509,138 research outputs found
Optimal Composition Ordering Problems for Piecewise Linear Functions
In this paper, we introduce maximum composition ordering problems. The input
is real functions and a constant
. We consider two settings: total and partial compositions. The
maximum total composition ordering problem is to compute a permutation
which maximizes , where .
The maximum partial composition ordering problem is to compute a permutation
and a nonnegative integer which maximize
.
We propose time algorithms for the maximum total and partial
composition ordering problems for monotone linear functions , which
generalize linear deterioration and shortening models for the time-dependent
scheduling problem. We also show that the maximum partial composition ordering
problem can be solved in polynomial time if is of form
for some constants , and . We
finally prove that there exists no constant-factor approximation algorithm for
the problems, even if 's are monotone, piecewise linear functions with at
most two pieces, unless P=NP.Comment: 19 pages, 4 figure
On the Necessary Memory to Compute the Plurality in Multi-Agent Systems
We consider the Relative-Majority Problem (also known as Plurality), in
which, given a multi-agent system where each agent is initially provided an
input value out of a set of possible ones, each agent is required to
eventually compute the input value with the highest frequency in the initial
configuration. We consider the problem in the general Population Protocols
model in which, given an underlying undirected connected graph whose nodes
represent the agents, edges are selected by a globally fair scheduler.
The state complexity that is required for solving the Plurality Problem
(i.e., the minimum number of memory states that each agent needs to have in
order to solve the problem), has been a long-standing open problem. The best
protocol so far for the general multi-valued case requires polynomial memory:
Salehkaleybar et al. (2015) devised a protocol that solves the problem by
employing states per agent, and they conjectured their upper bound
to be optimal. On the other hand, under the strong assumption that agents
initially agree on a total ordering of the initial input values, Gasieniec et
al. (2017), provided an elegant logarithmic-memory plurality protocol.
In this work, we refute Salehkaleybar et al.'s conjecture, by providing a
plurality protocol which employs states per agent. Central to our
result is an ordering protocol which allows to leverage on the plurality
protocol by Gasieniec et al., of independent interest. We also provide a
-state lower bound on the necessary memory to solve the problem,
proving that the Plurality Problem cannot be solved within the mere memory
necessary to encode the output.Comment: 14 pages, accepted at CIAC 201
Spectral ordering and 2-switch transformations
We address the problem of ordering trees with the same degree sequence by their spectral radii. To achieve that, we consider 2-switch transformations which preserve the degree sequence and establish when the index decreases. Our main contribution is to determine a total ordering of a particular family by their indices according to a given parameter related to sizes in the tree
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