30,425 research outputs found
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Revisiting Interval Graphs for Network Science
The vertices of an interval graph represent intervals over a real line where
overlapping intervals denote that their corresponding vertices are adjacent.
This implies that the vertices are measurable by a metric and there exists a
linear structure in the system. The generalization is an embedding of a graph
onto a multi-dimensional Euclidean space and it was used by scientists to study
the multi-relational complexity of ecology. However the research went out of
fashion in the 1980s and was not revisited when Network Science recently
expressed interests with multi-relational networks known as multiplexes. This
paper studies interval graphs from the perspective of Network Science
Designing Coalition-Proof Reverse Auctions over Continuous Goods
This paper investigates reverse auctions that involve continuous values of
different types of goods, general nonconvex constraints, and second stage
costs. We seek to design the payment rules and conditions under which
coalitions of participants cannot influence the auction outcome in order to
obtain higher collective utility. Under the incentive-compatible
Vickrey-Clarke-Groves mechanism, we show that coalition-proof outcomes are
achieved if the submitted bids are convex and the constraint sets are of a
polymatroid-type. These conditions, however, do not capture the complexity of
the general class of reverse auctions under consideration. By relaxing the
property of incentive-compatibility, we investigate further payment rules that
are coalition-proof without any extra conditions on the submitted bids and the
constraint sets. Since calculating the payments directly for these mechanisms
is computationally difficult for auctions involving many participants, we
present two computationally efficient methods. Our results are verified with
several case studies based on electricity market data
Revisiting the Status of Speech Rhythm
Text-to-Speech synthesis offers an interesting manner of synthesising various knowledge components related to speech production. To a certain extent, it provides a new way of testing the coherence of our understanding of speech production in a highly systematic manner. For example, speech rhythm and temporal organisation of speech have to be well-captured in order to mimic a speaker correctly.
The simulation approach used in our laboratory for two languages supports our original hypothesis of multidimensionality and non-linearity in the production of speech rhythm. This paper presents an overview of our approach towards this issue, as it has been developed over the last years.
We conceive the production of speech rhythm as a multidimensional task, and the temporal organisation of speech as a key component of this task (i.e., the establishment of temporal boundaries and durations). As a result of this multidimensionality, text-to-speech systems have to accommodate a number of systematic transformations and computations at various levels. Our model of the temporal organisation of read speech in French and German emerges from a combination of quantitative and qualitative parameters, organised according to psycholinguistic and linguistic structures. (An ideal speech synthesiser would also take into account subphonemic as well as pragmatic parameters. However such systems are not yet available)
- …