Finite Domain Cuts for Minimum Bandwidth

Transcription

1 Finite Domain Cts for Minimm Bandwidth J. N. Hooker Carnegie Mellon University, USA Joint work with Alexandre Friere, Cid de Soza State University of Campinas, Brazil INFORMS 2013

2 Two Related Problems They differ only in the objective fnction. Minimm Linear Arrangement problem: Label vertices of a graph to minimize sm of edge lengths. Application: minimize wiring on a linear circit board 1 2 Min edge sm =

3 Two Related Problems They differ only in the objective fnction. Minimm Linear Arrangement problem: Label vertices of a graph to minimize sm of edge lengths. Application: minimize wiring on a linear circit board 1 2 Min edge sm = 14 3 min E i j is perm. of 1,, n 4 5

4 Two Related Problems They differ only in the objective fnction. Minimm Bandwidth problem: Label vertices of a graph to minimize maximm edge length. Application: sparse matrix calclation Application: ordering variables to minimize backtracking 1 2 Min bandwidth= 3 min max 3 E i j is perm. of 1,, n 4 5

5 Finite-Domain Cts Difficlt to find good bonds, polyhedral or otherwise. 0-1 models have weak relaxations. Idea: Use finite domain cts Formlate problem with finite domain variables rather than 0-1 variables. Find valid ineqalities. This was sefl for graph coloring. Bergman and Hooker, CPAIOR Let x i = color assigned to vertex i. Obtained tighter bonds in less time.

6 Finite-Domain Cts For linear arrangement and bandwidth, se edge length variables only. Unclear how to give problem a complete ineqality formlation. Bt we can derive valid cts. min E valid cts in E min w max valid cts in w,

7 Min Linear Arrangement This idea already applied to MLA problem. Sbstantial improvement over existing bonds. Caprara, Letchford, Salazar-González, IJOC Polytope hard to analyze. CLS stdied dominant of polytope instead. Facets fond for stars, etc. k = 5: k 2 4

8 Min Linear Arrangement This idea has already been applied to MLA problem. CLS generated cts for stars in complete graph. Filled in missing edges with paths in original graph. Reslting cts remain valid (not facet defining). Separation algorithm fond the paths. Stars did most of the work.

9 Min Linear Arrangement This idea has already been applied to MLA problem. CLS generated cts for stars in complete graph. Filled in missing edges with paths in original graph. Reslting cts remain valid (not facet defining). Separation algorithm fond the paths. Stars did most of the work. So far, or cts do not improve on stars + separation.

10 Min Bandwidth Missing edges trick does not work for min bandwidth. Reslting cts are not valid. We experiment with alternate approaches to finding cts. Focs on validity, not facets. Identify cts for large strctres that fill mch of the graph. Modify cts to restore validity after modifying strctres. Use conting argments.

11 Known Bonds Density bond is best-known bond. Chvátal (1970) NP-hard to compte. Not a polyhedral ct. S 1 w max SV ds ( ) diameter of S w bandwidth = 3

12 Known Bonds Alternate bond (call it CS bond) Caprara, Salazar-González, IJOC P-time to compte. Bt not a polyhedral ct. S 1 w min max vv S: vs d( v, S) max distance from v to vertex in S w min, bandwidth = 3

13 Or Proof Strategy for Bonds Find an pper bond on edge sm in a strctred sbgraph, as a fnction of bandwidth w: Then solve for bond on w: U( w) 1 w U Combine with lower bonds on edge sm: L( w) and with lower bonds de to CLS.

14 Path Cts Path on k vertices has max edge sm U( w) ( k 1) w So we have the ct 1 1 w U k 1 There are many long paths, bt the ct is very weak. For w = 5: w 1 5

15 Cycle Cts k-cycle has max edge sm kw 2 if k is even ( k 1) w if k is odd 0 For w = 5: 5 4 4w 2 7

16 Cycle Cts This yields the (sharp) bond 1 2 w if k k k 1 w if k k 1 Slightly stronger than for path is even is odd w 1 4

17 Cliqe Cts Cliqe of size k has max edge sm 2 k k w ( k 1)( k 2) if k is even k 1 k 1 w ( k 1)( k 3) if k is odd w4

18 Cliqe Cts Cliqe of size k has max edge sm 2 k k w ( k 1)( k 2) if k is even k 1 k 1 w ( k 1)( k 3) if k is odd 4 12 Achieve this max by sing k/2 smallest and k/2 largest vertex labels in {0,, w} for k even, similarly for k odd. 0 For w = 8: 1 8 6w4 6 7

