Mutated Near Optimal Vertex Cover Algorithm (NOVCA) Visualization on a Tile Display Sanjaya Gajurel and Roger Bielefeld High Performance Computing (HPC), Advanced Research Computing (ARC) IEEE Cluster 2015, Sep. 8-11, Chicago, IL, USA

ABSTRACT This paper describes the mutated version of extremely fast polynomial time algorithm, NOVCA (Near Optimal Vertex Cover Algorithm). NOVCA is based on the idea of including the vertex having higher degree in the cover. Mutation is introduced in NOVCA by randomly selecting any remaining vertex having degree greater than 1 in the cover as an exception.

INTRODUCTION

Figure 2—Graphs: 2D (right) and 3D (left) showing the animation on tile display for brock-200 instance of a graph at CWRU Viz-Wall; vertices in a cover havng negative value for scalar degree, are distinguished by red color from other vertices. The color coding of a vertex is based on current magnitude of its associated degree.

The Vertex Cover (VC) of a graph G(V,E) with vertex set V and edge set E is a subset of vertices C of V (C ⊆ V) such that every edge of G has at least one endpoint in C. In 1972, Richard [1] showed that identification of minimal VC in a graph is an NP-complete problem.

MUTATED NEAR OPTIMAL VERTEX COVER ALGORITHM (MNOVCA) NOVCA [2][3][4] (Fig. 1) is based on the concept that vertex cover candidates are those that are adjacent to the minimum degree vertex. It prevents the minimum degree vertex from being included in the cover. In case of a tie in a minimum degree vertex, the one having higher sum of the degrees of its adjacent vertices, is chosen. In MNVOCA, mutation is introduced by randomly including any of the remaining vertices, not in the cover, having degree at least greater than one. NOVCA always returns minimum cover for all sorts of random graphs including the Benchmark random graphs. MNOVCA tackles the family of benchmark graphs [5] having minimum cover consisting of lower degree vertices that defeats NOVCA's fundamental heuristics of including vertices of higher degree in the cover.

IMPLEMENTATION @ CWRU HPC & VIZ-WALL The NOVCA algorithm is re-written in VTK/Cxx [6], implemented in the CWRU High Performance Computing (HPC) Cluster, and visualized on the CWRU Viz-Wall. In the VTK/Cxx implementation, the output of the code is dumped as VTK files in VTK Unstructured (.vtu) format, where points and cells represent vertices and edges of the graph respectively. ParaView [7] that uses VTK under the hood as the data processing and rendering engine, then produces high resolution animation in a tile display (Fig. 2). Fig. 3 and Fig. 4 depict high resolution ParaView screenshots of 2D and 3D animations respectively.

Figure 3: Screenshot of 2D animation for brock-200 instance of a graph. The color coding of a vertex is based on current magnitude of its associated degree.

CONCLUSIONS

REFERENCES

The ability to visualize combinatorial optimization problems such as Vertex Cover on a large display (Fig. 2) helps researchers to observe and analyze their behavior at different stages of the algorithm. NOVCA, which always returns optimal value for small benchmark graphs, produces suboptimal results on some larger benchmark graphs. The animated visualization permitted detection of its patterns of failure on these graphs, which would have been nearly impossible to achieve using static renderings. Moreover, because the small displays can quickly get cluttered when size of the graph increases, there is a great benefit to using a larger high resolution tile display. This technique can be applied to other relevant fields not limited to combinatorial optimization.

1. KARP, R. 1972. Reducibility among combinatorial problems. In R. E. Miller and J. W. Thatcher (eds.). Complexity of Computer Computations, Plenum Press, NY, pp. 85-103. 2. GAJUREL, S., AND BIELEFELD, R. 2012. A Simple NOVCA: Near Optimal Vertex Cover Algorithm. Procedia Computer Science, vol. 9, pp 747-753. 3. GAJUREL, S., AND BIELEFELD, R. 2012. A Fast Near Optimal Vertex Cover Algorithm (NOVCA). IJEA, Vol. 3(1), pp 9-18. 4. GAJUREL, S., AND BIELEFELD, R. 2014., A Heuristic Approach to Fast NOVCA (Near Optimal Vertex Cover Algorithm), JCTA 5, pp 83-90.

ACKNOWLEDGEMENT This work made use of the High Performance Computing Resource in the Core Facility for Advanced Research Computing at Case Western Reserve University.

5. XU, K 2012. Vertex Cover Benchmark Instances (DIMACS and BHOSLIB). http://www.cs.hbg.psu.edu/benchmarks/vertex_cover.html. Figure 1: Example elucidating the stages in NOVCA algorithm. The blue colored nodes are the minimum degree vertices obtained through magic function where as the green colored nodes are the vertices included in a cover; V is the vertex ID, deg[v] is the degree of the vertex, and sum_adj_deg[v] is the sum of the degree of adjacent vertices to V.

Figure 2—Graphs:.

Figure 4: Screenshot of 3D animation for brock-200 instance of a graph. The color coding of a vertex is based on current magnitude of its associated degree.

6. The Visualization ToolKit (VTK), www.vtk.org 7. ParaView, www.paraview.org