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Approximation Algorithms for Minimum-Length Corridors

This paper discusses approximation algorithms for the Minimum-Length Corridor problem and its related problems. It presents a parameterized approximation algorithm and explores various selector functions.

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Approximation Algorithms for Minimum-Length Corridors

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  1. Version 10/09 Teofilo F. Gonzalez Approximation Algorithms for Minimum-Length Corridors Department of Computer Science University of California at Santa Barbara teo@cs.ucsb.edu (Co-author: Arturo González-Gutiérrez) Version 09/09

  2. Outline • Introduction • Preliminaries and Related Problems • Parameterized Approximation Algorithm Alg(S) • Selector Function S • Constant Ratio Approximation • Additional Results and Conclusion Version 09/09

  3. MLC MLC-R Introduction:Minimum-Length Corridor Problem (MLC) Version 09/09

  4. Introduction:Formal Definition of the MLC-R problem • INPUT: A pair (F,R), where F is a rectangular boundary partitioned into rectangles (or rooms) R={R1,R2,…,Rr}. • OUTPUT: A corridor consisting of a set of connected line segments each of which lies along the line segments that form F and/or the boundary of the rooms, that includes at least one point of F and at least one point from each of the rooms. • OBJECTIVE FUNCTION: Minimize the total edge length of the corridor. Version 09/09

  5. Introduction:Applications of the MLC Problem Network Communication in Metropolitan Areas Version 09/09

  6. Introduction:Origin of the MLC Problem • Demaine, E.D., and O’Rourke, J. Open problems from CCCG 2000. In Proceedings of the 13th. Canadian Conference on Computational Geometry (2001), (by Naoki Katoh). http://theory.csail.mit.edu/~edemaine/papers/CCCG2000Open/ • Eppstein, D. Some open problems in graph theory and computational geometry. http://www.ics.uci.edu/~eppstein/200-f01.pdf, November 2001. • Jin, L. Y., and Chong, O. W. The minimum touching tree problem. http://www.yewjin.com/research/MinimumTouchingTrees.pdf, National University of Singapore, School of Computing, 2003. • No polynomial time algorithm known • Not even a constant ratio approximation algorithm • Seems likely to be NP-complete but no proof known Version 09/09

  7. Related Problems:Outline • Tree Errand Cover (TEC) problem • Generalization of the Group Steiner Tree (GSTP) Problem Version 09/09

  8. Related Problems:Formal Definition of the TEC problem • INPUT: A connected undirected edge-weighted graph G=(V,E,w), where w:E→R+ is an edge-weight function; a non-empty set C, CV, of terminals; a non-empty set E={e1,e2,...,ek} of errands; a collectionC={C1, C2,…, Ck}, where CiC specifies the vertices where errand ei can be performed. • OUTPUT: A tree T(G,C)=(V’,E’), where E’  E and V’  V, such that for each errand ei there is at least one vertex vCi and vV’, and the total length is minimized GST problem: C={C1, C2,…, Ck} is a partition of C Version 09/09

  9. R3 R2 R1 R6 R5 R4 R8 R7 R9 Related Problems:MLC-RTEC E={Ri | 0 ≤ i ≤ 9} C={Ci | 0 ≤ i ≤ 9} v4 v2 v3 v1 v6 v8 v7 v5 v9 v10 v12 v11 v13 v16 v14 v15 v17 v20 v18 v19 Version 09/09

  10. Related Problems:TEC Problem Performance Ratio • Slavik, P. (1998) • The TEC problem can be approximated to within a ratio of 2ρ in polynomial time, when each errand is assigned to at most ρ vertices. • For the MLC problem there are errands that may be assigned to an arbitrary number of vertices. • GST problem • (k-1) OPT (Ihler, E., 1991) • ( 1 + ln(k/2) ) k0.5 OPT (Bateman, C. D. et al., 1997) • Polynomial time O(kα)-approximation algorithms, for arbitrarily small values of α>0 (Helvig, C. S. et al.,2001) • These results do not generate a constant ratio approximation for the MLC and MLC-R problems. Version 09/09

  11. MLC MLC-R p-MLC-R Parameterized Approximation Algorithm: Alg(S)Hierarchy of the MLC Problem Version 09/09

  12. Parameterized Approximation Algorithm Alg(S):Outline MLC-R α p-MLC-R • Parameterized algorithm Alg(S) for the p-MLC-R problem. Version 09/09

