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 # edge-disjoint -paths

A T-path Not a T-path

• Paths

Graph # edge-disjoint -paths

𝜈 𝐻, 𝑈 = 7 𝜈 𝐻, 𝑈 = 6 𝜈 𝐻, 𝑈 = 6

An Upper Bound

• ∈

∈

• subpartition
• 𝒴
• ∈

Theorem [Lovász (1976), Cherkassky (1977)]

An Upper Bound

𝜈 𝐻, 𝑈 = 6 𝑒 𝑌 = 12

∈

𝜈 𝐻, 𝑈 = 6 𝑒 𝑌 = 14

∈

𝜈 𝐻, 𝑈 = 7 𝑒 𝑌 = 14

∈

A Tighter Upper Bound

∈

• subpartition

# odd degree components in

• ∈

Mader’s Theorem

∈

• subpartition

𝒴

• ∈

Theorem [Mader 1978] # odd degree components in

• ∈

𝜈 𝐻, 𝑈 = 6 𝜆 𝒴 = 12

Original Papers by Mader

• W. Mader: Über die Maximalzahl kantendisjunkter
• Wege,
• Archiv. Math., 30 (1978), pp.325--336.
• W. Mader: Über die Maximalzahl kreuzungsfreier
• Wege,
• Archiv. Math., 31 (1978), pp.387--402.

edge-disjoint cross-free

• penly 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 Mader (1978) Inner Eulerian edge-disjoint -paths Lovász (1976), Cherkassky (1977) Vertex-disjoint -paths Gallai (1961) Matching 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:

Augmenting Walk

𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢 𝑠 𝑠 𝑠 𝑠 𝑠 𝑢 𝑢 𝑢 𝑤 𝑣 𝑢 𝑡 𝑢 𝑠 𝑤 𝑣 𝑡 𝑢 𝑠 𝑤 𝑣

Augmenting Walk

𝑡 𝑢 𝑠 𝑡 𝑡 𝑢 𝑢 𝑠𝑠 𝑡 𝑢 𝑠 𝑠𝑡𝑢𝑠 𝑡 𝑢 𝑠 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢

Augmenting Walk

Auxiliary Labeled Graph

𝑟 𝑠 𝑟 𝑢 𝑟 𝑠 𝑡 𝑡 𝑢 𝑠 𝑟 𝑡 𝑡 𝑠 𝑠 𝑡 𝑠 𝑡 𝑡 𝑢 𝑢 𝑢 𝑢 𝑢 𝑟 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑡 𝑢 𝑟 𝑢 𝑟 𝑠 𝑡 𝑟 𝑠 𝑠

Augmenting walk:

• Between terminals
• No consecutive symbols
• Uses edge at most once,
• Uses selfloop at most once,
• Uses edge at most twice,

at most once in each direction

𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢

Augmenting Walk

Auxiliary Labeled Graph Augmenting walk:

• Between terminals
• No consecutive symbols
• Uses edge at most once,
• Uses selfloop at most once,
• Uses edge at most twice,

at most once in each direction

𝑟 𝑠 𝑟 𝑢 𝑟 𝑠 𝑡 𝑡 𝑢 𝑠 𝑟 𝑡 𝑡 𝑠 𝑠 𝑡 𝑠 𝑡 𝑡 𝑢 𝑢 𝑢 𝑢 𝑢 𝑟 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑡 𝑢 𝑟 𝑢 𝑟 𝑠 𝑡 𝑟 𝑠 𝑠 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢

Augmenting Walk

Auxiliary Labeled Graph

𝑡 𝑠 𝑟

Symmetric Difference

𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢 𝑟 𝑠 𝑟 𝑢 𝑟 𝑠 𝑡 𝑡 𝑢 𝑠 𝑟 𝑡 𝑡 𝑠 𝑠 𝑡 𝑠 𝑡 𝑡 𝑢 𝑢 𝑢 𝑢 𝑢 𝑟 𝑠 𝑠 𝑠 𝑠 𝑠 𝑠 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑟 𝑡 𝑢 𝑟 𝑢 𝑟 𝑠 𝑡 𝑟 𝑠 𝑠

Augmenting Walk

Auxiliary Labeled Graph

𝑡 𝑢 𝑠 𝑟 𝑢 𝑢 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟

Symmetric Difference

𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢

Augmentation

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

Augmentation

Auxiliary Labeled Graph Symmetric Difference

𝑡 𝑢 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢

Augmentation

Auxiliary Labeled Graph Symmetric Difference

𝑡 𝑠 𝑟 𝑡 𝑢 𝑠 𝑟

Shortcut

𝑢 𝑡 𝑢 𝑡 𝑡 𝑢 𝑢 𝑡 𝑢 𝑡 𝑢 𝑡 𝑡 𝑡 𝑢 𝑢 𝑢

Shortcut Operations

𝑡 𝑢 𝑡 𝑢 𝑡 𝑢 ≠ 𝑢 𝑡 𝑢 ≠ 𝑢

Shortcut Operations

𝑡 𝑢 ≠ 𝑢 ≠ 𝑡 𝑡 𝑢 ≠ 𝑢 ≠ 𝑡

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 𝑠 𝑠 𝑡 𝑠 𝑠 𝑠 𝑡 𝑡 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

• ∈

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

• ∈

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.