Holographic Conformal Window - A Bottom Up Approach

We propose a five-dimensional framework for modeling the background geometry associated to ordinary Yang-Mills (YM) as well as to nonsupersymmetric gauge theories possessing an infrared fixed point with fermions in various representations of the underlying gauge group. The model is based on the improved holographic approach, on the string theory side, and on the conjectured all-orders beta function for the gauge theory one. We first analyze the YM gauge theory. We then investigate the effects of adding flavors and show that, in the holographic description of the conformal window, the geometry becomes AdS when approaching the ultraviolet and the infrared regimes. As the number of flavors increases within the conformal window we observe that the geometry becomes more and more of AdS type over the entire energy range.Comment: 20 Pages, 3 Figures. v2: references adde

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Online minimum spanning tree with advice

In the online minimum spanning tree problem, a graph is revealed vertex by vertex; together with every vertex, all edges to vertices that are already known are given, and an online algorithm must irrevocably choose a subset of them as a part of its solution. The advice complexity of an online problem is a means to quantify the information that needs to be extracted from the input to achieve good results. For a graph of size n, we show an asymptotically tight bound of \u398(n log n) on the number of advice bits to produce an optimal solution for any given graph. For particular graph classes, e.g., with bounded degree or a restricted edge weight function, we prove that the upper bound can be drastically reduced; e.g., 5(n 12 1) advice bits allow to compute an optimal result if the weight function is the Euclidean distance; if the graph is complete, even a logarithmic number suffices. Some of these results make use of the optimality of Kruskal\u2019s algorithm for the offline setting. We also study the trade-off between the number of advice bits and the achievable competitive ratio. To this end, we perform a reduction from another online problem to obtain a linear lower bound on the advice complexity for any near-optimal solution. Using our results from the advice complexity finally allows us to give a lower bound on the expected competitive ratio of any randomized online algorithm for the problem

Disjoint path allocation with sublinear advice

We study the disjoint path allocation problem. In this setting, a path P of length L is given, and a sequence of subpaths of P arrives online, one in every time step. Each such path requests a permanent connection between its two end-vertices. An online algorithm can admit or reject such a request; in the former case, none of the involved edges can be part of any other connection. We investigate how much additional binary information (called “advice”) can help to obtain a good solution. It is known that, with roughly log2log2L advice bits, it can be guaranteed that a log2L-competitive solution is computed. In this paper, we prove the surprising result that, with L1−ε advice bits, it is not possible to obtain a solution with a competitive ratio better than (δlog2L)/2, where 0<δ<ε<1. This shows an interesting threshold behavior of the problem. A fairly good competitive ratio, namely log2L, can be obtained with very few advice bits. However, any increase of the advice does not help any further until an almost linear number of advice bits is supplied. Then again, it is also known that linear advice allows for optimality

Advice Complexity of the Online Search Problem

The online search problem is a fundamental problem in finance. The numerous direct applications include searching for optimal prices for commodity trading and trading foreign currencies. In this paper, we analyze the advice complexity of this problem. In particular, we are interested in identifying the minimum amount of information needed in order to achieve a certain competitive ratio. We design an algorithm that reads b bits of advice and achieves a competitive ratio of (M/m)^(1/(2^b+1)) where M and m are the maximum and minimum price in the input. We also give a matching lower bound. Furthermore, we compare the power of advice and randomization for this problem