13,105 research outputs found
On Revenue Maximization with Sharp Multi-Unit Demands
We consider markets consisting of a set of indivisible items, and buyers that
have {\em sharp} multi-unit demand. This means that each buyer wants a
specific number of items; a bundle of size less than has no value,
while a bundle of size greater than is worth no more than the most valued
items (valuations being additive). We consider the objective of setting
prices and allocations in order to maximize the total revenue of the market
maker. The pricing problem with sharp multi-unit demand buyers has a number of
properties that the unit-demand model does not possess, and is an important
question in algorithmic pricing. We consider the problem of computing a revenue
maximizing solution for two solution concepts: competitive equilibrium and
envy-free pricing.
For unrestricted valuations, these problems are NP-complete; we focus on a
realistic special case of "correlated values" where each buyer has a
valuation v_i\qual_j for item , where and \qual_j are positive
quantities associated with buyer and item respectively. We present a
polynomial time algorithm to solve the revenue-maximizing competitive
equilibrium problem. For envy-free pricing, if the demand of each buyer is
bounded by a constant, a revenue maximizing solution can be found efficiently;
the general demand case is shown to be NP-hard.Comment: page2
Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries
Suppose that an -simplex is partitioned into convex regions having
disjoint interiors and distinct labels, and we may learn the label of any point
by querying it. The learning objective is to know, for any point in the
simplex, a label that occurs within some distance from that point.
We present two algorithms for this task: Constant-Dimension Generalised Binary
Search (CD-GBS), which for constant uses queries, and Constant-Region Generalised Binary
Search (CR-GBS), which uses CD-GBS as a subroutine and for constant uses
queries.
We show via Kakutani's fixed-point theorem that these algorithms provide
bounds on the best-response query complexity of computing approximate
well-supported equilibria of bimatrix games in which one of the players has a
constant number of pure strategies. We also partially extend our results to
games with multiple players, establishing further query complexity bounds for
computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in
Theorem 6, adds footnotes with additional comments and fixes typo
Fast Algorithms at Low Temperatures via Markov Chains
For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time O(n log n) for the Potts model and O(n^2 log n) for the hard-core model, in contrast to typical running times of n^{O(log Delta)} for algorithms based on Barvinok\u27s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces
The complexity of approximating conservative counting CSPs
We study the complexity of approximately solving the weighted counting
constraint satisfaction problem #CSP(F). In the conservative case, where F
contains all unary functions, there is a classification known for the case in
which the domain of functions in F is Boolean. In this paper, we give a
classification for the more general problem where functions in F have an
arbitrary finite domain. We define the notions of weak log-modularity and weak
log-supermodularity. We show that if F is weakly log-modular, then #CSP(F)is in
FP. Otherwise, it is at least as difficult to approximate as #BIS, the problem
of counting independent sets in bipartite graphs. #BIS is complete with respect
to approximation-preserving reductions for a logically-defined complexity class
#RHPi1, and is believed to be intractable. We further sub-divide the #BIS-hard
case. If F is weakly log-supermodular, then we show that #CSP(F) is as easy as
a (Boolean) log-supermodular weighted #CSP. Otherwise, we show that it is
NP-hard to approximate. Finally, we give a full trichotomy for the arity-2
case, where #CSP(F) is in FP, or is #BIS-equivalent, or is equivalent in
difficulty to #SAT, the problem of approximately counting the satisfying
assignments of a Boolean formula in conjunctive normal form. We also discuss
the algorithmic aspects of our classification.Comment: Minor revisio
Unparticle physics and Higgs phenomenology
Recently, conceptually new physics beyond the Standard Model has been
proposed, where a hidden conformal sector provides ``unparticle'' which couples
to the Standard Model sector through higher dimensional operators in low energy
effective theory. Among several possibilities, we focus on operators involving
unparticle, the Higgs doublet and the gauge bosons. Once the Higgs doublet
develops the vacuum expectation value, the conformal symmetry is broken and as
a result, the mixing between unparticle and Higgs boson emerges. We find that
this mixing can cause sizable shifts for the couplings between Higgs boson and
a pair of gluons and photons, because these couplings exist only at the loop
level in the Standard Model. These Higgs couplings are the most important ones
for the Higgs boson search at the CERN Large Hadron Collider, and the
unparticle physics effects may be observed together with the discovery of Higgs
boson.Comment: 5 pages, 4 figures; version to appear in Phys. Lett.
Conformative Filtering for Implicit Feedback Data
Implicit feedback is the simplest form of user feedback that can be used for
item recommendation. It is easy to collect and is domain independent. However,
there is a lack of negative examples. Previous work tackles this problem by
assuming that users are not interested or not as much interested in the
unconsumed items. Those assumptions are often severely violated since
non-consumption can be due to factors like unawareness or lack of resources.
Therefore, non-consumption by a user does not always mean disinterest or
irrelevance. In this paper, we propose a novel method called Conformative
Filtering (CoF) to address the issue. The motivating observation is that if
there is a large group of users who share the same taste and none of them have
consumed an item before, then it is likely that the item is not of interest to
the group. We perform multidimensional clustering on implicit feedback data
using hierarchical latent tree analysis (HLTA) to identify user `tastes' groups
and make recommendations for a user based on her memberships in the groups and
on the past behavior of the groups. Experiments on two real-world datasets from
different domains show that CoF has superior performance compared to several
common baselines
Increasing the Fisher Information Content in the Matter Power Spectrum by Non-linear Wavelet Weiner Filtering
We develop a purely mathematical tool to recover some of the information lost
in the non-linear collapse of large-scale structure. From a set of 141
simulations of dark matter density fields, we construct a non-linear Weiner
filter in order to separate Gaussian and non-Gaussian structure in wavelet
space. We find that the non-Gaussian power is dominant at smaller scales, as
expected from the theory of structure formation, while the Gaussian counterpart
is damped by an order of magnitude on small scales. We find that it is possible
to increase the Fisher information by a factor of three before reaching the
translinear plateau, an effect comparable to other techniques like the linear
reconstruction of the density field.Comment: 7 pages, 6 figures. Accepted for publication in The Astrophysical
Journa
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