Maders Theorem on Edge-Disjoint -Paths Satoru Iwata (University of - PowerPoint PPT Presentation

Mader’s Theorem on Edge-Disjoint -Paths Satoru Iwata (University of Tokyo) Joint work with Yu Yokoi (NII)

-Paths Graph A T -path # edge-disjoint -paths Not a T -path

-Paths Graph 𝜈 𝐻, 𝑈 = 6 𝜈 𝐻, 𝑈 = 7 𝜈 𝐻, 𝑈 = 6 # edge-disjoint -paths

An Upper Bound -subpartition � �∈� � � � � �∈� Theorem [Lovász (1976), Cherkassky (1977)] � 𝒴 �∈�

An Upper Bound � 𝑒 𝑌 � = 14 � 𝑒 𝑌 � = 12 � 𝑒 𝑌 � = 14 �∈� �∈� �∈� 𝜈 𝐻, 𝑈 = 7 𝜈 𝐻, 𝑈 = 6 𝜈 𝐻, 𝑈 = 6

A Tighter Upper Bound -subpartition � �∈� # odd degree components in � �∈�

Mader’s Theorem -subpartition � �∈� # odd degree components in 𝜆 𝒴 = 12 � �∈� Theorem [Mader 1978] 𝜈 𝐻, 𝑈 = 6 � 𝒴 �∈�

Original Papers by Mader • W. Mader: Über die Maximalzahl kantendisjunkter -Wege, Archiv. Math., 30 (1978), pp.325--336. edge-disjoint • W. Mader: Über die Maximalzahl kreuzungsfreier -Wege, Archiv. Math., 31 (1978), pp.387--402. cross-free openly disjoint

Proofs from Books • R. Diestel: Graph Theory. --- Section for Mader’s theorems. --- No Proofs. • B. Korte & J. Vygen: Combinatorial Optimization: Theory and Algorithms --- No Mentions. • A. Frank: Connections in Combinatorial Optimization. --- Theorem of Lovász and Cherkassky. • A. Schrijver: Combinatorial Optimization: Polyhedra and Efficiency --- A short proof on openly disjoint -paths. --- Reduction via line graphs.

Hierarchy of Frameworks Matroid Matching Lovász (1980) Openly disjoint paths Mader (1978) Edge-disjoint -paths Vertex-disjoint -paths Mader (1978) Gallai (1961) Matching Inner Eulerian edge-disjoint -paths Lovász (1976), Cherkassky (1977) Tutte (1947), Berge (1958)

Previous Works • Lovász (1980): Reduction to matroid matching. • Karzanov (1993, 1997): Minimum cost edge-disjoint -paths. • Schrijver (2001): Short proof for openly disjoint -paths. • Schrijver (2003): Reduction to linear matroid parity. • Keijsper, Pendavingh, Stougie (2006): LP formulation of maximum edge-disjoint -paths. • Hirai and Pap (2014): Weighted maximization with tree metric.

Our Contribution • A constructive proof of Mader’s theorem on edge-disjoint -paths. • A combinatorial algorithm for finding maximum edge-disjoint -paths. Running time:

𝑤 Augmenting Walk 𝑡 𝑢 𝑤 𝑠 𝑡 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢 𝑣 𝑡 𝑢 𝑢 𝑤 𝑢 𝑠 𝑠 𝑠 𝑠 𝑢 𝑢 𝑠 𝑢 𝑡 𝑣 𝑢 𝑠 𝑣

Augmenting Walk 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑠𝑠 𝑠 𝑠 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑠𝑡𝑢𝑠 𝑠

