Diego Delle Donne & Javier Marenco Computer Science department, FCEN, University of Buenos Aires.

1. Introduction Modelling the problem A polyhedral study of the minimum-adjacency vertex coloring problem ILP model Polyhedral study Branch & Cut Computational results Final remarks Diego Delle Donne & Javier Marenco Computer Science department, FCEN, University of Buenos Aires. Sciences Institute, National University of General Sarmiento. A polyhedral study of the minimum-adjacency vertex coloring problem

2. Introduction Cellular GSM networks: ¿How do communications work? Hand-over A polyhedral study of the minimum-adjacency vertex coloring problem

3. Introduction Possible problems: Co-channel interference (same channel) Adjacent-channel interference A polyhedral study of the minimum-adjacency vertex coloring problem

4. Introduction Covering an area using a network: … but the number of available channels is limited… A polyhedral study of the minimum-adjacency vertex coloring problem

5. Introduction Possible schema for a frequency assignment: 1. Assign one channel to each antenna avoiding co-channel interference. 2. Minimize adjacent-channel interference. A polyhedral study of the minimum-adjacency vertex coloring problem

6. Introduction Considerations: Most common antennas cover 120º and there is often more than one antenna within a sector Control channels (BCCH) Vs. transmit channels (TCH) There are many studies on this problem, but little work on exact approaches. We ommit other characteristics of the problem: blocked channels, minimum distances, etc. A polyhedral study of the minimum-adjacency vertex coloring problem

7. Introduction Modelling the problem ILP model Polyhedral study Branch & Cut Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

8. Modelling the problem G = (V, E) C = {c1, c2, … , ct} Goal: Find a coloring of G using colors from C, minimizing the number of adjacent vertices receiving adjacent colors (NP-H). A polyhedral study of the minimum-adjacency vertex coloring problem

9. Modelling the problem ILP model Polyhedral study Branch & Cut Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

10. ILP model Considered models: Stable model (Méndez Díaz and Zabala, 2001) Orientation model (Grötschel et. al., 1998) Distance model Representatives model (Campêlo, Corrêa and Frota, 2004) A polyhedral study of the minimum-adjacency vertex coloring problem

11. : Stable model ILP model xvcrepresents whether color c is assigned to vertex v or not zvw asserts whether vertices v and w receive adjacent colors or not A polyhedral study of the minimum-adjacency vertex coloring problem

12. ILP model Polyhedral study Branch & Cut Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

13. Polyhedral study Definition: Let be a clique of G and let Q = {c1,…,cq} be a set of consecutive colors. We define the Consecutive Colors Clique Inequality to be: q-1 adjacencies Theorem: The Consecutive Colors Clique Inequalities are valid for PS(G,C) and, if |C| > , |C|>|Q|,|C|>|K|+ 4 and |K|> , they are also facet-defining for PS(G,C). x x x x x x x Q = c1 c2 …… cq Removes 2 adjacencies Removes 1 adjacency A polyhedral study of the minimum-adjacency vertex coloring problem

14. Polyhedral study Definition: Let be a clique of G and , p non-adjacent sets of consecutive colors. We define the Multi Consecutive Colors Clique Inequality to be: Theorem: The Multi Consecutive Colors Clique Inequalities are valid for PS(G,C). C = … A polyhedral study of the minimum-adjacency vertex coloring problem

15. Polyhedral study Definition: Let be a clique of G and a vertex from the clique. Let be a set of consecutive colors. We define the 3-Colors Inner Clique Inequality to be: Theorem: The 3-Colors Inner Clique Inequalities are valid for PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-defining for PS(G,C). Q = {c1, c2,c3} A polyhedral study of the minimum-adjacency vertex coloring problem

16. Polyhedral study Definition: Let be a clique of G and a vertex from the clique. Let be a set of consecutive colors. We define the 3-Colors Outer Clique Inequality to be: Theorem: The 3-Colors Outer Clique Inequalities are valid for PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-defining for PS(G,C). Q = {c1, c2,c3} A polyhedral study of the minimum-adjacency vertex coloring problem

