The Maximum Carpool Matching Problem Gilad Kutiel - - PowerPoint PPT Presentation

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The Maximum Carpool Matching Problem

Gilad Kutiel gkutiel@cs.technion.ac.il CSR 2017

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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The Maximum Carpool Matching Problem (informal)

◮ A group of people want to carpool from point A to point B ◮ We want to maximize the total welfare ◮ How they should be matched?

♀ ♀ ♂

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Formal Definition

Input: (D, w, c)

◮ D = (V , A)

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

◮ P ∩ D = ∅, P ∪ D = V

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

◮ P ∩ D = ∅, P ∪ D = V ◮ M ⊆ A ∩ (P × D)

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

◮ P ∩ D = ∅, P ∪ D = V ◮ M ⊆ A ∩ (P × D) ◮ ∀p ∈ P, degM

• ut(p) ≤ 1

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

◮ P ∩ D = ∅, P ∪ D = V ◮ M ⊆ A ∩ (P × D) ◮ ∀p ∈ P, degM

• ut(p) ≤ 1

◮ ∀d ∈ D, degM in (d) ≤ c(d)

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

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Formal Definition

Input: (D, w, c)

◮ D = (V , A) ◮ w : A → R+ ◮ c : V → N

Matching: (P, D, M)

◮ P ∩ D = ∅, P ∪ D = V ◮ M ⊆ A ∩ (P × D) ◮ ∀p ∈ P, degM

• ut(p) ≤ 1

◮ ∀d ∈ D, degM in (d) ≤ c(d)

1 2 1 4 2 3 3

2 3 4 3 2 1 2 1 2 2 1 1

We seek for a matching that maximizes w(M) =

• e∈M

w(e)

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Maximum Spanning Star Forest (MSSF)

◮ A special case of the Maximum Carpool Matching

problem (MCM)

2 3 4 1 2 2 3 2 1 2 1 1

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Maximum Spanning Star Forest (MSSF)

◮ A special case of the Maximum Carpool Matching

problem (MCM)

◮ Undirected graph

2 3 4 1 2 2 3 2 1 2 1 1

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Maximum Spanning Star Forest (MSSF)

◮ A special case of the Maximum Carpool Matching

problem (MCM)

◮ Undirected graph ◮ No capacities

2 3 4 1 2 2 3 2 1 2 1 1

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Maximum Spanning Star Forest (MSSF)

◮ A special case of the Maximum Carpool Matching

problem (MCM)

◮ Undirected graph ◮ No capacities

2 3 4 1 2 2 3 2 1 2 1 1

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Hardness

◮ MSSF (and thus MCM) is APX-hard

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves ◮ Minimize the number of centers

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves ◮ Minimize the number of centers ◮ (Minimum Dominating Set problem)

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves ◮ Minimize the number of centers ◮ (Minimum Dominating Set problem)

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves ◮ Minimize the number of centers ◮ (Minimum Dominating Set problem)

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Hardness

◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight)

◮ Maximize the number of leaves ◮ Minimize the number of centers ◮ (Minimum Dominating Set problem)

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Previous Work (MSSF)

Approximation Ratio Year

6 7 8 9 10 0.25 .5 0.75 1

Nguyen, C. Thach, et al. SODA 2007 Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Athanassopoulos, Stavros, et al. MFCS 2009

Hardness Unweighted Vertex Weighted Edge Weighted

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Previous Work (MSSF)

Approximation Ratio Year

6 7 8 9 10 0.25 .5 0.75 1

Nguyen, C. Thach, et al. SODA 2007 Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Athanassopoulos, Stavros, et al. MFCS 2009

Hardness Unweighted Vertex Weighted Edge Weighted

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Previous Work (MSSF)

Approximation Ratio Year

6 7 8 9 10 0.25 .5 0.75 1

Nguyen, C. Thach, et al. SODA 2007 Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Athanassopoulos, Stavros, et al. MFCS 2009

Hardness Unweighted Vertex Weighted Edge Weighted

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Previous Work (MSSF)

Approximation Ratio Year

6 7 8 9 10 0.25 .5 0.75 1

Nguyen, C. Thach, et al. SODA 2007 Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Athanassopoulos, Stavros, et al. MFCS 2009

Hardness Unweighted Vertex Weighted Edge Weighted

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Previous Work (MSSF)

Approximation Ratio Year

6 7 8 9 10 0.25 .5 0.75 1

Nguyen, C. Thach, et al. SODA 2007 Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Athanassopoulos, Stavros, et al. MFCS 2009

Hardness Unweighted Vertex Weighted Edge Weighted

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Previous Work (MCM)

◮ Heuristics + experiments

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Previous Work (MCM)

◮ Heuristics + experiments

◮ Solution’s quality matters

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Previous Work (MCM)

◮ Heuristics + experiments

◮ Solution’s quality matters

◮ No approximation algorithm

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Previous Work (MCM)

◮ Heuristics + experiments

◮ Solution’s quality matters

◮ No approximation algorithm ◮ The algorithms for MSSF do not generalize to MCM

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Our Result (MCM)

◮ First approximation algorithms

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Our Result (MCM)

◮ First approximation algorithms

◮ 1/2-approximation (unweighted)

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Our Result (MCM)

◮ First approximation algorithms

◮ 1/2-approximation (unweighted) ◮ 1/3-approximation (weighted)

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Maximum Carpool Matching

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MSSF on Trees

◮ Optimally solvable (using DP)

? ? ? ? ? ?

