An Ant-Based Algorithm for the Minimum Vertex Cover Problem

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3 Ant-Based MVC Algorithm 12Input: Graph G = (V, E)Output: A vertex cover C of Gbegin// initialization stageC ← ∅bestC ← Vwhile a degree 1 vertex u exists // preprocessing stepv ← neighbor(u)C ← C ∪ {v}V ← V − {u, v}remove all edges incident with v from Eend-whilecopyE ← Eposition ants on randomly selected edgesinitialize the pheromone level of all verticeswhile stopping criteria not met// exploration stageE ← copyEfor step = 1 to numOfStepsfor each ant a // more details in Figure 3.4select an edge (u, v) for the ant a to move to and move it thereupdate the pheromone level for all verticesend-for// construction stage – more details in Figure 3.5compute coverFactor for all verticeswhile E is not emptyadd the vertex u with highest coverFactor to Cremove all edges incident with u from Erecompute coverFactor for all verticesend-while// local optimization – more details in Figure 3.7 and Figure 3.8C ← RebuildCover(C)C ← RedundantVertexRemoval(C)if |C| < |bestC| thenbestC ← Cevaporate pheromone level globallyend-whilereturn bestCendFigure 3.1. An Overview of the AB–MVC Algorithm.
3 Ant-Based MVC Algorithm 13a ••c• e•b• dFigure 3.2. Cascading Effect of the Preprocessing Step Solves the Graph•ftesting, the number of ants to explore the graph is set to one tenth of the numberof edges in G. However, we have a maximum and minimum limit on the numberof ants to avoid crowding or sparseness. The number of steps each ant will take inone cycle is set to one fourth of the number of vertices in G, so that there will beenough exploration in each cycle. Next, the ants are distributed across randomlyselected edges, such that no edge contains more than one ant. However, it shouldbe noted that there can be edges with no ants on them. Also, every ant has atabu list attached to it, so that it can remember the last ten vertices that it hascrossed. This list is set to null at the start of the algorithm. This completes theinitialization stage and the ants are ready to explore the graph.3.3 ExplorationIn the exploration stage, the ants tend to learn about the problem space by movingaround the graph, passing from one edge to another. The objective of this stageis to explore the graph and identify potentially good vertices that can be used forconstructing a vertex cover in the next stage.Every edge in the graph has a value called the movement factor, defined withrespect to the position of an ant. Let (s, u) and (u, v) be two edges in the graph.Let the pheromone level of a vertex v be denoted by τ[v]. Let an ant be present inthe edge (s, u). Then, the movement factor of the edge (u, v) with respect to the
Page 1 and 2: We approve the thesis of Muthubhara
Page 3 and 4: The Pennsylvania State UniversityTh
Page 5 and 6: AbstractA vertex cover for a graph
Page 7 and 8: 3.6.1 Rebuild Vertex Cover Techniqu
Page 9 and 10: List of TablesA.1 AB-MVC results on
Page 11 and 12: Chapter 1IntroductionA vertex cover
Page 13 and 14: Chapter 2Preliminaries2.1 Problem D
Page 15 and 16: 2 PRELIMINARIES 5Input: Graph G = (
Page 17 and 18: 2 PRELIMINARIES 7the degree constra
Page 19 and 20: 2 PRELIMINARIES 9and Dras have deve
Page 21: 3 Ant-Based MVC Algorithm 11The alg
Page 25 and 26: 3 Ant-Based MVC Algorithm 15vertice
Page 27 and 28: 3 Ant-Based MVC Algorithm 17the val
Page 29 and 30: 3 Ant-Based MVC Algorithm 193.6.1 R
Page 31 and 32: 3 Ant-Based MVC Algorithm 21are mor
Page 33 and 34: Chapter 4Experimental ResultsIn thi
Page 35 and 36: 4 Experimental Results 25Figure 4.2
Page 37 and 38: 4 Experimental Results 27edge densi
Page 39 and 40: 5 Conclusion 29a parametric study w
Page 41 and 42: 5 Conclusion 31Table A.2. AB-MVC re
Page 47 and 48: REFERENCES 37[9] Dorigo, M., V. Man

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