Augmenting Walk 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑡 𝑢 𝑡 Augmenting walk: 𝑢 𝑡 𝑢 𝑡 𝑢 - Between terminals 𝑠 - No consecutive symbols 𝑠 𝑟 𝑟 𝑢 𝑡 𝑠 𝑡 𝑟 𝑢 𝑡 𝑢 𝑟 - Uses edge at most once, 𝑠 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟 𝑟 𝑟 𝑠 𝑠 - Uses selfloop at most once, 𝑟 𝑟 𝑟 𝑟 𝑠 𝑠 𝑠 - Uses edge at most twice, 𝑠 𝑠 𝑟 𝑠 𝑟 at most once in each direction 𝑟 𝑠

Augmenting Walk 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑡 𝑢 𝑡 Augmenting walk: 𝑢 𝑡 𝑢 𝑡 𝑢 - Between terminals 𝑠 - No consecutive symbols 𝑠 𝑟 𝑟 𝑢 𝑡 𝑠 𝑡 𝑟 𝑢 𝑡 𝑢 𝑟 - Uses edge at most once, 𝑠 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟 𝑟 𝑟 𝑠 𝑠 - Uses selfloop at most once, 𝑟 𝑟 𝑟 𝑟 𝑠 𝑠 𝑠 - Uses edge at most twice, 𝑠 𝑠 𝑟 𝑠 𝑟 at most once in each direction 𝑠 𝑟

Augmenting Walk 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑠 𝑠 𝑟 𝑟 𝑢 𝑡 𝑠 𝑡 𝑟 𝑢 𝑡 𝑢 𝑟 𝑠 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟 𝑟 𝑟 𝑠 𝑠 𝑟 𝑟 𝑟 𝑟 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑟 𝑟 𝑠 𝑟 𝑠 𝑟 Symmetric Difference

Augmenting Walk 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 𝑡 𝑢 𝑢 𝑟 𝑟 𝑠 𝑠 𝑠 𝑟 Symmetric Difference

Augmentation Edge-disjoint -paths Augmenting Walk in the Auxiliary Labeled Graph Edge-disjoint -paths

Augmentation 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑟 𝑟 𝑠 𝑠 Symmetric Difference

Augmentation 𝑡 𝑢 𝑡 𝑢 𝑢 𝑡 Auxiliary Labeled Graph 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑟 𝑟 𝑠 𝑠 Symmetric Difference Shortcut

Shortcut Operations ≠ 𝑢 𝑡 𝑡 𝑢 𝑢 ≠ 𝑢 𝑡 𝑡 𝑢 𝑢

Shortcut Operations ≠ 𝑢 ≠ 𝑡 𝑡 𝑢 ≠ 𝑢 ≠ 𝑡 𝑡 𝑢

Validity of Augmentation Edge-disjoint -paths Augmenting walk w/o shortcuts has edge-disjoint -paths. : Inner Eulerian Apply the theorem of Lovász & Cherkassky

Validity of Augmentation 𝑡 𝑡 𝑡 � � �

Validity of Augmentation 𝑡 𝑡 𝑡

Validity of Augmentation 𝑡 𝑡 𝑠 𝑠 𝑡 𝑡 𝑠 𝑠 𝑠 𝑠 𝑠

Tightness Edge-disjoint -paths No Augmenting Walks in the Auxiliary Labeled Graph : -subpartition such that . � �∈�

Tightness Edge-disjoint -paths The last symbol in the admissible walk from to . � � 𝑡 𝑢 𝑠 𝑟

Tightness No edge between and for A -path between and is disjoint from with At most one edge leaves a connected component of � �∈� � �∈�

Summary • A constructive proof of Mader’s theorem on edge-disjoint -paths. • A combinatorial algorithm for finding maximum edge-disjoint -paths. S. Iwata and Y. Yokoi: A blossom algorithm for maximum edge-disjoint -paths, METR 2019-16. https://www.keisu.t.u-tokyo.ac.jp/research/techrep/y2019/

Future Directions • A combinatorial algorithm for minimum cost edge-disjoint -paths. • A combinatorial algorithm for the integer free multiflow problem. • A combinatorial algorithm for maximum openly disjoint -paths w/o reduction to matroid parity.