15,008 research outputs found
A Candidate Access Structure for Super-polynomial Lower Bound on Information Ratio
The contribution vector (convec) of a secret sharing scheme is the vector of all share sizes divided by the secret size. A measure on the convec (e.g., its maximum or average) is considered as a criterion of efficiency of secret sharing schemes, which is referred to as the information ratio.
It is generally believed that there exists a family of access structures such that the information ratio of any secret sharing scheme realizing it is , where the parameter stands for the number of participants. The best known lower bound, due to Csirmaz (1994), is . Closing this gap is a long-standing open problem in cryptology.
Using a technique called \emph{substitution}, we recursively construct a family of access structures by starting from that of Csirmaz, which might be a candidate for super-polynomial information ratio. We provide support for this possibility by showing that our family has information ratio , assuming the truth of a well-stated information-theoretic conjecture, called the \emph{substitution conjecture}. The substitution method is a technique for composition of access structures, similar to the so called block composition of Boolean functions, and the substitution conjecture is reminiscent of the Karchmer-Raz-Wigderson conjecture on depth complexity of Boolean functions. It emerges after introducing the notion of convec set for an access structure, a subset of -dimensional real space, which includes all achievable convecs. We prove some topological properties about convec sets and raise several open problems
Preference Elicitation in Matching Markets Via Interviews: A Study of Offline Benchmarks
The stable marriage problem and its extensions have been
extensively studied, with much of the work in the literature
assuming that agents fully know their own preferences over
alternatives. This assumption however is not always practical
(especially in large markets) and agents usually need
to go through some costly deliberation process in order to
learn their preferences. In this paper we assume that such
deliberations are carried out via interviews, where an interview
involves a man and a woman, each of whom learns
information about the other as a consequence. If everybody
interviews everyone else, then clearly agents can fully learn
their preferences. But interviews are costly, and we may
wish to minimize their use. It is often the case, especially
in practical settings, that due to correlation between agents’
preferences, it is unnecessary for all potential interviews to
be carried out in order to obtain a stable matching. Thus
the problem is to find a good strategy for interviews to be
carried out in order to minimize their use, whilst leading to a
stable matching. One way to evaluate the performance of an
interview strategy is to compare it against a na¨ıve algorithm
that conducts all interviews. We argue however that a more
meaningful comparison would be against an optimal offline
algorithm that has access to agents’ preference orderings under
complete information. We show that, unless P=NP, no
offline algorithm can compute the optimal interview strategy
in polynomial time. If we are additionally aiming for a
particular stable matching (perhaps one with certain desirable
properties), we provide restricted settings under which
efficient optimal offline algorithms exist
Nilpotent Networks and 4D RG Flows
Starting from a general SCFT, we study the network of
SCFTs obtained from relevant deformations by nilpotent mass
parameters. We also study the case of flipper field deformations where the mass
parameters are promoted to a chiral superfield, with nilpotent vev. Nilpotent
elements of semi-simple algebras admit a partial ordering connected by a
corresponding directed graph. We find strong evidence that the resulting fixed
points are connected by a similar network of 4D RG flows. To illustrate these
general concepts, we also present a full list of nilpotent deformations in the
case of explicit SCFTs, including the case of a single
D3-brane probing a - or -type F-theory 7-brane, and 6D conformal
matter compactified on a , as described by a single M5-brane probing a
- or -type singularity. We also observe a number of numerical
coincidences of independent interest, including a collection of theories with
rational values for their conformal anomalies, as well as a surprisingly nearly
constant value for the ratio for the entire
network of flows associated with a given UV SCFT. The
submission also includes the full dataset of theories which
can be accessed with a companion script.Comment: v2: 73 pages, 12 figures, clarifications and references adde
On Compact Routing for the Internet
While there exist compact routing schemes designed for grids, trees, and
Internet-like topologies that offer routing tables of sizes that scale
logarithmically with the network size, we demonstrate in this paper that in
view of recent results in compact routing research, such logarithmic scaling on
Internet-like topologies is fundamentally impossible in the presence of
topology dynamics or topology-independent (flat) addressing. We use analytic
arguments to show that the number of routing control messages per topology
change cannot scale better than linearly on Internet-like topologies. We also
employ simulations to confirm that logarithmic routing table size scaling gets
broken by topology-independent addressing, a cornerstone of popular
locator-identifier split proposals aiming at improving routing scaling in the
presence of network topology dynamics or host mobility. These pessimistic
findings lead us to the conclusion that a fundamental re-examination of
assumptions behind routing models and abstractions is needed in order to find a
routing architecture that would be able to scale ``indefinitely.''Comment: This is a significantly revised, journal version of cs/050802
Constrained Monotone Function Maximization and the Supermodular Degree
The problem of maximizing a constrained monotone set function has many
practical applications and generalizes many combinatorial problems.
Unfortunately, it is generally not possible to maximize a monotone set function
up to an acceptable approximation ratio, even subject to simple constraints.
One highly studied approach to cope with this hardness is to restrict the set
function. An outstanding disadvantage of imposing such a restriction on the set
function is that no result is implied for set functions deviating from the
restriction, even slightly. A more flexible approach, studied by Feige and
Izsak, is to design an approximation algorithm whose approximation ratio
depends on the complexity of the instance, as measured by some complexity
measure. Specifically, they introduced a complexity measure called supermodular
degree, measuring deviation from submodularity, and designed an algorithm for
the welfare maximization problem with an approximation ratio that depends on
this measure.
In this work, we give the first (to the best of our knowledge) algorithm for
maximizing an arbitrary monotone set function, subject to a k-extendible
system. This class of constraints captures, for example, the intersection of
k-matroids (note that a single matroid constraint is sufficient to capture the
welfare maximization problem). Our approximation ratio deteriorates gracefully
with the complexity of the set function and k. Our work can be seen as
generalizing both the classic result of Fisher, Nemhauser and Wolsey, for
maximizing a submodular set function subject to a k-extendible system, and the
result of Feige and Izsak for the welfare maximization problem. Moreover, when
our algorithm is applied to each one of these simpler cases, it obtains the
same approximation ratio as of the respective original work.Comment: 23 page
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