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Maintaining Perfect Matchings at Low Cost
Jannik Matuschke Ulrike Schmidt-Kraepelin Jos´ e Verschae
KU Leuven TU Berlin Universidad de O’Higgins
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Maintaining Perfect Matchings at Low Cost Jannik Matuschke Ulrike - - PowerPoint PPT Presentation
Maintaining Perfect Matchings at Low Cost Jannik Matuschke Ulrike - - PowerPoint PPT Presentation
Maintaining Perfect Matchings at Low Cost Jannik Matuschke Ulrike Schmidt-Kraepelin Jos e Verschae KU Leuven TU Berlin Universidad de OHiggins 1 1 but with two more nodes: 1 Two-Stage Robust Model Stage 1 Input:
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but with two more nodes:
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Two-Stage Robust Model
Stage 1 Input: complete weighted graph G1 Task: find matching M1 in G1 s.t.
- c(M1) ≤ α · c(O1)
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Two-Stage Robust Model
Stage 1 Input: complete weighted graph G1 Task: find matching M1 in G1 s.t.
- c(M1) ≤ α · c(O1)
Stage 2 Input: 2k additional nodes, inducing G2 Task: find matching M2 in G2 s.t.
- c(M2) ≤ α · c(O2)
- |M1 \ M2| ≤ β · k
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Two-Stage Robust Model
Stage 1 Input: complete weighted graph G1 Task: find matching M1 in G1 s.t.
- c(M1) ≤ α · c(O1)
Stage 2 Input: 2k additional nodes, inducing G2 Task: find matching M2 in G2 s.t.
- c(M2) ≤ α · c(O2)
- |M1 \ M2| ≤ β · k
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Q: Does there exist a robust matching M1 with α, β ∈ O(1)?
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Related Work
Recoverable Robustness [Liebchen et al. 2007] Online MST with Recourse [Megow et al. 2012] Online (Bipartite) Matching
- metric matching without recourse
[Nayyar and Raghvendra 2017]
- with recourse, no cost
[Bernstein et al. 2018]
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Results
Setting 1: Arbitrary Metric, k known G1 contains a robust matching M1 with α = 3 and β = 1. Setting 2: On the Line, k unknown G1 contains a robust matching M1 with α = 10 and β = 2. 2k: number of arrivals α: approximation factor β: recourse factor
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Stage 1
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Heavy Paths and Gain
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Heavy Paths and Gain
For matching X and path P: P is X-heavy, if c(P ∩ X) ≥ 2 · c(P \ X).
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Heavy Paths and Gain
e(P) For matching X and path P: P is X-heavy, if c(P ∩ X) ≥ 2 · c(P \ X).
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Heavy Paths and Gain
For matching X and path P: P is X-heavy, if c(P ∩ X) ≥ 2 · c(P \ X). Define gainX(P) := c(P ∩ X) − c(P \ X).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Computing Stage 1 Matching
Set X := O1 and M1 := ∅ While(X = ∅) (i) Find X-heavy path P maximizing gainX(P). (ii) X ← X∆P and M1 ← M1 ∪ e(P).
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Properties of M1
1) Selected paths form a laminar family.
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Properties of M1
1) Selected paths form a laminar family. 2) Along the laminar hierarchy, paths are alternatingly O1-heavy ( ) O1-light ( )
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Properties of M1
1) Selected paths form a laminar family. 2) Along the laminar hierarchy, paths are alternatingly O1-heavy ( ) O1-light ( ) 3) Costs of nested paths decrease exponentially.
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Properties of M1
1) Selected paths form a laminar family. 2) Along the laminar hierarchy, paths are alternatingly O1-heavy ( ) O1-light ( ) 3) Costs of nested paths decrease exponentially. Lemma: c(M1) ≤ 3c(O1).
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Stage 2
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Stage 2: Replying to 2k Arrivals
O1∆O2 contains k paths (request intervals).
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Delete edges from M1 according to their greedy order: 1) a heavy edge for each request interval 2) a light edge for each gap between request intervals
(precise definition uses laminar structure)
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Stage 2: Replying to 2k Arrivals
O1∆O2 contains k paths (request intervals).
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Delete edges from M1 according to their greedy order: 1) a heavy edge for each request interval 2) a light edge for each gap between request intervals
(precise definition uses laminar structure)
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Stage 2: Replying to 2k Arrivals
O1∆O2 contains k paths (request intervals).
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Delete edges from M1 according to their greedy order: 1) a heavy edge for each request interval 2) a light edge for each gap between request intervals
(precise definition uses laminar structure)
Add a min cost matching on exposed nodes.
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Edge Removal: Example ✗ e∗
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A By greedy construction, A is not O1-heavy.
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Edge Removal: Example ✗ e∗
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A By greedy construction, A is not O1-heavy. O2 ∩ A = A \ O1 (optimum is inverted in request interval)
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Edge Removal: Example ✗ e∗
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A By greedy construction, A is not O1-heavy. O2 ∩ A = A \ O1 (optimum is inverted in request interval) Therefore: c(A ∩ O2) = c(A \ O1) ≥ 1 3c(A)
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Conclusion
We can construct matchings that can quickly adapt to changing input:
- in arbitrary metrics, when number of arrivals is known
- on the line, when number of arrivals is not known
Open Questions
- Can results for line be extended to general metric spaces?
- Can this be extended to the online setting with α, β ∈ O(1)?
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