Polynomial time and parameterized approximation algorithms for boxicity

The boxicity (cubicity) of a graph G, denoted by box(G) (respectively cub(G)), is the minimum integer k such that G can be represented as the intersection graph of axis parallel boxes (cubes) in ℝ k . The problem of computing boxicity (cubicity) is known to be inapproximable in polynomial time even for graph classes like bipartite, co-bipartite and split graphs, within an O(n 0.5 − ε ) factor for any ε > 0, unless NP = ZPP. We prove that if a graph G on n vertices has a clique on n − k vertices, then box(G) can be computed in time n22O(k2logk) . Using this fact, various FPT approximation algorithms for boxicity are derived. The parameter used is the vertex (or edge) edit distance of the input graph from certain graph families of bounded boxicity - like interval graphs and planar graphs. Using the same fact, we also derive an O(nloglogn√logn√) factor approximation algorithm for computing boxicity, which, to our knowledge, is the first o(n) factor approximation algorithm for the problem. We also present an FPT approximation algorithm for computing the cubicity of graphs, with vertex cover number as the parameter.

BELIEF PROPAGATION FOR OPTIMAL EDGE COVER IN THE RANDOM COMPLETE GRAPH

We apply the objective method of Aldous to the problem of finding the minimum-cost edge cover of the complete graph with random independent and identically distributed edge costs. The limit, as the number of vertices goes to infinity, of the expected minimum cost for this problem is known via a combinatorial approach of Hessler and Wastlund. We provide a proof of this result using the machinery of the objective method and local weak convergence, which was used to prove the (2) limit of the random assignment problem. A proof via the objective method is useful because it provides us with more information on the nature of the edge's incident on a typical root in the minimum-cost edge cover. We further show that a belief propagation algorithm converges asymptotically to the optimal solution. This can be applied in a computational linguistics problem of semantic projection. The belief propagation algorithm yields a near optimal solution with lesser complexity than the known best algorithms designed for optimality in worst-case settings.

Parameterized Algorithms for MAX COLORABLE INDUCED SUBGRAPH Problem on Perfect Graphs

We address the parameterized complexity ofMaxColorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a n(O(q)) algorithm on chordal graphs. We first observe that the problem is W2]-hard parameterized by q, even on split graphs. However, when parameterized by l, the number of vertices in the solution, we give two fixed-parameter tractable algorithms. The first algorithm runs in time 5.44(l) (n+#alpha(G))(O(1)) where #alpha(G) is the number of maximal independent sets of the input graph. The second algorithm runs in time q(l+o()l())n(O(1))T(alpha) where T-alpha is the time required to find a maximum independent set in any induced subgraph of G. The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in l alone (whenever T-alpha is a polynomial in n), since q <= l for all non-trivial situations. Finally, we show that (under standard complexitytheoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense: (a) On split graphs, we do not expect a polynomial kernel if q is a part of the input. (b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q >= 2.

Rainbow Connection Number of Graph Power and Graph Products

Rainbow connection number, rc(G), of a connected graph G is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same (note that the coloring need not be proper). In this paper we study the rainbow connection number with respect to three important graph product operations (namely the Cartesian product, the lexicographic product and the strong product) and the operation of taking the power of a graph. In this direction, we show that if G is a graph obtained by applying any of the operations mentioned above on non-trivial graphs, then rc(G) a parts per thousand currency sign 2r(G) + c, where r(G) denotes the radius of G and . In general the rainbow connection number of a bridgeless graph can be as high as the square of its radius 1]. This is an attempt to identify some graph classes which have rainbow connection number very close to the obvious lower bound of diameter (and thus the radius). The bounds reported are tight up to additive constants. The proofs are constructive and hence yield polynomial time -factor approximation algorithms.

Graph Modification Problems: A Modern Perspective

We give an overview of recent results and techniques in parameterized algorithms for graph modification problems.

Distributed flow algorithms for scalable similarity visualization

We describe simple yet scalable and distributed algorithms for solving the maximum flow problem and its minimum cost flow variant, motivated by problems of interest in objects similarity visualization. We formulate the fundamental problem as a convex-concave saddle point problem. We then show that this problem can be efficiently solved by a first order method or by exploiting faster quasi-Newton steps. Our proposed approach costs at most O(|ε|) per iteration for a graph with |ε| edges. Further, the number of required iterations can be shown to be independent of number of edges for the first order approximation method. We present experimental results in two applications: mosaic generation and color similarity based image layouting. © 2010 IEEE.

algorithms for computing weak bisimulation equivalence

IEEE Comp Soc, IFIP, Tianjin Normal Univ

Parallel optimisation algorithms for multilevel mesh partitioning

Three parallel optimisation algorithms, for use in the context of multilevel graph partitioning of unstructured meshes, are described. The first, interface optimisation, reduces the computation to a set of independent optimisation problems in interface regions. The next, alternating optimisation, is a restriction of this technique in which mesh entities are only allowed to migrate between subdomains in one direction. The third treats the gain as a potential field and uses the concept of relative gain for selecting appropriate vertices to migrate. The results are compared and seen to produce very high global quality partitions, very rapidly. The results are also compared with another partitioning tool and shown to be of higher quality although taking longer to compute.