  13. p p S(2OC+) I p-MLC-R I p-MLC-RS Parameterized Approximation Algorithm Alg(S):Selector Function S and the p-MLC-RS problem Version 09/09

  14. p-MLC-R problem p-MLC-RS problem Feasible solutions of the p-MLC-R problem Feasible solutions after relaxing (LP) the set of feasible solutions of the p-MLC-RS problem Feasible solutions after rounding the solution found Parameterized Approximation Algorithm Alg(S):Approximation Technique Version 09/09

  15. Parameterized Approximation Algorithm Alg(S):For p-MLC-R problem t(I)≤2 kS rS opt(I) I=(p,F,R) p-MLC-R IS=(p,F,R,S) p-MLC-RS S J=(G=(V,E,w),C)  TEC JS=(G=(V,E,w),CS)  TECS Invoke Slavik’s Algorithm: t(I) ≤ 2 k opt(I); k = max{|V(Ri)|} Invoke Slavik’s Algorithm: t(IS) ≤ 2 kS opt(IS); kS = |S| t(IS)≤ rS t(I) opt(IS) ≤ rS opt(I) opt(IS) ≤ t(IS) ≤ rS opt(I) t(I) = t(IS) ≤ 2 kS rS opt(I) Version 09/09

  16. Selector Function SOutline • S selects four corners: S(4C) • Definition of Special Point • S selects special points: S(+) • S selects two adjacent corners and one special point: S(2AC+) • S selects two opposite corners and one special point: S(2OC+) Version 09/09

  17. Selector Function S:S selects four corners: S(4C) kS(4C)=4 t(IS(4C)(j)) ≤ rS(4C) t(I(j)) Version 09/09

  18. How about if we select the middle vertex? t(IS(j)) ≤ rS t(I(j)) Version 09/09

  19. Selector Function S:Definition of Special Point p SpP Version 09/09

  20. 2 2 2 2 3 2 3 3 2 Selector Function S:Definition of Special Point v1 v3 v2 v4 R2 R3 R1 v6 v8 v7 v5 R6 v9 CD(v11,R)= CD(R5,R)= 2 v10 v12 v11 R4 R8 v13 v16 v15 v14 R7 R9 R5 v17 v20 v18 v19 Version 09/09

  21. Selector Function S:S selects special point: S(+) kS(+)=1 t(IS(+)(j)) ≤ rS(+) t(I(j)) Version 09/09

  22. 1 2 3 j Selector Function S:S selects special points: S(+) kS(+)=1 t(IS(+)(j)) ≤ rS(+) t(I(j)) Version 09/09

  23. Selector Function S:S selects two adjacent corners and one special point: S(2AC+) kS(2AC+)=3 t(IS(2AC+)(j)) ≤ rS(2AC+) t(I(j)) Version 09/09

  24. Selector Function S:Gathering properties to construct S Special Point Corners: Opposite ones Version 09/09

  25. Constant Ratio Approximation:Outline • S selects two opposite corners and a special point: S(2OC+) • kS(2OC+)=3 and prove that rS(2OC+) ≤ 5 • Ncpe/cpe rectangles and tour • Paths type-1 and type-2 for adjacent ncpe rectangles • CD(SpPi,R) is bounded by a portion of the corridor • Type-1 • Type-2 Version 09/09

  26. p I p-MLC-R Constant Ratio Approximation:S selects two opposite corners and a special point: S(2OC+) kS(2OC+)=3 t(IS(2OC+)) ≤ rS(2OC+) t(I) TR, BL, SpP T(I) T(IS) Version 09/09

  27. Constant Ratio Approximation:Ncpe/cpe rectangles and tour 18 17 19 21 16 7 20 5 8 6 9 10 11 4 3 2 12 13 1 14 15 Version 09/09

  28. SpP3 Constant Ratio Approximation: kS(2OC+)=3 t(IS(2OC+)) ≤ rS(2OC+) t(I) rS(2OC+) ≤ 5 Version 09/09

  29. ni-1 ni-1 SpPi SpPi Constant Ratio Approximation:Paths type-1 and type-2 for adjacent ncpe rectangles ni+1 ni ni Version 09/09

