Lower Bounds for On-line Interval Coloring with Vector and Cardinality Constraints

We propose two strategies for Presenter in the on-line interval graph coloring games. Specifically, we consider a setting in which each interval is associated with a $d$-dimensional vector of weights and the coloring needs to satisfy the $d$-dimensional bandwidth constraint, and the $k$-cardinality constraint. Such a variant was first introduced by Epstein and Levy and it is a natural model for resource-aware task scheduling with $d$ different shared resources where at most $k$ tasks can be scheduled simultaneously on a single machine. The first strategy forces any on-line interval coloring algorithm to use at least $(5m-3)\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of intervals. The second strategy forces any on-line interval coloring algorithm to use at least $\lfloor\frac{5m}{2}\rfloor\frac{d}{\log d + 3}$ different colors on an $m(\frac{d}{k} + \log{d} + 3)$-colorable set of unit intervals

Threshold Saturation in Spatially Coupled Constraint Satisfaction Problems

We consider chains of random constraint satisfaction models that are spatially coupled across a finite window along the chain direction. We investigate their phase diagram at zero temperature using the survey propagation formalism and the interpolation method. We prove that the SAT-UNSAT phase transition threshold of an infinite chain is identical to the one of the individual standard model, and is therefore not affected by spatial coupling. We compute the survey propagation complexity using population dynamics as well as large degree approximations, and determine the survey propagation threshold. We find that a clustering phase survives coupling. However, as one increases the range of the coupling window, the survey propagation threshold increases and saturates towards the phase transition threshold. We also briefly discuss other aspects of the problem. Namely, the condensation threshold is not affected by coupling, but the dynamic threshold displays saturation towards the condensation one. All these features may provide a new avenue for obtaining better provable algorithmic lower bounds on phase transition thresholds of the individual standard model

An exact decomposition approach for the real-time Train Dispatching problem (v.2)

-Trains movements on a railway network are regulated by official timetables. Deviations and delays occur quite often in practice, demanding fast re-scheduling and re-routing decisions in order to avoid conflicts and minimize overall delay. This is the real-time train dispatching problem. In contrast with the classic ""holistic"" approach, we show how to decompose the problem into smaller subproblems associated with the line and the stations. The decomposition is the basis for a master-slave solution algorithm, in which the master problem is associated with the line and the slave problem is associated with the stations. The two subproblems are modeled as mixed integer linear programs, with their specific sets of variables and constraints. Similarly to the classical Bender's decomposition approach, the slave and the master communicate through suitable feasibility cuts in the variables of the master. By applying our approach to a number of real-life instances from single and double-track lines in Italy, we were able to (quickly) find optimal or near-optimal solutions, with impressive improvements over the performances of the current operating control systems. The new approach will be put in operation in such lines for an extensive on-field test-campaign as of April 2013. Follows SINTEF Technical Report A2327

A new graph perspective on max-min fairness in Gaussian parallel channels

In this work we are concerned with the problem of achieving max-min fairness in Gaussian parallel channels with respect to a general performance function, including channel capacity or decoding reliability as special cases. As our central results, we characterize the laws which determine the value of the achievable max-min fair performance as a function of channel sharing policy and power allocation (to channels and users). In particular, we show that the max-min fair performance behaves as a specialized version of the Lovasz function, or Delsarte bound, of a certain graph induced by channel sharing combinatorics. We also prove that, in addition to such graph, merely a certain 2-norm distance dependent on the allowable power allocations and used performance functions, is sufficient for the characterization of max-min fair performance up to some candidate interval. Our results show also a specific role played by odd cycles in the graph induced by the channel sharing policy and we present an interesting relation between max-min fairness in parallel channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information Theory on August the 6th, 200