8,695 research outputs found
The Complexity of Computing Minimal Unidirectional Covering Sets
Given a binary dominance relation on a set of alternatives, a common thread
in the social sciences is to identify subsets of alternatives that satisfy
certain notions of stability. Examples can be found in areas as diverse as
voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08]
proved that it is NP-hard to decide whether an alternative is contained in some
inclusion-minimal upward or downward covering set. For both problems, we raise
this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and
provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other
natural problems regarding minimal or minimum-size covering sets are hard or
complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of
our results is that neither minimal upward nor minimal downward covering sets
(even when guaranteed to exist) can be computed in polynomial time unless P=NP.
This sharply contrasts with Brandt and Fischer's result that minimal
bidirectional covering sets (i.e., sets that are both minimal upward and
minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure
Resource efficient redundancy using quorum-based cycle routing in optical networks
In this paper we propose a cycle redundancy technique that provides optical
networks almost fault-tolerant point-to-point and multipoint-to-multipoint
communications. The technique more importantly is shown to approximately halve
the necessary light-trail resources in the network while maintaining the
fault-tolerance and dependability expected from cycle-based routing. For
efficiency and distributed control, it is common in distributed systems and
algorithms to group nodes into intersecting sets referred to as quorum sets.
Optimal communication quorum sets forming optical cycles based on light-trails
have been shown to flexibly and efficiently route both point-to-point and
multipoint-to-multipoint traffic requests. Commonly cycle routing techniques
will use pairs of cycles to achieve both routing and fault-tolerance, which
uses substantial resources and creates the potential for underutilization.
Instead, we intentionally utilize redundancy within the quorum cycles for
fault-tolerance such that almost every point-to-point communication occurs in
more than one cycle. The result is a set of cycles with 96.60% - 99.37% fault
coverage, while using 42.9% - 47.18% fewer resources.Comment: 17th International Conference on Transparent Optical Networks
(ICTON), 5-9 July 2015. arXiv admin note: substantial text overlap with
arXiv:1608.05172, arXiv:1608.0516
Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling
A distance labeling scheme is an assignment of bit-labels to the vertices of
an undirected, unweighted graph such that the distance between any pair of
vertices can be decoded solely from their labels. An important class of
distance labeling schemes is that of hub labelings, where a node
stores its distance to the so-called hubs , chosen so that for
any there is belonging to some shortest
path. Notice that for most existing graph classes, the best distance labelling
constructions existing use at some point a hub labeling scheme at least as a
key building block. Our interest lies in hub labelings of sparse graphs, i.e.,
those with , for which we show a lowerbound of
for the average size of the hubsets.
Additionally, we show a hub-labeling construction for sparse graphs of average
size for some , where is the
so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced
matchings in dense graphs. This implies that further improving the lower bound
on hub labeling size to would require a
breakthrough in the study of lower bounds on , which have resisted
substantial improvement in the last 70 years. For general distance labeling of
sparse graphs, we show a lowerbound of , where is the communication complexity of the
Sum-Index problem over . Our results suggest that the best achievable
hub-label size and distance-label size in sparse graphs may be
for some
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