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MSSF on Trees

◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight

? ? ? ? ? ?

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MSSF on Trees

◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight

? ? ? ? ? ?

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MSSF on Trees

◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight

? ? ? ? ? ?

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MSSF on Trees

◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight

? ? ? ? ? ?

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MSSF on Trees

◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ◮ 1/2-approximation algorithm to MSSF (Max Spanning Tree ≥ OPT)

? ? ? ? ? ?

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Relaxed Matching

◮ Every vertex can be a driver and a passenger at the same time

1 2 5 3 2

1 2 2 2 3 1 3

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Relaxed Matching

◮ Every vertex can be a driver and a passenger at the same time

1 2 5 3 2

1 2 2 2 3 1 3

s t

c = 1 w = 0 c = 1 w = 2 c = 1 w = 1 c = 1 w = 3 c = 3 w = 0

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1)

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Relaxed Matching (Observations)

◮ A relaxed matching is a pseudo-forest (out degree ≤ 1) ◮ Can be partitioned into 3 feasible solutions ◮ One has at least 1/3 of the weight ◮ 1/3-approximation algorithm to MCM (Relaxed Matching ≥ OPT)

1 2 3 4 5 6 7

? ? ? ? ? ? ?

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Maximum Carpool Matching Unweighted

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general)

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general)

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general)

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general) ◮ Improve ⇐

⇒ a passenger has two incoming arcs from unmatched vertices

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general) ◮ Improve ⇐

⇒ a passenger has two incoming arcs from unmatched vertices

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general) ◮ Improve ⇐

⇒ a passenger has two incoming arcs from unmatched vertices

◮ Make it maximal

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (1/2-approximation algorithm)

◮ Start with any maximal solution (bad in general) ◮ Improve ⇐

⇒ a passenger has two incoming arcs from unmatched vertices

◮ Make it maximal

1 1 3 1 2 3 4 1 2 3

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Unweighted MCM (Analysis)

◮ Load every arc with 2$

1$ 1$ 1$ 1$ 1$ 2$ 3$

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Unweighted MCM (Analysis)

◮ Load every arc with 2$

1$ 1$ 1$ 1$ 1$ 2$ 3$

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution

1$ 1$ 1$ 1$ 1$ 2$ 3$

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists)

? ? ? ? ? ? ?

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists)

? ? ? ? ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices

? ? ? ? ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices ◮ Can be paid by their destination

? ? ? ? ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices ◮ Can be paid by their destination

? ? ? ? ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices ◮ Can be paid by their destination

? ? ? 1$ ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices ◮ Can be paid by their destination

? ? ? 1$ ? ? ?

X X

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Unweighted MCM (Analysis)

◮ Load every arc with 2$ ◮ Consider another solution ◮ Each vertex pays (if it can) for its outgoing arc (if exists) ◮ Left with outgoing arcs from unmatched vertices ◮ Can be paid by their destination

? ? ? ? ? ? ?

X X

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Summary

◮ MSSF

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Summary

◮ MSSF

◮ APX-hard problem (10/11)

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

◮ MCM

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

◮ MCM

◮ Generalization of MSSF

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

◮ MCM

◮ Generalization of MSSF ◮ First approximation algorithms:

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

◮ MCM

◮ Generalization of MSSF ◮ First approximation algorithms: ◮ 1/2-approximation to the unweighted variant

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Summary

◮ MSSF

◮ APX-hard problem (10/11) ◮ No better than 1/2-approximation algorithm is known

◮ MCM

◮ Generalization of MSSF ◮ First approximation algorithms: ◮ 1/2-approximation to the unweighted variant ◮ 1/3-approximation to the general problem

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Followups

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

s t

c = 1 w = 0 c = 1 w = 4 c = 1 w = 1 c = 3 w = 0

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

s t

c = 1 w = 0 c = 1 w = 4 c = 1 w = 1 c = 3 w = 0

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

◮ f (D) = OPT(D) is submodular (David Adjiashvili, personal communication)

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Fixed Matching

◮ P and D are given (tractable) 2 1 2 1 1 1 2 3

3 1 1 3 4 1 3 1 2 3 2 2 2

s t

c = 1 w = 0 c = 1 w = 4 c = 1 w = 1 c = 3 w = 0

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

◮ f (D) = OPT(D) is submodular (David Adjiashvili, personal communication) ◮ 1 2-approximation

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Group Carpool

◮ Some people insists to carpool together

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Group Carpool

◮ Some people insists to carpool together ◮ Not submodular anymore

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Group Carpool

◮ Some people insists to carpool together ◮ Not submodular anymore ◮ Still admits a 1 2-approximation

20/21

Open Question Better than 1/2-approximation?

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Thank You !