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The Hypergraph Network Simplex Algorithm & Railway Optimization

Ralf Borndörfer

joint work with Isabel Beckenbach and Markus Reuther

4th ISM-ZIB-IMI MODAL Workshop on Mathematical Optimization and Data Analysis Tokyo, ISM, 27.03.2019

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 1

Directed Hypergraphs

Definition A directed hypergraph is a pair 𝐼 = (𝑊, 𝒝) where 𝑊 is a finite set of vertices and 𝒝 is a family of hyperarcs. A hyperarc a ∈ 𝒝 is a pair 𝑏 = (𝑢, ℎ) of disjoint sets 𝑢, ℎ ⊆ 𝑊 sets of vertices, at least one of them non-empty; 𝑢 ⊆ 𝑊 is called the tail of 𝑏, ℎ ⊆ 𝑊 is the head.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 2

Graph-based Hypergraphs

Definition Let 𝐸 = (𝑊, 𝐵) be a simple directed draph. A directed hypergraph based on 𝐸 is a pair 𝐼 = (𝑊, 𝒝) where 𝒝 ⊆ 2𝐵 is a set of non-empty subsets 𝑏 ⊆ 𝐵 of vertex-disjoint arcs. A directed hypergraph based on some graph 𝐸 is called graph-based. Remark A graph-based directed hypergraph is a directed hypergraph: For 𝑏 ∈ 𝒝 let 𝑢 𝑏 ≔ {𝑤 ∈ 𝑊: ∃ 𝑤, 𝑥 ∈ 𝑏} and ℎ 𝑏 ≔ 𝑥 ∈ 𝑊: ∃ 𝑤, 𝑥 ∈ 𝑏 .

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 3

Definition Let 𝐼 = (𝑊, 𝒝) be a directed hypergraph based on a directed graph 𝐸, 𝑑 ∈ ℝ𝒝 a vector of costs, and 𝑐 ∈ ℝ𝑊 of demands s.t. 𝑐𝑈1 = 0. The minimum cost hyperflow problem (MCH) is the linear program min ෍

𝑏∈𝒝

𝑑𝑏𝑦𝑏 ෍

𝑏∈𝒝:𝑤∈ℎ 𝑏

𝑦𝑏 − ෍

𝑏∈𝒝:𝑤∈𝑢 𝑏

𝑦𝑏 = 𝑐𝑤 ∀𝑤 ∈ 𝑊 𝑦 ≥ 0. A vector 𝑦 ∈ ℝ𝒝 that is feasible for this LP is a hyperflow (in 𝐼) (actually a circulation).

The Minimum Cost Hyperflow Problem

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 4

The Minimum Cost Hyperflow Problem

►

In contrast to the graph case, there might not exist an integral min cost hyperfow, even if all data is integral (see example later).

►

Finding a minimum cost integral hyperflow is NP-hard, even if the hyperarcs consist of at most two arcs.

►

If the underlying digraph 𝐸 is connected and 𝐵 ⊆ 𝒝, i.e., all arcs

• f the underlying digraph are also hyperarcs, then

෍

𝑏∈𝒝:𝑤∈ℎ 𝑏

𝑦𝑏 − ෍

𝑏∈𝒝:𝑤∈𝑢 𝑏

𝑦𝑏 = 𝑐𝑤 ∀𝑤 ∈ 𝑊 𝑦 ≥ 0. has a solution if and only if 𝑐𝑈1 = 0.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 5

A Hypernetwork Simplex Algorithm

Earlier work

• R. Cambini, G. Gallo, and M. G. Scutellà: Flows on hypergraphs.

Mathematical Programming 78.2, p. 195-217 (1997). However

►

We heavily use that we work on graph-based hypergraphs.

►

The algorithm for our setting is simpler and closer to the original network simplex. Reference

• I. Beckenbach: Matchings and Flows in Hypergraphs. PhD thesis,

Freie Universität Berlin (2019).

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Characterizing the Bases

Assumption The underlying digraph 𝐸 is connected and 𝐵 ⊆ 𝒝.

►

Let 𝑁 ∈ 0, ±1 𝑊×𝒝 be the incidence matrix of 𝐼. The minimum cost hyperflow problem can then be written as min 𝑑𝑈𝑦, 𝑁𝑦 = 𝑐, 𝑦 ≥ 0.

