135 research outputs found
Meixner class of non-commutative generalized stochastic processes with freely independent values I. A characterization
Let be an underlying space with a non-atomic measure on it (e.g.
and is the Lebesgue measure). We introduce and study a
class of non-commutative generalized stochastic processes, indexed by points of
, with freely independent values. Such a process (field),
, , is given a rigorous meaning through smearing out
with test functions on , with being a
(bounded) linear operator in a full Fock space. We define a set
of all continuous polynomials of , and then define a con-commutative
-space by taking the closure of in the norm
, where is the vacuum in the Fock
space. Through procedure of orthogonalization of polynomials, we construct a
unitary isomorphism between and a (Fock-space-type) Hilbert space
, with
explicitly given measures . We identify the Meixner class as those
processes for which the procedure of orthogonalization leaves the set invariant. (Note that, in the general case, the projection of a
continuous monomial of oder onto the -th chaos need not remain a
continuous polynomial.) Each element of the Meixner class is characterized by
two continuous functions and on , such that, in the
space, has representation
\omega(t)=\di_t^\dag+\lambda(t)\di_t^\dag\di_t+\di_t+\eta(t)\di_t^\dag\di^2_t,
where \di_t^\dag and \di_t are the usual creation and annihilation
operators at point
Free Meixner states
Free Meixner states are a class of functionals on non-commutative polynomials
introduced in math.CO/0410482. They are characterized by a resolvent-type form
for the generating function of their orthogonal polynomials, by a recursion
relation for those polynomials, or by a second-order non-commutative
differential equation satisfied by their free cumulant functional. In this
paper, we construct an operator model for free Meixner states. By combinatorial
methods, we also derive an operator model for their free cumulant functionals.
This, in turn, allows us to construct a number of examples. Many of these
examples are shown to be trivial, in the sense of being free products of
functionals which depend on only a single variable, or rotations of such free
products. On the other hand, the multinomial distribution is a free Meixner
state and is not a product. Neither is a large class of tracial free Meixner
states which are analogous to the simple quadratic exponential families in
statistics.Comment: 30 page
Social welfare in one-sided matchings: Random priority and beyond
We study the problem of approximate social welfare maximization (without
money) in one-sided matching problems when agents have unrestricted cardinal
preferences over a finite set of items. Random priority is a very well-known
truthful-in-expectation mechanism for the problem. We prove that the
approximation ratio of random priority is Theta(n^{-1/2}) while no
truthful-in-expectation mechanism can achieve an approximation ratio better
than O(n^{-1/2}), where n is the number of agents and items. Furthermore, we
prove that the approximation ratio of all ordinal (not necessarily
truthful-in-expectation) mechanisms is upper bounded by O(n^{-1/2}), indicating
that random priority is asymptotically the best truthful-in-expectation
mechanism and the best ordinal mechanism for the problem.Comment: 13 page
On the Complexity of -Closeness Anonymization and Related Problems
An important issue in releasing individual data is to protect the sensitive
information from being leaked and maliciously utilized. Famous privacy
preserving principles that aim to ensure both data privacy and data integrity,
such as -anonymity and -diversity, have been extensively studied both
theoretically and empirically. Nonetheless, these widely-adopted principles are
still insufficient to prevent attribute disclosure if the attacker has partial
knowledge about the overall sensitive data distribution. The -closeness
principle has been proposed to fix this, which also has the benefit of
supporting numerical sensitive attributes. However, in contrast to
-anonymity and -diversity, the theoretical aspect of -closeness has
not been well investigated.
We initiate the first systematic theoretical study on the -closeness
principle under the commonly-used attribute suppression model. We prove that
for every constant such that , it is NP-hard to find an optimal
-closeness generalization of a given table. The proof consists of several
reductions each of which works for different values of , which together
cover the full range. To complement this negative result, we also provide exact
and fixed-parameter algorithms. Finally, we answer some open questions
regarding the complexity of -anonymity and -diversity left in the
literature.Comment: An extended abstract to appear in DASFAA 201
Network Creation Games: Think Global - Act Local
We investigate a non-cooperative game-theoretic model for the formation of
communication networks by selfish agents. Each agent aims for a central
position at minimum cost for creating edges. In particular, the general model
(Fabrikant et al., PODC'03) became popular for studying the structure of the
Internet or social networks. Despite its significance, locality in this game
was first studied only recently (Bil\`o et al., SPAA'14), where a worst case
locality model was presented, which came with a high efficiency loss in terms
of quality of equilibria. Our main contribution is a new and more optimistic
view on locality: agents are limited in their knowledge and actions to their
local view ranges, but can probe different strategies and finally choose the
best. We study the influence of our locality notion on the hardness of
computing best responses, convergence to equilibria, and quality of equilibria.