  30. Constant Ratio Approximation:CD(SpPi,R) is bounded by a portion of the corridor: Type-1 Version 09/09

  31. Constant Ratio Approximation:CD(SpPi,R) is bounded by a portion of the corridor : Type-1 Version 09/09

  32. Constant Ratio Approximation:CD(SpPi,R) is bounded by a portion of the corridor: Type-1 Version 09/09

  33. Constant Ratio Approximation:CD(SpPi,R) is bounded by a portion of the corridor : Type-1 Version 09/09

  34. ni-1 ni+1 Constant Ratio Approximation:CD(SpPi,R) is bounded by a portion of the corridor: Type-2 ni ni ni ni Version 09/09

  35. Constant Ratio Approximation:CD(SpPi,R)=CD(ni,R) ≤ li-1+hi+li Version 09/09

  36. Additional Results and Conclusions:Outline • MLCk problem, c-gons, c≤k • Rectangular group-TSP • Edges may be visited more than once • Other results Version 09/09

  37. Additional Results and Conclusions:MLCk problem Rectilinear c-gons for c≤k, and c≥6 Version 09/09

  38. Additional Results and Conclusions:New results • Hans Bodlaender1, Corinne Feremans2, Alexander Grigoriev2, Eelko Penninkx1, René Sitters3, Thomas Wolle4. On the Minimum Corridor Connection Problem and Other Generalized Geometric Problems. In 4th. Workshop on Approximation and Online Algorithms (WAOA) Zurich, Switzerland, September 2006. • NP-completeness of MLC problem • Geographic Clustering Problem: • NP-complete? • PTAS: (1+ε) OPT in time n(log n)O(1/ε) • α-fatness rooms: • NP-complete? • (16/α)-1 OPT; 0≤ α≤1 • If all the rooms are squares then α=1 and the solution is 15 times the optimal one. 1Utrecht University,2Maastricht University,3Max-Planck-Institute for Computer Science 4National ICT Australia Ltd. Version 09/09

  39. Additional Results and Conclusions:Rectangular group-TSP • Instead a tree we have a tour: 2*30=60 • Slavik, P. (1998): • The Errand (Tour) Scheduling problem can be approximated to within a factor of 3ρ/2 in polynomial time, when each errand is assigned to at most ρ vertices. Version 09/09

  40. Additional Results and Conclusions:Rectangular group-TSP • Mark de Berga, Joachim Gudmundssonb, Mathew J. Katzc, Christos Levcopoulosd, Mark H. Overmarse, A. Frank van der Stappenete. TSP with neighborhoods of varying sizes. J. of Algorithms 57 (2005) 22-36. • 1200 α3 times the optimal, α≥1 • 93 times the optimal when the regions are squares aTU Eindhoven, bNICTA Sydney, cBen-Gurion University, dLund University, eUtrecht University Version 09/09

  41. Additional Results and Conclusions: • TRA-MLC,TRA-MLC-R,p-MLC-R,MLC-R problems are NP-complete • p-MLC-RS for S(2OC+),S(4C+) are NP-complete • Solution of the p-MLC-R and MLC-R problem is at most 2ksrs=2*3*5=30 times the optimal solution. • Solution of the MLCk problem is at most 15k-10 times the optimal solution • The approximation ratio is a constant! Version 09/09

  42. Publications… • A. Gonzalez-Gutierrez, T.F. Gonzalez, Complexity of the Minimum-Length Corridor Problem, Computational Geometry: Theory and Applications, 37, 2 (2007),72-103. • A. Gonzalez-Gutierrez, T.F. Gonzalez, Approximation Algorithms for the Minimum-Length Corridor and Related Problems, UCSB CS TR 2007-03 (to appear ACM TALG). Version 09/09

  43. Thank you! • Questions • Comments • Suggestions Version 09/09

  44. Introduction:Applications of the MLC Problem • Circuit Board Layout Design • Wires for power supply • Wires for clock signal • Building Wiring Design • Optical Fiber for Data Communication Networks • Wires for Electrical Networks Version 09/09

  45. Example… FFFFF Version 09/09

  46. Example… TTTTT Version 09/09

  47. Example… FFTFF Version 09/09

  48. Parameterized Approximation Algorithm Alg(S) Version 09/09

  49. Parameterized Approximation Algorithm Alg(S) Version 09/09

  50. Related Problems: Tree Vertex Cover • INPUT: A connected undirected edge-weighted graph G=(V,E,w), where w:E→+ is an edge-weight function. • OUTPUT: A tree T=(V’,E’), where E’E, and V’V is a vertex cover (i.e. every edge in G includes at least one vertex in V’) and the total edge-weight is minimized. Version 09/09

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