►

If 𝐶 = {𝑏1, … , 𝑏𝑙}, then let 𝑁⋅𝐶 ≔ (𝑁⋅𝑏1, … , 𝑁⋅𝑏𝑙).

►

𝑠𝑙 𝑁 = 𝑊 − 1

►

𝐶 is a basis if and only if 𝑠𝑙 𝑁⋅𝐶 = 𝐶 = 𝑊 − 1.

►

If 𝐶 is a basis, then 𝐼 𝐶 ∩ 𝐵 = (𝑊, 𝐶 ∩ 𝐵) is a forest that contains 𝑊| − 𝐶 ∩ 𝐵 = 𝑊 − 𝐶 − 𝐶 ∖ 𝐵 = |𝐶 ∖ 𝐵 + 1 components.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 7

Characterizing the Bases

►

Let 𝐶 ⊆ 𝒝 be s.t. 𝐶 = 𝑊 − 1, 𝐶1 ≔ 𝐶 ∩ 𝐵, 𝐶2 ≔ 𝐶 ∖ 𝐵.

►

𝐼[𝐶1] is a forest with 𝐶2 + 1 components.

►

For every tree of 𝐼[𝐶1], choose a root 𝑠 and denote its tree by 𝑈

𝑠.

►

Let 𝑆 be the set of all such roots.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 8

Characterizing the Bases

►

If 𝐶 is a basis and 𝑠

1, 𝑠 2 ∈ 𝑆 two roots, the system

𝑁⋅𝐶𝑔 = −𝑓𝑠1 + 𝑓𝑠2, 𝑦 ≥ 0 has a unique solution; we can send 1 unit of flow from 𝑠

1 to 𝑠 2.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 9

Characterizing the Bases

►

If 𝐶 is a basis and 𝑠

1, 𝑠 2 ∈ 𝑆 two roots, the system

𝑁⋅𝐶𝑔 = −𝑓𝑠1 + 𝑓𝑠2, 𝑔 ≥ 0 has a unique solution; we can (in a unique way) send 1 unit of flow from 𝑠

1 to 𝑠 2 in 𝐼 𝐶 ∩ 𝐵 .

►

The unique flow of 1 unit from an arbitrary fixed root 𝑠∗ to some

• ther other root 𝑠 ≠ 𝑠∗ is called elementary.

►

We can send 1 unit of flow from 𝑠

1 to 𝑠 2 via an arbitrary

intermediate root 𝑠∗, i.e., from 𝑠

1 to 𝑠∗ to 𝑠 2.

►

We can also (easily) send 1 unit of flow inside of a tree.

►

Any flow in 𝐼 𝐶 ∩ 𝐵 is a superposition of elementary flows and flows on trees.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 10

The Elementary Hyperflow Matrix

►

Let 𝐶 be a basis, 𝑈

𝑠, 𝑠 ∈ 𝑆, the rooted trees, 𝑠∗ ∈ 𝑆 a fixed root.

►

For every 𝑠 ∈ 𝑆 ∖ 𝑠∗ there is a unique hyperflow 𝑔

𝑠 in 𝑊, 𝐶 that

transports 1 unit from 𝑠∗ to 𝑠.

►

Let 𝐺 ≔ 𝑔

𝑠 𝑠∈𝑆∖{𝑠∗} ∈ ℝ𝐶×𝑆∖ 𝑠∗ be the elementary hyperflow matrix

whose 𝑠-th column contains this flow.

►

𝐺 is easily reconstructed from 𝐺|𝐶2 = 𝐺𝐶2⋅ by recomputing the flows on the trees (but this takes time).

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 11

The Elementary Hyperflow Matrix

Example 𝑠∗ = 𝑠

1,

𝑔

𝑠2|𝐶2 = 1/2

1/4

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The Elementary Hyperflow Matrix

Example 𝑠∗ = 𝑠

1,

𝐺𝐶2⋅ = 𝑔

𝑠2, 𝑔 𝑠3 𝐶2⋅ = 1/2

1/2 1/4 −1/4

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 13

The Elementary Hyperflow Matrix

Example 𝑠∗ = 𝑠

1,

𝐹𝑆∖ 𝑠1 ⋅ = 1 2 1 −2 , 𝐺𝐶2⋅ = 1/2 1/2 1/4 −1/4

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 14

෍

𝑏∈𝐶

− 𝑊 𝑈

𝑠2 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠2 ∩ 𝑢 𝑏

𝑔

𝑠2𝑏 = 1

෍

𝑏∈𝐶

− 𝑊 𝑈

𝑠3 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠3 ∩ 𝑢 𝑏

𝑔

𝑠2𝑏 = 0

Characterizing the Bases

►

Define 𝐹 ∈ ℤ𝑆×𝐶2 as 𝐹𝑠𝑏 ≔ 𝑊 𝑈

𝑠 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠 ∩ 𝑢 𝑏 .