Moreover, we compare the strength of local versus non-local strategy-changes.
Our results address the gap between the original model and the worst case
locality variant. On the bright side, our efficiency results are in line with
observations from the original model, yet we have a non-constant lower bound on
the price of anarchy.Comment: An extended abstract of this paper has been accepted for publication
in the proceedings of the 40th International Conference on Mathematical
Foundations on Computer Scienc
Social Welfare in One-Sided Matching Mechanisms
We study the Price of Anarchy of mechanisms for the well-known problem of
one-sided matching, or house allocation, with respect to the social welfare
objective. We consider both ordinal mechanisms, where agents submit preference
lists over the items, and cardinal mechanisms, where agents may submit
numerical values for the items being allocated. We present a general lower
bound of on the Price of Anarchy, which applies to all
mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and
Random Priority, achieve a matching upper bound. We extend our lower bound to
the Price of Stability of a large class of mechanisms that satisfy a common
proportionality property, and show stronger bounds on the Price of Anarchy of
all deterministic mechanisms
The Firefighter Problem: A Structural Analysis
We consider the complexity of the firefighter problem where b>=1 firefighters
are available at each time step. This problem is proved NP-complete even on
trees of degree at most three and budget one (Finbow et al.,2007) and on trees
of bounded degree b+3 for any fixed budget b>=2 (Bazgan et al.,2012). In this
paper, we provide further insight into the complexity landscape of the problem
by showing that the pathwidth and the maximum degree of the input graph govern
its complexity. More precisely, we first prove that the problem is NP-complete
even on trees of pathwidth at most three for any fixed budget b>=1. We then
show that the problem turns out to be fixed parameter-tractable with respect to
the combined parameter "pathwidth" and "maximum degree" of the input graph
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation
On Linear Congestion Games with Altruistic Social Context
We study the issues of existence and inefficiency of pure Nash equilibria in
linear congestion games with altruistic social context, in the spirit of the
model recently proposed by de Keijzer {\em et al.} \cite{DSAB13}. In such a
framework, given a real matrix specifying a particular
social context, each player aims at optimizing a linear combination of the
payoffs of all the players in the game, where, for each player , the
multiplicative coefficient is given by the value . We give a broad
characterization of the social contexts for which pure Nash equilibria are
always guaranteed to exist and provide tight or almost tight bounds on their
prices of anarchy and stability. In some of the considered cases, our
achievements either improve or extend results previously known in the
literature
Matching Dynamics with Constraints
We study uncoordinated matching markets with additional local constraints
that capture, e.g., restricted information, visibility, or externalities in
markets. Each agent is a node in a fixed matching network and strives to be
matched to another agent. Each agent has a complete preference list over all
other agents it can be matched with. However, depending on the constraints and
the current state of the game, not all possible partners are available for
matching at all times. For correlated preferences, we propose and study a
general class of hedonic coalition formation games that we call coalition
formation games with constraints. This class includes and extends many recently
studied variants of stable matching, such as locally stable matching, socially
stable matching, or friendship matching. Perhaps surprisingly, we show that all
these variants are encompassed in a class of "consistent" instances that always
allow a polynomial improvement sequence to a stable state. In addition, we show
that for consistent instances there always exists a polynomial sequence to
every reachable state. Our characterization is tight in the sense that we
provide exponential lower bounds when each of the requirements for consistency
is violated. We also analyze matching with uncorrelated preferences, where we
obtain a larger variety of results. While socially stable matching always
allows a polynomial sequence to a stable state, for other classes different
additional assumptions are sufficient to guarantee the same results. For the
problem of reaching a given stable state, we show NP-hardness in almost all
considered classes of matching games.Comment: Conference Version in WINE 201
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