference version to appear in the proceedings of SIROCCO 201
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
Locally Optimal Load Balancing
This work studies distributed algorithms for locally optimal load-balancing:
We are given a graph of maximum degree , and each node has up to
units of load. The task is to distribute the load more evenly so that the loads
of adjacent nodes differ by at most .
If the graph is a path (), it is easy to solve the fractional
version of the problem in communication rounds, independently of the
number of nodes. We show that this is tight, and we show that it is possible to
solve also the discrete version of the problem in rounds in paths.
For the general case (), we show that fractional load balancing
can be solved in rounds and discrete load
balancing in rounds for some function , independently of the
number of nodes.Comment: 19 pages, 11 figure
Optimal pebbling number of graphs with given minimum degree
Consider a distribution of pebbles on a connected graph . A pebbling move
removes two pebbles from a vertex and places one to an adjacent vertex. A
vertex is reachable under a pebbling distribution if it has a pebble after the
application of a sequence of pebbling moves. The optimal pebbling number
is the smallest number of pebbles which we can distribute in such a
way that each vertex is reachable. It was known that the optimal pebbling
number of any connected graph is at most , where
is the minimum degree of the graph. We strengthen this bound by showing that
equality cannot be attained and that the bound is sharp. If
then we further improve the bound to
. On the other hand, we show that for
arbitrary large diameter and any there are infinitely many graphs
whose optimal pebbling number is bigger than
A local 2-approximation algorithm for the vertex cover problem
We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in (Î + 1)2 synchronous communication rounds, where Î is the maximum degree of the graph. For Î = 3, we give a 2-approximation algorithm also for the weighted version of the problem.Peer reviewe
Local algorithms : Self-stabilization on speed
Non peer reviewe