Improper colouring of weighted grid and hexagonal graphs

We study a weighted improper colouring problem on graph, and in particular of triangular and hexagonal grid graphs. This problem is motivated by a frequency allocation problem. We propose approximation algorithms to compute such colouring

Improper colouring of weighted grid and hexagonal graphs

International audienceWe study a weighted improper colouring problem motivated by a frequency allocation problem. It consists of associating to each vertex a set of p(v) (weight) distinct colours (frequencies), such that the set of vertices having a given colour induces a graph of degree at most k (the case k = 0 corresponds to a proper coloring). The objective is to minimize the number of colors. We propose approximation algorithms to compute such colouring for general graphs. We apply these to obtain good approximation ratio for grid and hexagonal graphs. Furthermore we give exact results for the 2-dimensional grid and the triangular lattice when the weights are all the same

Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Defective Coloring on Classes of Perfect Graphs

In Defective Coloring we are given a graph $G$ and two integers $\chi_d$, $\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum degree induced by any color class is at most $\Delta^*$. We show that this natural generalization of Coloring is much harder on several basic graph classes. In particular, we show that it is NP-hard on split graphs, even when one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* = 1$). Together with a simple treewidth-based DP algorithm this completely determines the complexity of the problem also on chordal graphs. We then consider the case of cographs and show that, somewhat surprisingly, Defective Coloring turns out to be one of the few natural problems which are NP-hard on this class. We complement this negative result by showing that Defective Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that it is in P for trivially perfect graphs; and that it admits a sub-exponential time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded

Prioritization of Strategies to Overcome Barriers for Cleaner and Energy Efficient Alternatives in Urban Transportation - Multi-criteria Approach

Adoption of cleaner and energy efficient technologies (CEETs) in urban transport experiences certain barriers and deriving a set of policies to remove/reduce barrier in the case of Delhi and Mumbai transport systems was attempted in this study. A set of policy alternatives and measures (PAMs) were identified for each barrier and a pool of barriers PAMs for all barriers were identified which were finally analysed for their potential based on 4 important criteria namely administrative costs, financial burden, human resource benefits, administrative backup and political acceptability. Based on aggregated multi-criteria assessment, the policy of distinct colouring scheme for alternate fuel vehicles (AFVs) stood first followed by awareness campaigns to the drivers, training programs to the workers, single window/priority check points, financial incentives and task force to carry out check. To realize the completeness, potential of PAMs in handling barriers was analysed considering not only a set of criteria but also their potential in handling more than one barrier. In overall ranking, policy to develop partnerships among major stakeholders and awareness campaigns to the drivers showed highest potential in removing barriers for the adoption of CEETs. Based on the ranking under both approaches a set of seven policy measures and alternatives were selected to remove barriers to CEETS and they are partnership between the Government, public sector undertakings and private actors in proving better infrastructure; Financial incentives like free or priority parking, separate lanes for alternative fuel vehicles and free inspection and maintenance; Task force to carry our checks; Heavy fines on defaulters; Distinct colour coding for AFVs; Demonstration of AFVs and their advantages; and Awareness campaigns to drivers. This set of PAMs would be able to control all seven pre-identified barriers to the adoption of CEETs in Delhi and Mumbai urban transportation systems.Barriers, CEETs, multi-criteria, policy analysis, urban transport

Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks

We present a distributed algorithm that finds a maximal edge packing in O(Δ + log* W) synchronous communication rounds in a weighted graph, independent of the number of nodes in the network; here Δ is the maximum degree of the graph and W is the maximum weight. As a direct application, we have a distributed 2-approximation algorithm for minimum-weight vertex cover, with the same running time. We also show how to find an f-approximation of minimum-weight set cover in O(f2k2 + fk log* W) rounds; here k is the maximum size of a subset in the set cover instance, f is the maximum frequency of an element, and W is the maximum weight of a subset. The algorithms are deterministic, and they can be applied in anonymous networks.Peer reviewe

Markov chain sampling of the O(n)O(n) loop models on the infinite plane

It was recently proposed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro & Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the $O(n)$ loop gas models for $n \in (1,2]$. We argue that even though the Gibbs measure is non local, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the $O(n)$ models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.Comment: v2: added conclusion section, changes in introduction and appendice