Lemma

►

𝐹𝑆∖ 𝑠∗ ⋅ = 𝐺𝐶2⋅

−1

►

𝐶 basis ⟺ 𝐼 𝐶1 forest of 𝐶2 + 1 components ⋀ 𝑠𝑙 𝐺𝐶2⋅ = 𝐶2 ⟺ 𝐼 𝐶1 forest of 𝐶2 + 1 components ⋀ 𝑠𝑙 𝐹 = 𝐶2 .

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 15

Characterizing the Bases

Example 𝐹 = −2 1 2 1 −2 has rank 2 ⇒ 𝐶 is a basis

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 16

Characterizing the Bases

Example 𝐹 = −1 −1 2 2 −1 −1 has rank 1 ⇒ 𝐶 is not a basis

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 17

Elementary Hyperflow and Intersection Count Matrix

►

Let 𝐶 be a basis, 𝑈

𝑠, 𝑠 ∈ 𝑆, the rooted trees, 𝑠∗ ∈ 𝑆 a fixed root.

►

For every 𝑠 ∈ 𝑆 ∖ 𝑠∗ there is a unique hyperflow 𝑔

𝑠 in 𝑊, 𝐶 that

transports 1 unit from 𝑠∗ to 𝑠.

►

Let 𝐺 ≔ 𝑔

𝑠 𝑠∈𝑆∖{𝑠∗} ∈ ℝ𝐶×𝑆∖ 𝑠∗ be the elementary hyperflow matrix

whose 𝑠-th column contains this flow.

►

𝐺 is easily reconstructed from 𝐺|𝐶2 = 𝐺𝐶2⋅ by recomputing the flows on the trees.

►

Define an intersection count matrix 𝐹 ∈ ℤ𝑆×𝐶2 as 𝐹𝑠𝑏 ≔ 𝑊 𝑈

𝑠 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠 ∩ 𝑢 𝑏 .

►

𝐹𝑆∖ 𝑠∗ ⋅ = 𝐺𝐶2⋅

−1

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 18

The Hyperflow Network Simplex Algorithm

Input: Hypergraph 𝐼 = (𝑊, 𝒝) based on digraph 𝐸 = (𝑊, 𝐵), cost 𝑑 ≥ 0, demand 𝑐 s.t. 𝑐𝑈1 = 0, feasible basic (hyper)flow 𝑦 s.t. supp 𝑦 ⊆ 𝐵, associated basis 𝐶 and tree 𝑈

𝑠.

Output: (Fractional) Minimum cost hyperflow 𝑦. 1. (BTRAN) Solve 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈 and compute reduced costs

ҧ 𝑑𝑏 = 𝑑𝑏 − 𝜌(𝑊 ℎ 𝑏 + 𝜌(𝑊(𝑢 𝑏 ). 2. (PRICE) If ҧ 𝑑 ≥ 0 then output 𝑦, stop; else choose 𝑏in s.t. ҧ 𝑑𝑏𝑗𝑜 < 0. 3. (FTRAN) Solve 𝑁⋅𝐶𝑔 = −𝑁𝑏in. 4. (CHUZR) Choose 𝑏out ∈ argmin −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶 .

5. (UPDATE) 𝑦𝑏 ← ൞ −𝑦𝑏out/𝑔𝑏out, 𝑏 = 𝑏in 𝑦𝑏 − 𝑦𝑏out/𝑔𝑏out, 𝑏 ∈ 𝐶 𝑦𝑏, 𝑏 ∉ 𝐶 ∪ {𝑏in} , update 𝐶, 𝑆, {𝑈

𝑠}, 𝑁𝐶, 𝐺.

6. Go to 1.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 19

(FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

1 unit of flow on entering arc 𝑏in creates at tree 𝑈

𝑠 demand

𝐹𝑠𝑏in ≔ −( 𝑊 𝑈

𝑠 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠 ∩ 𝑢 𝑏 )

►

Compute hyperflow on basic hyperarcs as superposition of elementary flows 𝑔|𝐶2 = 𝐺𝐶2⋅𝐹𝑆∖{𝑠∗}𝑏in.

