689 research outputs found
List version of (,1)-total labellings
The (,1)-total number of a graph is the width of the
smallest range of integers that suffices to label the vertices and the edges of
such that no two adjacent vertices have the same label, no two incident
edges have the same label and the difference between the labels of a vertex and
its incident edges is at least . In this paper we consider the list version.
Let be a list of possible colors for all . Define
to be the smallest integer such that for every list
assignment with for all , has a
(,1)-total labelling such that for all . We call the (,1)-total labelling choosability and
is list -(,1)-total labelable. In this paper, we present a conjecture on
the upper bound of . Furthermore, we study this parameter for paths
and trees in Section 2. We also prove that for
star with in Section 3 and for outerplanar graph with in Section 4.Comment: 11 pages, 2 figure
Finite-Time Convergent Algorithms for Time-Varying Distributed Optimization
This paper focuses on finite-time (FT) convergent distributed algorithms for
solving time-varying distributed optimization (TVDO). The objective is to
minimize the sum of local time-varying cost functions subject to the possible
time-varying constraints by the coordination of multiple agents in finite time.
We first provide a unified approach for designing finite/fixed-time convergent
algorithms to solve centralized time-varying optimization, where an auxiliary
dynamics is introduced to achieve prescribed performance. Then, two classes of
TVDO are investigated included unconstrained distributed consensus optimization
and distributed optimal resource allocation problems (DORAP) with both
time-varying cost functions and coupled equation constraints. For the previous
one, based on nonsmooth analysis, a continuous-time distributed discontinuous
dynamics with FT convergence is proposed based on an extended zero-gradient-sum
method with a local auxiliary subsystem. Different from the existing methods,
the proposed algorithm does not require the initial state of each agent to be
the optimizer of the local cost function. Moreover, the provided algorithm has
a simpler structure without estimating the global information and can be used
for TVDO with nonidentical Hessians. Then, an FT convergent distributed
dynamics is further obtained for time-varying DORAP by dual transformation.
Particularly, the inverse of Hessians is not required from a dual perspective,
which reduces the computation complexity significantly. Finally, two numerical
examples are conducted to verify the proposed algorithms
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