The Maximum Carpool Matching Problem Gilad Kutiel gkutiel@cs.technion.ac.il CSR 2017 1/21
The Maximum Carpool Matching Problem (informal) ◮ A group of people want to carpool from point A to point B 2/21
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 2/21
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? 2/21
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? ♀ ♀ ♂ 2/21
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? ♀ ��� �� ��� � ♀ ♂ � � 2/21
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? ♀ ��� �� ��� � ♀ ♂ � � 2/21
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? � ♀ ��� �� ��� � ♀ ♂ � � � � 2/21
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? � ♀ ��� �� ��� � ♀ ♂ � � � � 2/21
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? � ♀ ��� �� ��� � ♀ ♂ � � � � 2/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) 1 3 4 2 1 3 4 2 2 1 1 2 3 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 1 3 4 2 2 1 1 2 3 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N 1 3 4 2 2 1 1 2 3 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 1 1 2 3 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 ◮ P ∩ D = ∅ , P ∪ D = V 1 1 2 3 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 ◮ P ∩ D = ∅ , P ∪ D = V 1 1 2 3 ◮ M ⊆ A ∩ ( P × D ) 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 ◮ P ∩ D = ∅ , P ∪ D = V 1 1 2 3 ◮ M ⊆ A ∩ ( P × D ) ◮ ∀ p ∈ P , deg M out ( p ) ≤ 1 2 3 2 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 ◮ P ∩ D = ∅ , P ∪ D = V 1 1 2 3 ◮ M ⊆ A ∩ ( P × D ) ◮ ∀ p ∈ P , deg M out ( p ) ≤ 1 2 3 2 ◮ ∀ d ∈ D , deg M in ( d ) ≤ c ( d ) 3/21
Formal Definition Input: ( D , w , c ) 2 1 1 ◮ D = ( V , A ) ◮ w : A → R + 1 3 4 2 ◮ c : V → N Matching: ( P , D , M ) 1 3 4 2 2 ◮ P ∩ D = ∅ , P ∪ D = V 1 1 2 3 ◮ M ⊆ A ∩ ( P × D ) ◮ ∀ p ∈ P , deg M out ( p ) ≤ 1 2 3 2 ◮ ∀ d ∈ D , deg M in ( d ) ≤ c ( d ) We seek for a matching that maximizes � w ( M ) = w ( e ) e ∈ M 3/21
Maximum Spanning Star Forest (MSSF) ◮ A special case of the Maximum Carpool Matching problem (MCM) 1 1 3 4 2 2 2 1 1 2 3 2 4/21
Maximum Spanning Star Forest (MSSF) ◮ A special case of the Maximum Carpool Matching problem (MCM) ◮ Undirected graph 1 1 3 4 2 2 2 1 1 2 3 2 4/21
Maximum Spanning Star Forest (MSSF) ◮ A special case of the Maximum Carpool Matching problem (MCM) ◮ Undirected graph ◮ No capacities 1 1 3 4 2 2 2 1 1 2 3 2 4/21
Maximum Spanning Star Forest (MSSF) ◮ A special case of the Maximum Carpool Matching problem (MCM) ◮ Undirected graph ◮ No capacities 1 1 3 4 2 2 2 1 1 2 3 2 4/21
Hardness ◮ MSSF (and thus MCM) is APX-hard 5/21
Hardness ◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight) 5/21
Hardness ◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight) ◮ Maximize the number of leaves 5/21
Hardness ◮ MSSF (and thus MCM) is APX-hard ◮ Unweighted MSSF (uniform weight) ◮ Maximize the number of leaves ◮ Minimize the number of centers 5/21
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) 5/21
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) 5/21
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) 5/21
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) 5/21
Previous Work (MSSF) Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Approximation Ratio 1 Hardness Unweighted Vertex Weighted 0.75 Edge Weighted .5 Nguyen, C. Thach, et al. SODA 2007 0.25 Athanassopoulos, Stavros, et al. MFCS 2009 Year 6 7 8 9 10 6/21
Previous Work (MSSF) Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Approximation Ratio 1 Hardness Unweighted Vertex Weighted 0.75 Edge Weighted .5 Nguyen, C. Thach, et al. SODA 2007 0.25 Athanassopoulos, Stavros, et al. MFCS 2009 Year 6 7 8 9 10 6/21
Previous Work (MSSF) Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Approximation Ratio 1 Hardness Unweighted Vertex Weighted 0.75 Edge Weighted .5 Nguyen, C. Thach, et al. SODA 2007 0.25 Athanassopoulos, Stavros, et al. MFCS 2009 Year 6 7 8 9 10 6/21
Previous Work (MSSF) Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Approximation Ratio 1 Hardness Unweighted Vertex Weighted 0.75 Edge Weighted .5 Nguyen, C. Thach, et al. SODA 2007 0.25 Athanassopoulos, Stavros, et al. MFCS 2009 Year 6 7 8 9 10 6/21
Previous Work (MSSF) Chen, Ning, et al. APPROX 2007 Deeparnab Chakrabarty et al. FOCS 2008 Approximation Ratio 1 Hardness Unweighted Vertex Weighted 0.75 Edge Weighted .5 Nguyen, C. Thach, et al. SODA 2007 0.25 Athanassopoulos, Stavros, et al. MFCS 2009 Year 6 7 8 9 10 6/21
Previous Work (MCM) ◮ Heuristics + experiments 7/21
Previous Work (MCM) ◮ Heuristics + experiments ◮ Solution’s quality matters 7/21
Previous Work (MCM) ◮ Heuristics + experiments ◮ Solution’s quality matters ◮ No approximation algorithm 7/21
Previous Work (MCM) ◮ Heuristics + experiments ◮ Solution’s quality matters ◮ No approximation algorithm ◮ The algorithms for MSSF do not generalize to MCM 7/21
Our Result (MCM) ◮ First approximation algorithms 8/21
Our Result (MCM) ◮ First approximation algorithms ◮ 1/2-approximation (unweighted) 8/21
Our Result (MCM) ◮ First approximation algorithms ◮ 1/2-approximation (unweighted) ◮ 1/3-approximation (weighted) 8/21
Maximum Carpool Matching 9/21
MSSF on Trees ◮ Optimally solvable (using DP) ? ? ? ? ? ? 10/21
MSSF on Trees ◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ? ? ? ? ? ? 10/21
MSSF on Trees ◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ? ? ? ? ? ? 10/21
MSSF on Trees ◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ? ? ? ? ? ? 10/21
MSSF on Trees ◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ? ? ? ? ? ? 10/21
MSSF on Trees ◮ Optimally solvable (using DP) ◮ At least 1/2 of the weight ◮ 1/2-approximation algorithm to MSSF (Max Spanning Tree ≥ OPT) ? ? ? ? ? ? 10/21
Relaxed Matching ◮ Every vertex can be a driver and a passenger at the same time 2 5 2 1 3 1 2 2 1 2 3 3 11/21
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