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The Hub Location Problem: A Geometric Rounding Approach

The Hub Location Problem: A Geometric Rounding Approach. Dongdong Ge, Yinyu Ye, Jiawei Zhang. Methods and Results. Solve the Linear Program Relaxation; Apply Geometry Rounding Previous Results 1) 2-approximation for The fixed-hub single allocation

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The Hub Location Problem: A Geometric Rounding Approach

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  1. The Hub Location Problem: A Geometric Rounding Approach Dongdong Ge, Yinyu Ye, Jiawei Zhang

  2. Methods and Results • Solve the Linear Program Relaxation; Apply Geometry Rounding • Previous Results 1) 2-approximation for The fixed-hub single allocation problem 2) 1/r for single-minded combinatorial auction max(2/3r, B/(B+r)) for MUCA

  3. New Results • Estimation for opening cost for typical resource allocation problems • Non-metric UFLP (logn, 1)-opt • Hub location with opening cost: logn-opt for equilateral case first result on this problem • Set cover: logn? Yes. f ? Conjeture…. • New 1/r proof for CA

  4. Analysis on UFLP • F: Facilities, |F|=k. • C: customers, |C|=n • : : Customers correlated to Facility i

  5. Analysis on UFLP Non-metric facility location problem: min Subject to:

  6. The Hub Location Problem • Hub-&-Spoke network design (airline) • Two-level network • Hubs: fully connected • Pairwise demands between cities • Possible path: City->hub(->hub)->city

  7. City k City j Hub t Hub s City i

  8. City k City j Hub t Hub s City i

  9. City k City j Hub t Hub s City i

  10. Model: the Fixed-Hub Single Allocation Problem • Hubs are fixed; • No opening cost; • Uncapacitated; • Each city is assigned to exactly one hub; • Hardness • 2-hub: poly time solvable; min-cut • 3-hub: NP-hard (Sohn & Park)

  11. Heuristic Approaches • p-hub median problem (O’Kelly) • Mathematical formulation • Routing: location and flow information • Local search; simulated annealing

  12. Mathematical Formulation • : demands between city i and j • C: cities i, j • H: hubs s, t • : distance between city i and hub s • : distance between hub s and t • : discount factor • xis: binary assignment variable

  13. The Quadratic Formulation • FHSAP-QP (QSAP) minimize subject to O’Kelly et al.

  14. Linear Formulation: MILP1 minimize subject to By Campbell

  15. A Flow Description: MILP2 minimize subject to By Ernst & Krishnamoorthy

  16. Implicit Flow equality Lemma 1Let and be defined as in Formulation FHSAP-MILP2. For any and , The lemma implies a new aggregation constraint:

  17. Modified flow formulation: MILP3 minimize subject to

  18. LP-based algorithm • After solving the LP relaxation FHSAP-LP1 (or 3), how do we assign city to a hub based on a fractional assignment vector: ? • An efficient rounding procedure needs: i) control the linear cost. ii) Maintain correlations among assignment vectors.(Round the same assignment vectors to the same vertex.)

  19. Geometric Rounding Algorithm Geometric Rounding Algorithm (GRA) ________________________________________________________________________ • Solve FHSAP-LP1 (LP2 or LP3). Get an optimal solution x*. • Generate a random vector u, which follows uniform distribution in . • For each xi*=(xi1 , xi2 , … , xik ), if ufalls into Axi,s, let xis=1; other components be 0.

  20. A Graph Illustration hub A (1, 0, 0) City 1 X=(x1, x2, x3) 2 3 B (0,1,0) C (0,0,1)

  21. hub A (1, 0, 0) City 1: x=(x1, x2, x3) u Ax,1 B (0,1,0) C (0,0,1)

  22. hub A (1, 0, 0) X=(x1, x2, x3) City 1: x Ax,1 B (0,1,0) C (0,0,1)

  23. hub A (1, 0, 0) City 1 2 3 B (0,1,0) C (0,0,1)

  24. hub A (1, 0, 0) City 1 2 3 B (0,1,0) C (0,0,1)

  25. A Feasible Assignment Scenario hub A (1, 0, 0) 1 2 City 1, 2 Hub A City 3  Hub B B (0,1,0) C (0,0,1) 3

  26. Geometric Rounding Algorithm Geometric Rounding Algorithm (GRA) ________________________________________________________________________ • Solve FHSAP-LP1 (LP2 or LP3). Get an optimal solution x*. • Generate a random vector u, which follows uniform distribution in . • For each xi*=(xi1 , xi2 , … , xik ), if ufalls into Axi,s, let xis=1; other components be 0.

  27. Worst Case Analysis of GRA The original quadratic formulation: . Let and

  28. Theoretical Analysis: Equilateral Case 1 2 1 2 2 1 center 1 2 2 2 Star Shape Equilateral

  29. Geometric Rounding: Linear cost Lemma 2For any and any : , . This lemma implies that the rounding generates an assignment maintaining the expected value of the vectors. Theorem 1For any

  30. Geometric Rounding: Quadratic cost Intuition: if two non-vertex points and are close in distance, then the rounded points and should not be too far from each other in expectation in order to maintain the quadratic cost. For any and , define . Then we have Theorem 2For any , randomly round and to vertices and in by the procedure in FHSAP-GRA, then

  31. Worst Case Analysis of GRA: II Theorem 3Assume that for all , then, for FHSAP-LP1, And for FHSAP-LP3, Geometric Rounding generates the optimal linear cost and at most twice the quadratic cost in expectation for both formulations

  32. General Case Analysis • Scale all hub edges. Assume the distortion r: 2r-approximation. r =(the longest hub edge)/(the shortest one) • 2-approximation algorithm for 3-hub: either embed into a line or an equilateral.

  33. Running time: LP1: hours. LP2: seconds. LP3 minutes Simulation Results: 10 Hubs; 100 Cities

  34. Simulation Results: 10 Hubs; 1000 Cities

  35. Future Research • Geometric Rounding and its applications • Quadratic semi-assignment problem • Game theory i) Combinatorial auction ii) Sponsored search • The hub location problems with constraints (capacity; opening cost)

  36. Thank you!

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