17. Polyhedral study Definition: Let be a clique of G and a vertex from the clique. Let be a set of consecutive colors. We define the 4-Colors Vertex Clique Inequality to be: Theorem: The 4-Colors Vertex Clique Inequalities are valid for PS(G,C) and, if |C| > and |C| > |K| + 4, they are also facet-defining for PS(G,C). Q = {c1, c2,c3 ,c4} A polyhedral study of the minimum-adjacency vertex coloring problem

18. Polyhedral study Definition: Let be a clique of G and an available color. We define the Clique Inequality (Méndez Díaz and Zabala, 2001) to be: Theorem: The Clique Inequalities are valid for PS(G,C) and, if K is a maximal clique and |C| > , they are also facet-defining for PS(G,C). A polyhedral study of the minimum-adjacency vertex coloring problem

19. Polyhedral study Branch & Cut Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

20. Branch & Cut Separation algorithms (searching for cliques): Backtracking Limit on the exploration of the backtracking tree Bounds for cutting whole branches Returns: first N, best N, all Greedy heuristic Returns N cliques (at most one for each starting vertex) A polyhedral study of the minimum-adjacency vertex coloring problem

21. C = Branch & Cut Extending intervals for the MCCK Inequality: A polyhedral study of the minimum-adjacency vertex coloring problem

22. Branch & Cut Other used technics: Primal bounds: Construction of feasible solutions by rounding techniques Subproblem reduction by logical implications Selection of the branching variable A polyhedral study of the minimum-adjacency vertex coloring problem

23. Branch & Cut Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

24. Computational results Context: Test instances extracted from EUCLID CALMA Project. Test set of 30 instances (chosen from the preliminary experimentation) Pentium IV with 1Gb of RAM memory with one microprocessor running at 1.8 Ghz. Each family of inequalities doesn’t seem to work well when used individually. Combining the MCCK and the K inequalities gives a very strong cutting plane phase for the branch & cut A polyhedral study of the minimum-adjacency vertex coloring problem

25. Computational results In this chart we can see the average time taken by different combinations of families with a base combination of MCCK+K. We also show the type of separation used to search for violating cliques. A polyhedral study of the minimum-adjacency vertex coloring problem

26. Computational results If we zoom in the section corresponding to the backtracking separation we can see that the best combination for these test set is the base combination of inequalities (MCCK+K), searching cliques with the “best” parameter. A polyhedral study of the minimum-adjacency vertex coloring problem

27. Computational results Next we test different values for the number of cliques returned by the backtracking separation (for MCCK+K with “N best cliques” separation). We also show the node limit for the exploration of the backtracking tree: 150, 300, 450 and 600 nodes. A polyhedral study of the minimum-adjacency vertex coloring problem

28. Computational results Zooming again we can see that the best times are achieved using a number between 14 and 22 cliques (actually the best time was achieved using 20 cliques) with a node limit of 150. A polyhedral study of the minimum-adjacency vertex coloring problem

29. Computational results Results: Adding other families to the MCCK+K combination only worsens the resolution times. The best parametrization for the Branch & Cut was achieved using backtracking with a 150 node limit on the bactracking tree exploration and returning the best 20 found cliques. Now with this combination of inequalities and the best parametrization found, we will compare our running times against CPLEX’s. A polyhedral study of the minimum-adjacency vertex coloring problem

30. Computational results A polyhedral study of the minimum-adjacency vertex coloring problem

31. Computational results A polyhedral study of the minimum-adjacency vertex coloring problem

32. Computational results Final remarks A polyhedral study of the minimum-adjacency vertex coloring problem

33. Final remarks Conclusions: The associated polytope has a very interesting combinatorial structure. The Branch & Cut seems to be very efficient and the proposed inequalities seem to help in a decisive manner. The experimentation shows that “MCCK+K+best 20” would be the best parametrization for the branch & cut algorithm. A polyhedral study of the minimum-adjacency vertex coloring problem

34. Final remarks Future work: Conclude the experiments testing different values for other branch & cut parameters (skip factor, cutting phase iterations, etc). Deepen the study of the other models. Incorporate to this study the characteristics setted aside at the beginning. A polyhedral study of the minimum-adjacency vertex coloring problem

35. Thank you! Diego Delle Donne & Javier Marenco Computer Science department, FCEN, Univeristy of Buenos Aires. Sciences Institute, National University of General Sarmiento. A polyhedral study of the minimum-adjacency vertex coloring problem