A multilevel algorithm for force-directed graph drawing

We describe a heuristic method for drawing graphs which uses a multilevel technique combined with a force-directed placement algorithm. The multilevel process groups vertices to form clusters, uses the clusters to define a new graph and is repeated until the graph size falls below some threshold. The coarsest graph is then given an initial layout and the layout is successively refined on all the graphs starting with the coarsest and ending with the original. In this way the multilevel algorithm both accelerates and gives a more global quality to the force- directed placement. The algorithm can compute both 2 & 3 dimensional layouts and we demonstrate it on a number of examples ranging from 500 to 225,000 vertices. It is also very fast and can compute a 2D layout of a sparse graph in around 30 seconds for a 10,000 vertex graph to around 10 minutes for the largest graph. This is an order of magnitude faster than recent implementations of force-directed placement algorithms.

A combined evolutionary search and multilevel optimisation approach to graph-partitioning

The graph-partitioning problem is to divide a graph into several pieces so that the number of vertices in each piece is the same within some defined tolerance and the number of cut edges is minimised. Important applications of the problem arise, for example, in parallel processing where data sets need to be distributed across the memory of a parallel machine. Very effective heuristic algorithms have been developed for this problem which run in real-time, but it is not known how good the partitions are since the problem is, in general, NP-complete. This paper reports an evolutionary search algorithm for finding benchmark partitions. A distinctive feature is the use of a multilevel heuristic algorithm to provide an effective crossover. The technique is tested on several example graphs and it is demonstrated that our method can achieve extremely high quality partitions significantly better than those found by the state-of-the-art graph-partitioning packages.

JOSTLE: multilevel graph partitioning software: an overview

In this chapter we look at JOSTLE, the multilevel graph-partitioning software package, and highlight some of the key research issues that it addresses. We first outline the core algorithms and place it in the context of the multilevel refinement paradigm. We then look at issues relating to its use as a tool for parallel processing and, in particular, partitioning in parallel. Since its first release in 1995, JOSTLE has been used for many mesh-based parallel scientific computing applications and so we also outline some enhancements such as multiphase mesh-partitioning, heterogeneous mapping and partitioning to optimise subdomain shape

Scalable and efficient graph colouring in 3 dimensions using emergence engineering principles

This paper describes ways in which emergence engineering principles can be applied to the development of distributed applications. A distributed solution to the graph-colouring problem is used as a vehicle to illustrate some novel techniques. Each node acts autonomously to colour itself based only on its local view of its neighbourhood, and following a simple set of carefully tuned rules. Randomness breaks symmetry and thus enhances stability. The algorithm has been developed to enable self-configuration in wireless sensor networks, and to reflect real-world configurations the algorithm operates with 3 dimensional topologies (reflecting the propagation of radio waves and the placement of sensors in buildings, bridge structures etc.). The algorithm’s performance is evaluated and results presented. It is shown to be simultaneously highly stable and scalable whilst achieving low convergence times. The use of eavesdropping gives rise to low interaction complexity and high efficiency in terms of the communication overheads.

Goal Recognition through Goal Graph Analysis

We present a novel approach to goal recognition based on a two-stage paradigm of graph construction and analysis. First, a graph structure called a Goal Graph is constructed to represent the observed actions, the state of the world, and the achieved goals as well as various connections between these nodes at consecutive time steps. Then, the Goal Graph is analysed at each time step to recognise those partially or fully achieved goals that are consistent with the actions observed so far. The Goal Graph analysis also reveals valid plans for the recognised goals or part of these goals. Our approach to goal recognition does not need a plan library. It does not suffer from the problems in the acquisition and hand-coding of large plan libraries, neither does it have the problems in searching the plan space of exponential size. We describe two algorithms for Goal Graph construction and analysis in this paradigm. These algorithms are both provably sound, polynomial-time, and polynomial-space. The number of goals recognised by our algorithms is usually very small after a sequence of observed actions has been processed. Thus the sequence of observed actions is well explained by the recognised goals with little ambiguity. We have evaluated these algorithms in the UNIX domain, in which excellent performance has been achieved in terms of accuracy, efficiency, and scalability.

Core-based Methodology: An Automated Approach for Implementing a Complete System From Algorithms to a Heterogeneous Network Including FPGAs

In this paper, we present a methodology for implementing a complete Digital Signal Processing (DSP) system onto a heterogeneous network including Field Programmable Gate Arrays (FPGAs) automatically. The methodology aims to allow design refinement and real time verification at the system level. The DSP application is constructed in the form of a Data Flow Graph (DFG) which provides an entry point to the methodology. The netlist for parts that are mapped onto the FPGA(s) together with the corresponding software and hardware Application Protocol Interface (API) are also generated. Using a set of case studies, we demonstrate that the design and development time can be significantly reduced using the methodology developed.