2 research outputs found
Local Conflict Coloring
Locally finding a solution to symmetry-breaking tasks such as
vertex-coloring, edge-coloring, maximal matching, maximal independent set,
etc., is a long-standing challenge in distributed network computing. More
recently, it has also become a challenge in the framework of centralized local
computation. We introduce conflict coloring as a general symmetry-breaking task
that includes all the aforementioned tasks as specific instantiations ---
conflict coloring includes all locally checkable labeling tasks from
[Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two
parameters and , where the former measures the amount of freedom given
to the nodes for selecting their colors, and the latter measures the number of
constraints which colors of adjacent nodes are subject to.We show that, in the
standard LOCAL model for distributed network computing, if l/d \textgreater{}
\Delta, then conflict coloring can be solved in rounds in -node graphs with maximum degree
, where ignores the polylog factors in . The
dependency in~ is optimal, as a consequence of the lower
bound by [Linial, SIAM J. Comp. 1992] for -coloring. An important
special case of our result is a significant improvement over the best known
algorithm for distributed -coloring due to [Barenboim, PODC 2015],
which required rounds. Improvements for other
variants of coloring, including -list-coloring,
-edge-coloring, -coloring, etc., also follow from our general
result on conflict coloring. Likewise, in the framework of centralized local
computation algorithms (LCAs), our general result yields an LCA which requires
a smaller number of probes than the previously best known algorithm for
vertex-coloring, and works for a wide range of coloring problems
The Distance Matching Problem
This paper introduces the \emph{-distance matching problem}, in which we
are given a bipartite graph with , a weight
function on the edges and an integer . The goal is to find a
maximum weight subset of the edges satisfying the following two
conditions: i) the degree of every node of is at most one in , ii) if
, then . The question arises naturally, for
example, in various scheduling problems.
We show that the problem is NP-complete in general and admits a simple
-approxi\-mation. We give an FPT algorithm parameterized by and also
settle the case when the size of is constant. From an approximability point
of view, we show that the integrality gap of the natural integer programming
model is at most , and give an LP-based approximation
algorithm for the weighted case with the same guarantee. A combinatorial
-approximation algorithm is also presented. Several greedy
approaches are considered, in particular, a local search algorithm that
achieves an approximation ratio of for any constant
in the unweighted case. The novel approaches used in the analysis of the
integrality gap and the approximation ratio of locally optimal solutions might
be of independent combinatorial interest