►

Compute associated flow on trees 𝑔|𝐶1 and set 𝑔 = 𝑔|𝐶1 𝑔|𝐶2 .

►

𝑁⋅𝐶𝑔 = −𝑁⋅𝑏𝑗𝑜

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 20

𝑏1 𝑏2

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

𝑏in is the red arc

►

𝐹𝑆𝑏in = 1,0, −1 𝑈

►

𝑔

𝑏1

𝑔

𝑏2

= 𝐺 ⋅ 𝐹𝑆∖ 𝑠∗ 𝑏in = 1/2 1/2 1/4 −1/4 −1 = −1/2 1/4

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 21

𝑏1 𝑏2

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

Compute demand on tree vertices induced by flow on basic hyerparcs.

►

Total demands (including demands on entering hyperarc) sum up to 0 on every tree.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 22

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

Compute flow on each tree by reverse BFS.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 23

The Hyperflow Network Simplex Algorithm

Input: Hypergraph 𝐼 = (𝑊, 𝒝) based on digraph 𝐸 = (𝑊, 𝐵), cost 𝑑 ≥ 0, demand 𝑐 s.t. 𝑐𝑈1 = 0, feasible basic (hyper)flow 𝑦 s.t. supp 𝑦 ⊆ 𝐵, associated basis 𝐶 and tree 𝑈

𝑠.

Output: (Fractional) Minimum cost hyperflow 𝑦. 1. (BTRAN) Solve 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈 and compute reduced costs

ҧ 𝑑𝑏 = 𝑑𝑏 − 𝜌(𝑊 ℎ 𝑏 + 𝜌(𝑊(𝑢 𝑏 ). 2. (PRICE) If ҧ 𝑑 ≥ 0 then output 𝑦, stop; else choose 𝑏in s.t. ҧ 𝑑𝑏𝑗𝑜 < 0. 3. (FTRAN) Solve 𝑁⋅𝐶𝑔 = −𝑁𝑏in. 4. (CHUZR) Choose 𝑏out ∈ argmin −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶 .

5. (UPDATE) 𝑦𝑏 ← ൞ −𝑦𝑏out/𝑔𝑏out, 𝑏 = 𝑏in 𝑦𝑏 − 𝑦𝑏out/𝑔𝑏out, 𝑏 ∈ 𝐶 𝑦𝑏, 𝑏 ∉ 𝐶 ∪ {𝑏in} , update 𝐶, 𝑆, {𝑈

𝑠}, 𝑁𝐶, 𝐺.

6. Go to 1.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 24

(BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

Set 𝜌′𝑆: = 0 and extend to potential 𝜌′ s.t. all tree arcs have reduced cost zero.

►

The basic hyperarcs have (preliminary) reduced costs ҧ 𝑑𝑏

′ = 𝑑𝑏 − 𝜌′(𝑊 ℎ 𝑏

+ 𝜌′(𝑊(𝑢 𝑏 ).𝑏.

►

Adjust potentials at trees (except 𝑠∗, i.e., 𝜌𝑠∗ = 0) such that basic hyperarcs get zero reduced costs by setting 𝜌𝑠 ≔ ҧ 𝑑𝐶2

′𝑈𝐺𝐶2𝑠

and 𝜌𝑊 𝑈

𝑠 ← 𝜌𝑊 𝑈 𝑠

′

+ 𝜌𝑠 ⋅ 1.

►

ҧ 𝑑𝑏= ҧ 𝑑𝑏

′ = 0,

𝑏 ∈ 𝐶1

►

ҧ 𝑑𝑏= ҧ 𝑑𝑏

′ − σ𝑠∈𝑆∖ 𝑠∗ 𝜌𝑠 𝑊 𝑈 𝑠 ∩ ℎ 𝑏

− 𝑊 𝑈

𝑠 ∩ 𝑢 𝑏

= ҧ 𝑑𝑏

′ − σ𝑠∈𝑆∖ 𝑠∗

ҧ 𝑑𝐶2

′𝑈𝐺𝐶2𝑠𝐹𝑏𝑠

= ҧ 𝑑𝑏

′ − ҧ

𝑑𝑏

′ = 0,

𝑏 ∈ 𝐶2

►

𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈 ⋅

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 25

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

Set 𝜌′𝑆: = 0 and extend to potentials 𝜌′ such that all tree arcs have reduced cost zero.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 26

