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On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods

On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods. Reza Dorrigiv (Dalhousie U) Bob Fraser (U of Waterloo) Meng He (Dalhousie U) Shahin Kamali (U of Waterloo) Akitoshi Kawamura (U of Tokyo) Alex López -Ortiz (U of Waterloo) Diego Seco (U of A Coruña ).

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On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods

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  1. On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods Reza Dorrigiv (Dalhousie U) Bob Fraser (U of Waterloo) Meng He (Dalhousie U) ShahinKamali (U of Waterloo) Akitoshi Kawamura (U of Tokyo) Alex López-Ortiz (U of Waterloo) Diego Seco (U of A Coruña)

  2. MST – Minimum Spanning Tree . . . . . . . .

  3. (Min weight) MST with neighborhoods . . . . . . . . . . . . . . . MSTN .

  4. Max weight MST with NEIGHBORHOODS . . . . . . max-MSTN .

  5. Max-MSTN is not these other things . . . . . . . . . . . . . . . . . . . max-MSTN . . max-maxST max-planar-maxST

  6. Related Work • MSTN has a PTAS on disjoint unit disks (Yang ‘07) • Hardness of MSTN was open (Yang ‘08) • NP-hard if disks may intersect (Löffler & van Kreveld ‘10) • Travelling Salesman Problem with Neighborhoods (TSPN) (Arkin& Hassin ‘94) • PTAS on disjoint unit disks (Dumitrescu & Mitchell ‘03) • General problem (overlapping disks of variable radii) is APX-hard (de Berg et al. ‘05) • Our Results • Approximation algorithm for max-MSTN • Parameterized algorithms for MSTN and max-MSTN • NP-hardness proofs for MSTN and max-MSTN

  7. Parameterized Algorithms • = separability of the instance • min distance between any two disks

  8. Parameterized max-MSTN Algorithm • – factor approximation by choosing disk centres . . . . . . . . . . . . . . . . . . . . . . . . Tc Tc’ Topt

  9. Parameterized max-MSTN Algorithm • – factor approximation by choosing disk centres . . . Consider this edge weight . . weight = . . . . . . . . . . . . . . . . . . . Tc Tc’ Topt

  10. Hardness of MSTN Need clause gadgets (with spinal path) Reduce from planar 3-SAT Need wires Need variable gadgets

  11. Hardness of MSTN clause (with spinal path) Reduce from planar 3-SAT • Create instance of MSTN so that: • Spinal path is part of solution • Clause gadgets join to only one wire • Weight of optimal solution may be precomputed • Weight of solution corresponding to a non-satisfiable instance is greater than optimal solution by a significant amount variable variable variable clause clause clause variable variable

  12. Hardness of MSTN Wires . . . . . . . . . . . . . . . . . . . . . . Clause gadget . To variable gadgets . . . . . . . . . . . . . . . All wires are part of an optimal solution . . Only one wire from the clause gadget is connected to a variable gadget . . .

  13. Hardness of MSTN . Variable Gadget . . . . . Spinal Path Spinal Path .

  14. HARDNESS OF MSTN Shortest path touching 2 disks path weight . unit distance

  15. Hardness of MSTN . . Variable Gadget . . . . . . . . . . “true” configuration . . . . . . . . . . . . . . . . . . . . . . . . Spinal Path Spinal Path Spinal Path Spinal Path . .

  16. Hardness of MSTN . . . • Weight of an optimal solution: • weight of all wires, including clause gadgets • weight of joining to all but m pairs in variable gadgets • weight of joining to m clause gadgets • What if there is no optimal solution? • At least one clause gadget is joined suboptimally. . . . . . . . . . . . . . . . . . . . . . . To variable gadgets . . . . . . . . . . . . . . . . . . . . . . Spinal Path . Spinal Path . .

  17. Hardness of MSTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spinal Path Spinal Path .

  18. Hardness of MSTN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  19. Hardness of MSTN . Opt Weight to connect to 4 disks & clause wire? . . . . . . . . . . . . . Redundancies!!! . . . . . How about connecting straight to the wire? How about connecting to a neighbour?

  20. On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods . Thanks! . . . . . .

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