19 Cliqe Cts This yields the (sharp) bond: 4 1 w 2 ( k 1)( k 2) if k is even k 3k 4 1 w ( k 3) if k is odd 2 ( k 1) 3 Strong ct, bt hard to find large cliqes w

20 Cliqe Cts We also have the (sharp) lower bond on edge sm k k ( k 1)( k 1) 6 3 wk1 Combined with LB on w, this yields which is sharp for a cliqe and same as density and CS bond. w 4

21 Cliqe Cycles Cycles of cliqes can provide good bonds for large strctres. Cycle of k 3-cliqes has max edge sm 2kw 1 For w = 5: w

22 Cliqe Cycles This yields the (sharp) bond For odd and even k 1 w 2k w 1 10

23 Cliqe Cycles We also have the (sharp) lower bond on edge sm 8( k1) Combined with LB on w, this yields 4 w 4 k For k = 5, this yields w 4 (sharp), stronger than density bond 3, same as CS bond

24 Cliqe Cycles Conjectre: Cycle of k 4-cliqes has max edge sm 4kw 2k 1 20w9

25 Cliqe Cycles This yields the bond w 1 2k 1 4k 4k w

26 Cliqe Cycles We also have the (sharp) lower bond on edge sm 16k15 Combined with LB on w, this yields For k = 5, this yields w 5 (sharp). Density and CS bond are also 5. w k 65

27 Cts when Edges Are Missing A certificate of a sharp bond on edge sm can yield a sharp bond when edges are removed. Start with a sharp pper bond U(w) on edge sm. For w = 5: 10w

28 Cts when Edges Are Missing A certificate of a sharp bond on edge sm can yield a sharp bond when edges are removed. Start with a sharp pper bond U(w) on edge sm. Let v be a vertex labeling that achieves edge sm U(w). 1 For w = 5: w

29 Cts when Edges Are Missing A certificate of a sharp bond on edge sm can yield a sharp bond when edges are removed. Start with a sharp pper bond U(w) on edge sm. Let v be a vertex labeling that achieves edge sm U(w). Remove m edges (i, j) for which v i v j = 1. 1 For w = 5: w

30 Cts when Edges Are Missing A certificate of sharp bond on edge sm can yield a sharp bond when edges are removed. Start with a sharp pper bond U(w) on edge sm. Let v be a vertex labeling that achieves edge sm U(w). Remove m edges (i, j) for which v i v j = 1. Now U(w) m is a sharp pper bond on edge sm. For w = 5: 10w2

31 Cts when Edges Are Missing Can get a valid (nonsharp) bond after removing any m edges U(w) m is an pper bond on edge sm. 10w4

32 Conting Argments Take any graph with 12 vertices. And any sbgraph with m edges. For any bandwidth w, compte maximm edge sm. Sm the first m edge labels starting with w. Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

33 Conting Argments For example, assming sbgraph has 7 edges For bandwidth z = 11 Max edge sm is = 66 Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

34 Conting Argments For example, assming sbgraph has 7 edges For bandwidth z = 10 Max edge sm is = 63 Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

35 Conting Argments For example, assming sbgraph has 7 edges For bandwidth z = 9 Max edge sm is = 59 Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

36 Min Bandwidth Conting Argments This yields family of valid cts of the form w 12 7-edge sbgraph of 12-vertex graph Edge sm in sbgraph

37 Min Bandwidth Conting Argments This yields family of valid cts of the form w 15-edge sbgraph of 12-vertex graph Edge sm in sbgraph

38 Conting Argments For stronger cts, we mst consider strctre of sbgraph. We will se degree constraints. Example: Sppose n = 12. Consider sbgraph with m = 7. Enforce degree constraints 2 vertices degree 4 6 vertices degree 1

39 Conting Argments Use greedy algorithm to get pper bond on edge sm For bandwidth z = 10 UB is = 65 (not 66) Max is 65 Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

40 Conting Argments Use greedy algorithm to get pper bond on edge sm For bandwidth z = 10 Max edge sm is = 61 (not 63) Max is 58 Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

41 Conting Argments Research qestion: For what strctres does greedy algorithm yield valid pper bond? Edge Label Max No. Edges Possible edges 11 1 (12,1) 10 2 (12,2) (11,1) 9 3 (12,3) (11,2) (10,1) 8 4 (12,4) (11,3) (10,2) (9,1) 7 5 (12,5) (11,4) (10,3) (9,2) (8,1) 6 6 (12,6) (11,5) (10,4) (9,3) (8,2) (7,1) etc.

42 Ongoing Research Explore other sbstrctres with dense regions. Develop cts based on certificates. Identify strctres for which greedy algorithm works in conting argment. Find separation algorithms. Comptational tests.