22,271 research outputs found
Characterization of revenue equivalence
The property of an allocation rule to be implementable in dominant strategies by a unique payment scheme is called \emph{revenue equivalence}. In this paper we give a characterization of revenue equivalence based on a graph theoretic interpretation of the incentive compatibility constraints. The characterization holds for any (possibly infinite) outcome space and many of the known results are immediate consequences. Moreover, revenue equivalence can be identified in cases where existing theorems are silent
On the bend number of circular-arc graphs as edge intersection graphs of paths on a grid
Golumbic, Lipshteyn and Stern \cite{Golumbic-epg} proved that every graph can
be represented as the edge intersection graph of paths on a grid (EPG graph),
i.e., one can associate with each vertex of the graph a nontrivial path on a
rectangular grid such that two vertices are adjacent if and only if the
corresponding paths share at least one edge of the grid. For a nonnegative
integer , -EPG graphs are defined as EPG graphs admitting a model in
which each path has at most bends. Circular-arc graphs are intersection
graphs of open arcs of a circle. It is easy to see that every circular-arc
graph is a -EPG graph, by embedding the circle into a rectangle of the
grid. In this paper, we prove that every circular-arc graph is -EPG, and
that there exist circular-arc graphs which are not -EPG. If we restrict
ourselves to rectangular representations (i.e., the union of the paths used in
the model is contained in a rectangle of the grid), we obtain EPR (edge
intersection of path in a rectangle) representations. We may define -EPR
graphs, , the same way as -EPG graphs. Circular-arc graphs are
clearly -EPR graphs and we will show that there exist circular-arc graphs
that are not -EPR graphs. We also show that normal circular-arc graphs are
-EPR graphs and that there exist normal circular-arc graphs that are not
-EPR graphs. Finally, we characterize -EPR graphs by a family of
minimal forbidden induced subgraphs, and show that they form a subclass of
normal Helly circular-arc graphs
On Linkedness of Cartesian Product of Graphs
We study linkedness of Cartesian product of graphs and prove that the product
of an -linked and a -linked graphs is -linked if the graphs are
sufficiently large. Further bounds in terms of connectivity are shown. We
determine linkedness of product of paths and product of cycles
Optimal redundancy against disjoint vulnerabilities in networks
Redundancy is commonly used to guarantee continued functionality in networked
systems. However, often many nodes are vulnerable to the same failure or
adversary. A "backup" path is not sufficient if both paths depend on nodes
which share a vulnerability.For example, if two nodes of the Internet cannot be
connected without using routers belonging to a given untrusted entity, then all
of their communication-regardless of the specific paths utilized-will be
intercepted by the controlling entity.In this and many other cases, the
vulnerabilities affecting the network are disjoint: each node has exactly one
vulnerability but the same vulnerability can affect many nodes. To discover
optimal redundancy in this scenario, we describe each vulnerability as a color
and develop a "color-avoiding percolation" which uncovers a hidden
color-avoiding connectivity. We present algorithms for color-avoiding
percolation of general networks and an analytic theory for random graphs with
uniformly distributed colors including critical phenomena. We demonstrate our
theory by uncovering the hidden color-avoiding connectivity of the Internet. We
find that less well-connected countries are more likely able to communicate
securely through optimally redundant paths than highly connected countries like
the US. Our results reveal a new layer of hidden structure in complex systems
and can enhance security and robustness through optimal redundancy in a wide
range of systems including biological, economic and communications networks.Comment: 15 page
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