New Complexity Results and Algorithms for ... - Soumya Basu · Soumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 15 / 21. Series Parallel

New Complexity Results and Algorithms forMinimum Tollbooth Problem

Soumya BasuThanasis Lianeas Evdokia Nikolova

Department of ECEThe University of Texas at Austin

WINE 2015, Amsterdam

Soumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 1 / 21


Introduction

Congestion Gridlock“ The combined annual cost of gridlock to these (U.S., U.K.,France and Germany) countries is expected to soar to$ 293.1 billion by 2030. . . ” –INRIX and the CEBR

How to tackle congestion?


Introduction


How to tackle congestion?Infrastructure growth is good.

But Not Always: Selfish users lead us to Braess’ ParadoxSolution: Influence the users by placing appropriate TollsNo Free Lunch: Each toll comes with large overhead


Introduction


How to tackle congestion?Infrastructure growth is good.But Not Always: Selfish users lead us to Braess’ ParadoxSolution: Influence the users by placing appropriate Tolls

No Free Lunch: Each toll comes with large overhead


Introduction


How to tackle congestion?Infrastructure growth is good.But Not Always: Selfish users lead us to Braess’ ParadoxSolution: Influence the users by placing appropriate TollsNo Free Lunch: Each toll comes with large overhead


Outline

1 Minimum Tollbooth Problem (MINTB)

2 Complexity ResultsSingle Commodity NetworkSingle Commodity Network with all Edges Used

3 MINTB in Series Parallel GraphBackgroundAlgorithm

4 Summary

5 Future Directions


System Model

Directed graph G(V ,E) with single commodity (s, t)Flow dependent latency, L = {è(·) : ∀e ∈ E}Traffic network, G = {G, (s, t),L}

Social Optimum (SO)SO flow o is a flow that minimizes social cost.

Nash Equilibrium (NE)A flow is said to be in NE iff property (1) and (2) holds:(1) All used paths cost L (2) All unused paths cost ≥ L


Problem Definition

Tolled edge cost: Under toll θ and flow f ,tolled edge cost ce = è(fe) + θe

Induce flow f : Under the tolled edge cost f is in NEInduced length: Under NE the tolled cost of used paths

Minimum Tollbooth Problem (MINTB)Given traffic network G and an optimal flow oFind toll with minimum support which induces o


MINTB: Where do we stand?

Formulated as Mixed integer linear program (MILP).Hearn et al., 1998Heuristics developed: Genetic Algorithm, LP relaxation.Hardwood et al., ’08; Bai et al., ’09In Multi-commodity networks it is NP-hard. Bai et al.,’08Design toll for inducing general flow. Harks et al.,’08

MINTBon

General Graphs and

Multicommodity Network

NP Hard

Figure: Complexity DiagramSoumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 7 / 21

Contributions

First APX-hardness:Single commodity networks with linear latencies.

MINTB with used edges only:Is MINTB efficiently solvable? No still NP hard.

First Exact Poly-time Algorithm:Algorithm for Series-Parallel graphs.


First APX hardness result

TheoremFor instances with linear latencies and single commodity, it isNP-hard to approximate the solution of MINTB by a factor of lessthan 1.1377.

MINTBon

General Graphs

APX Hard

Figure: Complexity Diagram


MINTB Reduction: Vertex Cover

I

JVertex Cover Instance

𝒆𝒌 = (𝒊, 𝒋)

Figure: MINTB Reduction



𝒂𝒊 𝒃𝒊 𝒄𝒊 𝒅𝒊

𝒆𝟏,𝒊 (1)

𝒆𝟒,𝒊 (3)

𝒆𝟑,𝒊 (1)𝒆𝟐,𝒊 (0)

Vertex gadget for 𝒗𝒊: 𝑮𝒊

𝒂𝒋 𝒃𝒋 𝒄𝒋 𝒅𝒋𝒆𝟏,𝒋 (1) 𝒆𝟑,𝒋 (1)𝒆𝟐,𝒋 (0)

Vertex gadget for 𝒗𝒋 : 𝑮𝒋𝒆𝟒,𝒋 (3)

