152,211 research outputs found
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
A J-Spectral Factorization Approach to ââ Control
Necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with ââ control, are derived using J-spectral factorization theory. The existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled J-spectral factorization problems, with the second factor providing a parametrization of all solutions to the model matching problem. The existence of the J-spectral factors is then shown to be equivalent to the existence of nonnegative definite, stabilizing solutions to two indefinite algebraic Riccati equations, allowing a state-space formula for a linear fractional representation of all controllers to be given. A virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework, and no additional auxiliary Riccati equations are required
Overhead-Aware Distributed CSI Selection in the MIMO Interference Channel
We consider a MIMO interference channel in which the transmitters and
receivers operate in frequency-division duplex mode. In this setting,
interference management through coordinated transceiver design necessitates
channel state information at the transmitters (CSI-T). The acquisition of CSI-T
is done through feedback from the receivers, which entitles a loss in degrees
of freedom, due to training and feedback. This loss increases with the amount
of CSI-T. In this work, after formulating an overhead model for CSI acquisition
at the transmitters, we propose a distributed mechanism to find for each
transmitter a subset of the complete CSI, which is used to perform interference
management. The mechanism is based on many-to-many stable matching. We prove
the existence of a stable matching and exploit an algorithm to reach it.
Simulation results show performance improvement compared to full and minimal
CSI-T.Comment: 5 pages, 2 figures. to appear at EUSIPCO 2015, Special Session on
Algorithms for Distributed Coordination and Learnin
Fictitious students creation incentives in school choice problems
We address the question of whether schools can manipulate the student-optimal stable mechanism by creating fictitious students in school choice problems. To this end, we introduce two different manipulation concepts, where one of them is stronger. We first demonstrate that the student-optimal stable mechanism is not even weakly fictitious student-proof under general priority structures. Then, we investigate the same question under acyclic priority structures. We prove that, while the student-optimal stable mechanism is not strongly fictitious student-proof even under the acyclicity condition, weak fictitious student-proofness is achieved under acyclicity. This paper, hence, shows a way to avoid the welfare detrimental fictitious students creation (in the weak sense) in terms of priority structures
Stable marriages and search frictions
Stable matchings are the primary solution concept for two-sided matching markets with nontransferable utility. We investigate the strategic foundations of stability in a decentralized matching market. Towards this end, we embed the standard marriage markets in a search model with random meetings. We study the limit of steady-state equilibria as exogenous frictions vanish. The main result is that convergence of equilibrium matchings to stable matchings is guaranteed if and only if there is a unique stable matching in the underlying marriage market. Whenever there are multiple stable matchings, sequences of equilibrium matchings converging to unstable, inefficient matchings can be constructed. Thus, vanishing frictions do not guarantee the stability and efficiency of decentralized marriage markets
Relativistic shells: Dynamics, horizons, and shell crossing
We consider the dynamics of timelike spherical thin matter shells in vacuum.
A general formalism for thin shells matching two arbitrary spherical spacetimes
is derived, and subsequently specialized to the vacuum case. We first examine
the relative motion of two dust shells by focusing on the dynamics of the
exterior shell, whereby the problem is reduced to that of a single shell with
different active Schwarzschild masses on each side. We then examine the
dynamics of shells with non-vanishing tangential pressure , and show that
there are no stable--stationary, or otherwise--solutions for configurations
with a strictly linear barotropic equation of state, , where
is the proper surface energy density and . For {\em
arbitrary} equations of state, we show that, provided the weak energy condition
holds, the strong energy condition is necessary and sufficient for stability.
We examine in detail the formation of trapped surfaces, and show explicitly
that a thin boundary layer causes the apparent horizon to evolve
discontinuously. Finally, we derive an analytical (necessary and sufficient)
condition for neighboring shells to cross, and compare the discrete shell model
with the well-known continuous Lema\^{\i}tre-Tolman-Bondi dust case.Comment: 25 pages, revtex4, 4 eps figs; published in Phys. Rev.
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