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

The basic hyperarcs have preliminary reduced costs ҧ 𝑑𝑏1

′

= 1 − (1 + 0) + (2 + 2) = 4- ҧ 𝑑𝑏2

′

= 1 − (1 + 1) + (1 - 1) = -1

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 27

𝑏1 𝑏2

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

Adjust potentials at tree roots (except 𝑠∗) such that basic hyperarcs get zero reduced costs by setting 𝜌𝑆∖ 𝑠1

𝑈

: = ҧ 𝑑𝐶2

′𝑈𝐺𝐶2⋅ = (4,-1) 1/2

1/2 1/4 −1/4 = 1.75,2.25 . and raising all tree potentials according to the roots.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 28

𝑏1 𝑏2

The Hyperflow Network Simplex Algorithm

Input: Hypergraph 𝐼 = (𝑊, 𝒝) based on digraph 𝐸 = (𝑊, 𝐵), cost 𝑑 ≥ 0, demand 𝑐 s.t. 𝑐𝑈1 = 0, feasible basic (hyper)flow 𝑦 s.t. supp 𝑦 ⊆ 𝐵, associated basis 𝐶 and tree 𝑈

𝑠.

Output: (Fractional) Minimum cost hyperflow 𝑦. 1. (BTRAN) Solve 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈 and compute reduced costs

ҧ 𝑑𝑏 = 𝑑𝑏 − 𝜌(𝑊 ℎ 𝑏 + 𝜌(𝑊(𝑢 𝑏 ). 2. (PRICE) If ҧ 𝑑 ≥ 0 then output 𝑦, stop; else choose 𝑏in s.t. ҧ 𝑑𝑏𝑗𝑜 < 0. 3. (FTRAN) Solve 𝑁⋅𝐶𝑔 = −𝑁𝑏in. 4. (CHUZR) Choose 𝑏out ∈ argmin −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶 .

5. (UPDATE) 𝑦𝑏 ← ൞ −𝑦𝑏out/𝑔𝑏out, 𝑏 = 𝑏in 𝑦𝑏 − 𝑦𝑏out/𝑔𝑏out, 𝑏 ∈ 𝐶 𝑦𝑏, 𝑏 ∉ 𝐶 ∪ {𝑏in} , update 𝐶, 𝑆, {𝑈

𝑠}, 𝑁𝐶, 𝐺.

