717 research outputs found
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
Generalized sequential tree-reweighted message passing
This paper addresses the problem of approximate MAP-MRF inference in general
graphical models. Following [36], we consider a family of linear programming
relaxations of the problem where each relaxation is specified by a set of
nested pairs of factors for which the marginalization constraint needs to be
enforced. We develop a generalization of the TRW-S algorithm [9] for this
problem, where we use a decomposition into junction chains, monotonic w.r.t.
some ordering on the nodes. This generalizes the monotonic chains in [9] in a
natural way. We also show how to deal with nested factors in an efficient way.
Experiments show an improvement over min-sum diffusion, MPLP and subgradient
ascent algorithms on a number of computer vision and natural language
processing problems
Optimal radio labelings of graphs
Let be the set of positive integers. A radio labeling of a graph
is a mapping such that
the inequality holds for
every pair of distinct vertices of , where and are
the diameter of and distance between and in , respectively. The
radio number of is the smallest number such that has radio
labeling with = . Das et al.
[Discrete Math. (2017) 855-861] gave a technique to find a lower
bound for the radio number of graphs. In [Algorithms and Discrete Applied
Mathematics: CALDAM 2019, Lecture Notes in Computer Science ,
springer, Cham, 2019, 161-173], Bantva modified this technique for finding an
improved lower bound on the radio number of graphs and gave a necessary and
sufficient condition to achieve the improved lower bound. In this paper, one
more useful necessary and sufficient condition to achieve the improved lower
bound for the radio number of graphs is given. Using this result, the radio
number of the Cartesian product of a path and a wheel graphs is determined.Comment: 12 pages, This is the final version accepted in Discrete Mathematics
Letters Journa
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