Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Power adjustment and scheduling in OFDMA femtocell networks

Densely-deployed femtocell networks are used to enhance wireless coverage in public spaces like office buildings, subways, and academic buildings. These networks can increase throughput for users, but edge users can suffer from co-channel interference, leading to service outages. This paper introduces a distributed algorithm for network configuration, called Radius Reduction and Scheduling (RRS), to improve the performance and fairness of the network. RRS determines cell sizes using a Voronoi-Laguerre framework, then schedules users using a scheduling algorithm that includes vacancy requests to increase fairness in dense femtocell networks. We prove that our algorithm always terminate in a finite time, producing a configuration that guarantees user or area coverage. Simulation results show a decrease in outage probability of up to 50%, as well as an increase in Jain's fairness index of almost 200%

Simulating the Impact of Traffic Calming Strategies

This study assessed the impact of traffic calming measures to the speed, travel times and capacity of residential roadways. The study focused on two types of speed tables, speed humps and a raised crosswalk. A moving test vehicle equipped with GPS receivers that allowed calculation of speeds and determination of speed profiles at 1s intervals were used. Multi-regime model was used to provide the best fit using steady state equations; hence the corresponding speed-flow relationships were established for different calming scenarios. It was found that capacities of residential roadway segments due to presence of calming features ranged from 640 to 730 vph. However, the capacity varied with the spacing of the calming features in which spacing speed tables at 1050 ft apart caused a 23% reduction in capacity while 350-ft spacing reduced capacity by 32%. Analysis showed a linear decrease of capacity of approximately 20 vphpl, 37 vphpl and 34 vphpl when 17 ft wide speed tables were spaced at 350 ft, 700 ft, and 1050 ft apart respectively. For speed hump calming features, spacing humps at 350 ft reduced capacity by about 33% while a 700 ft spacing reduced capacity by 30%. The study concludes that speed tables are slightly better than speed humps in terms of preserving the roadway capacity. Also, traffic calming measures significantly reduce the speeds of vehicles, and it is best to keep spacing of 630 ft or less to achieve desirable crossing speeds of less or equal to 15 mph especially in a street with schools nearby. A microscopic simulation model was developed to replicate the driving behavior of traffic on urban road diets roads to analyze the influence of bus stops on traffic flow and safety. The impacts of safety were assessed using surrogate measures of safety (SSAM). The study found that presence of a bus stops for 10, 20 and 30 s dwell times have almost 9.5%, 12%, and 20% effect on traffic speed reductions when 300 veh/hr flow is considered. A comparison of reduction in speed of traffic on an 11 ft wide road lane of a road diet due to curbside stops and bus bays for a mean of 30s with a standard deviation of 5s dwell time case was conducted. Results showed that a bus stop bay with the stated bus dwell time causes an approximate 8% speed reduction to traffic at a flow level of about 1400 vph. Analysis of the trajectories from bust stop locations showed that at 0, 25, 50, 75, 100, 125, 150, and 175 feet from the intersection the number of conflicts is affected by the presence and location of a curbside stop on a segment with a road diet

An optically thick inner corona geometry for the Very High State Galactic Black Hole XTE J1550-564

(truncated version) The X-ray spectra of Galactic binary systems in the very high state show both strong disk emission and a strong, steep tail to high energies. We use simultaneous optical-ASCA-RXTE data from the black hole transient XTE J1550-564 as a specific example, and show that these have disc spectra which are significantly lower in temperature than those seen from the same source at the same luminosity when in the disc dominated state. If these give a true picture of the disc then either the disc emissivity has reduced, and/or the disc truncates above the last stable orbit. However, it is often assumed that the tail is produced by Compton scattering, in which case its shape in these spectra requires that the Comptonising region is marginally optically thick (tau~2-3), and covers a large fraction of the inner disc. This will distort our view of the disc. We build a theoretical model of a Comptonising corona over an inner disc, and fit this to the data, but find that it still requires a large increase in inner disc radius for a standard disc emissivity. Instead it seems more probable that the disc emissivity changes in the presence of the corona. We implement the specific inner disc-corona coupling model of Svensson & Zdziarski (1994) and show that this can explain the low temperature/high luminosity disc emission seen in the very high state with only a small increase in radius of the disc. While this inferred disc truncation is probably not significant given the model uncertainties, it is consistent with the low frequency QPO and gives continuity of properties with the low/hard state spectra.Comment: MNRAS accepted version, with major expansion of the discussion to include comments on comptonisation, extreme broad iron lines and high frequency QPO's. 18 pages, 8 figure