6. Go to 1.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 29

UPDATE: Root Set & Elementary Hyperflow Matrix

►

New basis 𝐶′ ← 𝐶 ∪ 𝑏in ∖ 𝑏out , 𝐶′′ ← 𝐶 ∪ 𝑏in

►

Find new root set 𝑆′ s.t. 𝑆′ ∋ 𝑠∗ and 𝑆∆𝑆′ = 1

►

𝑁⋅𝐶𝑔 = −𝑁𝑏in ⇔ 𝑁⋅𝐶′′ 𝑔 1 =: 𝑁⋅𝐶′′𝑔′′ = 0

►

𝑔

𝑠 ′′ ← 𝑔 𝑠

0 , 𝑠 ∈ 𝑆 ∪ 𝑆′, where 𝑁𝐶𝑔

𝑠 = 𝑓𝑠 − 𝑓𝑠∗ if there is 𝑠 ∈ 𝑆′ ∖ 𝑆

►

𝐺

𝑏𝑠 ′ ← 𝑔 𝑠 ′′(𝑏) − 𝑔

𝑠 ′′ 𝑏out

𝑔 𝑏out 𝑔′′(𝑏),

a ∈ 𝐶′′, 𝑠 ∈ 𝑆′

►

𝐺𝑏out𝑠

′

= 0

►

𝑁⋅𝐶′𝐺𝐶′𝑠

′

= 𝑁⋅𝐶′′𝐺𝐶′′𝑠

′

= 𝑁⋅𝐶′′𝑔

𝑠 ′′ − 𝑔

𝑠 ′′ 𝑏out

𝑔 𝑏out 𝑁⋅𝐶′′𝑔′′

= 𝑁⋅𝐶′′𝑔

𝑠 ′′

= 𝑁⋅𝐶𝑔

𝑠

= 𝑓𝑠 − 𝑓𝑠∗

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 30

Example: UPDATE

𝐺 = −0.5, −0.5,0.5,0.5,0.5 𝑈

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 31

https://cloud.zib.de/s/HsBnZLsGGLXrpyi

Example: UPDATE

𝑔 = 0.5, −0.5,0.5,0.5, −0.5 𝑈

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 32

Example: UPDATE

►

𝐺 = −0.5, −0.5,0.5,0.5,0.5 𝑈, 𝑔 = 0.5, −0.5,0.5,0.5, −0.5 𝑈

►

Ratio is −

0.5 −0.5 = 1 ⇒ 𝐺𝑏in ′

= 1

►

𝐺𝐶∩𝐶′

′

= −0.5, −0,5,0.5,0.5 𝑈 − −1 ⋅ 0.5, −0.5,0.5,0.5 𝑈 = 0, −1,1,1 𝑈

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 33

1 1 1 1 1 1

• 1

The Hyperflow Network Simplex Algorithm

Input: Hypergraph 𝐼 = (𝑊, 𝒝) based on digraph 𝐸 = (𝑊, 𝐵), cost 𝑑 ≥ 0, demand 𝑐 s.t. 𝑐𝑈1 = 0, feasible basic (hyper)flow 𝑦 s.t. supp 𝑦 ⊆ 𝐵, associated basis 𝐶 and tree 𝑈

𝑠.

Output: (Fractional) Minimum cost hyperflow 𝑦. 1. (BTRAN) Solve 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈 and compute reduced costs

ҧ 𝑑𝑏 = 𝑑𝑏 − 𝜌(𝑊 ℎ 𝑏 + 𝜌(𝑊(𝑢 𝑏 ). 2. (PRICE) If ҧ 𝑑 ≥ 0 then output 𝑦, stop; else choose 𝑏in s.t. ҧ 𝑑𝑏𝑗𝑜 < 0. 3. (FTRAN) Solve 𝑁⋅𝐶𝑔 = −𝑁𝑏in. 4. (CHUZR) Choose 𝑏out ∈ argmin −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶 .

5. (UPDATE) 𝑦𝑏 ← ൞ −𝑦𝑏out/𝑔𝑏out, 𝑏 = 𝑏in 𝑦𝑏 − 𝑦𝑏out/𝑔𝑏out, 𝑏 ∈ 𝐶 𝑦𝑏, 𝑏 ∉ 𝐶 ∪ {𝑏in} , update 𝐶, 𝑆, {𝑈

𝑠}, 𝑁𝐶, 𝐺.

6. Go to 1.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 34

Example: Start with a Feasible Hyperflow

►

Only arcs/hyperarcs of the current basis 𝐶 are shown.

►

Flow 𝑦𝑏 = ቊ1, 𝑏 ∈ 𝐶 0, 𝑏 ∉ 𝐶

►

Costs 𝑑 = 1, demands 𝑐 as labeled.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 35

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

Set 𝜌′𝑆: = 0 and extend to potentials 𝜌′ such that all tree arcs have reduced cost zero.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 36

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

The basic hyperarcs have preliminary reduced costs ҧ 𝑑𝑏1

′

= 1 − (1 + 0) + (2 + 2) = 4- ҧ 𝑑𝑏2

′

= 1 − (1 + 1) + (1 - 1) = -1

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 37

𝑏1 𝑏2

Example: (BTRAN) 𝜌𝑈𝑁⋅𝐶 = 𝑑𝐶

𝑈

►

Adjust potentials at tree roots (except 𝑠∗) such that basic hyperarcs get zero reduced costs by setting 𝜌𝑆∖ 𝑠1

𝑈

: = ҧ 𝑑𝐶2

′𝑈𝐺𝐶2⋅ = (4,-1) 1/2

1/2 1/4 −1/4 = 1.75,2.25 . and raising all tree potentials according to the roots.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 38

𝑏1 𝑏2

Example: (PRICE) Choose ҧ 𝑑𝑏𝑗𝑜 < 0

►

Green arc has reduced cost 1 - 4:25 + 2 = -1.25 < 0.

►

Add this arc to the basis.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 39

𝑏1 𝑏2

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

𝑏in is the red arc

►

𝐹𝑆𝑏in = 1,0, −1 𝑈

►

𝑔

𝑏1

𝑔

𝑏2

= 𝐺 ⋅ 𝐹𝑆∖ 𝑠∗ 𝑏in = 1/2 1/2 1/4 −1/4 −1 = −1/2 1/4

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 40

𝑏1 𝑏2

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

Compute demand on tree vertices induced by flow on basic hyerparcs.

