Advances on Matroid Secretary Problems: Free Order Model and Laminar Case

The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondr\'ak [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9-approximation for the \emph{free order model}, a model suggested shortly after the introduction of the matroid secretary problem, and for which no constant-factor approximation was known so far. The free order model is a relaxed version of the original matroid secretary problem, with the only difference that one can choose the order in which secretaries are interviewed. Furthermore, we consider the classical matroid secretary problem for the special case of laminar matroids. Only recently, a constant-factor approximation has been found for this case, using a clever but rather involved method and analysis (Im and Wang, [SODA 2011]) that leads to a 16000/3-approximation. This is arguably the most involved special case of the matroid secretary problem for which a constant-factor approximation is known. We present a considerably simpler and stronger $3\sqrt{3}e\approx 14.12$-approximation, based on reducing the problem to a matroid secretary problem on a partition matroid

Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms

Constrained submodular maximization problems have long been studied, with near-optimal results known under a variety of constraints when the submodular function is monotone. The case of non-monotone submodular maximization is less understood: the first approximation algorithms even for the unconstrainted setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC '09, APPROX '09) show how to approximately maximize non-monotone submodular functions when the constraints are given by the intersection of p matroid constraints; their algorithm is based on local-search procedures that consider p-swaps, and hence the running time may be n^Omega(p), implying their algorithm is polynomial-time only for constantly many matroids. In this paper, we give algorithms that work for p-independence systems (which generalize constraints given by the intersection of p matroids), where the running time is poly(n,p). Our algorithm essentially reduces the non-monotone maximization problem to multiple runs of the greedy algorithm previously used in the monotone case. Our idea of using existing algorithms for monotone functions to solve the non-monotone case also works for maximizing a submodular function with respect to a knapsack constraint: we get a simple greedy-based constant-factor approximation for this problem. With these simpler algorithms, we are able to adapt our approach to constrained non-monotone submodular maximization to the (online) secretary setting, where elements arrive one at a time in random order, and the algorithm must make irrevocable decisions about whether or not to select each element as it arrives. We give constant approximations in this secretary setting when the algorithm is constrained subject to a uniform matroid or a partition matroid, and give an O(log k) approximation when it is constrained by a general matroid of rank k.Comment: In the Proceedings of WINE 201

Search for jet extinction in the inclusive jet-pT spectrum from proton-proton collisions at s=8 TeV

Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published articles title, journal citation, and DOI.The first search at the LHC for the extinction of QCD jet production is presented, using data collected with the CMS detector corresponding to an integrated luminosity of 10.7  fb−1 of proton-proton collisions at a center-of-mass energy of 8 TeV. The extinction model studied in this analysis is motivated by the search for signatures of strong gravity at the TeV scale (terascale gravity) and assumes the existence of string couplings in the strong-coupling limit. In this limit, the string model predicts the suppression of all high-transverse-momentum standard model processes, including jet production, beyond a certain energy scale. To test this prediction, the measured transverse-momentum spectrum is compared to the theoretical prediction of the standard model. No significant deficit of events is found at high transverse momentum. A 95% confidence level lower limit of 3.3 TeV is set on the extinction mass scale

Searches for electroweak neutralino and chargino production in channels with Higgs, Z, and W bosons in pp collisions at 8 TeV

Searches for supersymmetry (SUSY) are presented based on the electroweak pair production of neutralinos and charginos, leading to decay channels with Higgs, Z, and W bosons and undetected lightest SUSY particles (LSPs). The data sample corresponds to an integrated luminosity of about 19.5 fb(-1) of proton-proton collisions at a center-of-mass energy of 8 TeV collected in 2012 with the CMS detector at the LHC. The main emphasis is neutralino pair production in which each neutralino decays either to a Higgs boson (h) and an LSP or to a Z boson and an LSP, leading to hh, hZ, and ZZ states with missing transverse energy (E-T(miss)). A second aspect is chargino-neutralino pair production, leading to hW states with E-T(miss). The decays of a Higgs boson to a bottom-quark pair, to a photon pair, and to final states with leptons are considered in conjunction with hadronic and leptonic decay modes of the Z and W bosons. No evidence is found for supersymmetric particles, and 95% confidence level upper limits are evaluated for the respective pair production cross sections and for neutralino and chargino mass values

In vivo determination of human breast fat composition by 1H magnetic resonance spectroscopy at 7 T

Radiolog

Tight Bounds for the Cover Time of Multiple Random Walks

We study the cover time of multiple random walks. Given a graph G of n vertices, assume that k independent random walks start from the same vertex. The parameter of interest is the speed-up defined as the ratio between the cover time of one and the cover time of k random walks. Recently Alon et al. developed several bounds that are based on the quotient between the cover time and maximum hitting times. Their technique gives a speed-up of Ω(k) on many graphs, however, for many graph classes, k has to be bounded by O(log n). They also conjectured that, for any 1 � k � n, the speed-up is at most O(k) on any graph. As our main results, we prove the following: – We present a new lower bound on the speed-up that depends on the mixing-time. It gives a speed-up of Ω(k) onmanygraphs,evenifk is as large as n. – We prove that the speed-up is O(k log n) on any graph. Under rather mild conditions, we can also improve this bound to O(k), matching exactly the conjecture of Alon et al. – We find the correct order of the speed-up for any value of 1 � k � n on hypercubes, random graphs and expanders. For d-dimensional torus graphs (d>2), our bounds are tight up to a factor of O(log n). – Our findings also reveal a surprisingly sharp dichotomy on several graphs (including d-dim. torus and hypercubes): up to a certain threshold the speed-up is k, while there is no additional speed-up above the threshold

Improved Competitive Ratios for Submodular Secretary Problems (Extended Abstract)

The Classical Secretary Problem was introduced during the 60’s of the 20 th century, nobody is sure exactly when. Since its introduction, many variants of the problem have been proposed and researched. In the classical secretary problem, and many of its variant, the input (which is a set of secretaries, or elements) arrives in a random order. In this paper we apply to the secretary problem a simple observation which states that the random order of the input can be generated by independently choosing a random continuous arrival time for each secretary. Surprisingly, this simple observation enables us to improve the competitive ratio of several known and studied variants of the secretary problem. In addition, in some cases the proofs we provide assuming random arrival times are shorter and simpler in comparison to existing proofs. In this work we consider three variants of the secretary problem, all of which have the same objective of maximizing the value of the chosen set of secretaries given a monotone submodular function f. In the first variant we are allowed to hire a set of secretaries only if it is an independent set of a given partition matroid. The second variant allows us to choose any set of up to k secretaries. In the last and third variant, we can hire any set of secretaries satisfying a given knapsack constraint