►

Total demands (including demands on entering hyperarc) sum up to 0 on every tree.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 41

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

Compute flow on each tree by reverse BFS.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 42

Example: (CHUZR) 𝑏out ∈ argmin −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶

►

min −

𝑦𝑏 𝑔

𝑏 : 𝑔

𝑏 < 0, 𝑏 ∈ 𝐶 = 1.

►

𝑏out = red arc.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 43

Example: (FTRAN) 𝑁⋅𝐶𝑔 = −𝑁⋅𝑏in

►

𝑏in is the red arc

►

𝐹𝑆𝑏in = 1,0, −1 𝑈

►

𝑔

𝑏1

𝑔

𝑏2

= 𝐺 ⋅ 𝐹𝑆∖ 𝑠∗ 𝑏in = 1/2 1/2 1/4 −1/4 −1 = −1/2 1/4

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 44

𝑏1 𝑏2

Example: UPDATE

►

𝑦 ← 𝑦 + 1 ⋅ 𝑔.

►

Trees 𝑈

𝑠1 and 𝑈 𝑠3 change.

►

𝑆, 𝐶2, 𝐺 do not change.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 45

Summary

►

One iteration of the Hyperflow Network Simplex Algorithm can be implemented in 𝑃(σ deg 𝑤 + 𝑊 2).

►

If Bland’s rule is used, the algorithm is finite. Also polynomial?

►

Generalization of Network Simplex Method to Graph Based Hypergraphs.

►

Combinatorial: Each pivot consists of (fast) graph theoretical computations.

►

Can handle upper bounds on the variables.

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 46

InterCity Express (ICE) High Speed Train

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 47

Railway Constraints

Regularity Parking Maintenance Train Composition

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 48

Photos courtesy of DB Mobility Logistics AG

Timetable Regularity

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 49

Assignment Solution

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 50

Rotation Regularity

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 51

Modeling Regularity via Hyperedges

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 52

Hyperflow Model

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 53

Railway Constraints

Regularity Parking Maintenance Train Composition

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 54

Photos courtesy of DB Mobility Logistics AG

Rare Train Composition Example

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 55

Train Composition: Type, Order, Orientation

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 56

Hypergraph Model: Possible Train Compositions

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 57

Hypergraph Model: Arrival and Departure Nodes

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 58

Hypergraph Model: Single Traction

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 59

Hypergraph Model: Double Traction

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 60

Hypergraph Model: Triple Traction

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 61

Hypergraph Model: Pass-Through Connections

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 62

Hypergraph Model: Pass-Through Connections

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 63

Hypergraph Model: All Connections

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 64

The Coarse-to-Fine Method

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 65

Motivation: ICE Connections

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 66

Timetabled Trips: 1 Day

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 67

Graphics: JavaView, MATHEON F4

Timetabled Trips: Standard Week

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 68

Graphics: JavaView, MATHEON F4

Vehicle Rotation (1 Week)

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 69

We Tu Th Fr Sa Su Mo

Graphics: JavaView, MATHEON F4

Vehicle Rotation (6 Weeks)

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 70

We Tu Th Fr Sa Su Mo

Graphics: JavaView, MATHEON F4

Rotation Plan: Follow-on Trip Assignment

(Blue: Timetabled Trips, Red: Deadhead Trips)

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 71

Graphics: JavaView, MATHEON F4

Regularity of the timetable trips: Red: Trip done on one day of the week Blue: Trip done on all days of the week

Regular Timetable

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 72

Graphics: JavaView, MATHEON F4

Regular Rotations

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 73

Regularity of the deadhead trips: Yellow: regular Red: irregular Graphics: JavaView, MATHEON F4

Thank you for your attention

Ralf Borndörfer

Zuse Institute Berlin Freie Universität Berlin

• Takustr. 7

14195 Berlin Germany Fon (+49 30) 84185-243 Fax (+49 30) 84185-269 borndoerfer@zib.de http://www.zib.de/borndoerfer

Hypernetwork Simplex Algorithm and Railways | 4th ISM-ZIB-IMI MODAL Workshop 74

Image courtesy of DB Mobility Logistics AG