I


𝒆𝒌 = (𝒊, 𝒋)





𝒆𝟏,𝒊 (1)

𝒆𝟒,𝒊 (3)

𝒆𝟑,𝒊 (1)𝒆𝟐,𝒊 (0)


𝒂𝒋 𝒃𝒋 𝒄𝒋 𝒅𝒋𝒆𝟏,𝒋 (1) 𝒆𝟑,𝒋 (1)𝒆𝟐,𝒋 (0)


𝒈𝟏,𝒌 (0.5)

𝒈𝟐,𝒌 (0.5)Edge Gadget for 𝒆𝒌

I


𝒆𝒌 = (𝒊, 𝒋)





s t

𝒆𝟏,𝒊 (1)

𝒆𝟒,𝒊 (3)

𝒆𝟑,𝒊 (1)𝒆𝟐,𝒊 (0)


𝒂𝒋 𝒃𝒋 𝒄𝒋 𝒅𝒋𝒆𝟏,𝒋 (1) 𝒆𝟑,𝒋 (1)𝒆𝟐,𝒋 (0)


𝒈𝟏,𝒌 (0.5)


I


𝒆𝒌 = (𝒊, 𝒋)





s t

𝒆𝟏,𝒊 (1+0.5)

𝒆𝟒,𝒊 (3)

𝒆𝟑,𝒊 (1+0.5)𝒆𝟐,𝒊 (0)


𝒂𝒋 𝒃𝒋 𝒄𝒋 𝒅𝒋𝒆𝟏,𝒋 (1) 𝒆𝟑,𝒋 (1)𝒆𝟐,𝒋 (0+1)


𝒈𝟏,𝒌 (0.5)


I


𝒆𝒌 = (𝒊, 𝒋)




Given Vertex Cover instance Gvc with nvc verticesCreate a Single Commodity network G

Lemma 1

∃ A Vertex Cover of size x in Gvc ⇐⇒∃ Opt-inducing toll with support nvc + x in G.

TheoremIt is NP hard to approximate Minimum Vertex Cover to within a factor of1.3606. Dinur et al 2005


Are Unused Edges the Troublemaker?

Motivation: Less overhead present in edge removal.Model: Unused edges in social optimum flow are removed.

TheoremFor instances with linear latencies, it is NP-hard to solve MINTB evenif all edges are used by the optimal solution.

Reduction follows from PARTITION problem.


Complexity Diagram

MINTBon

General Graphs

APX Hard

General Graphs with Used edges

NP Hard

Figure: Complexity DiagramSoumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 13 / 21

Series Parallel Graphs

Series Parallel Graphs (SP)An SP graph is created by starting from a directed edge andinductively connecting two graphs in series or in parallel.

AB

C

Figure: SP Graph




AB

C

Figure: SP Graph and Parse Tree




AB

C

Figure: SP Graph


Algorithm for Parallel Link (PL) Graph

Algorithm 1Input: A parallel link networkOutput: Induce L with min support

1 Sort edges2 Append length ènd =∞

3 Max used edge cost `max

4 Create list {(η, `)}

Toll on edges 1, . . . , ηMaximally induce `

5 Place toll to induce L

1 2 3 4

ℓ1

ℓ2∞

Length

Edges

Unused

Used

Used

Used

S t

ℓ2ℓ1ℓ1ℓ0

ℓ0

Figure: Example




1 Sort edges2 Append length ènd =∞3 Max used edge cost `max

4 Create list {(η, `)}

Toll on edges 1, . . . , ηMaximally induce `


1 2 3 4

ℓ1

ℓ2∞

Length

Edges

Unused

Used

Used

Used

List

Edge Length1 ℓ12 ℓ23 ℓ24 ∞

ℓ0

Figure: Example: `max = `1,i0 = 1





4 Create list {(η, `)}Toll on edges 1, . . . , ηMaximally induce `


1 2 3 4

ℓ1

ℓ2∞

Length

Edges

Unused

Used

Used

Used

List


ℓ0

Figure: Example: `max = `1,i0 = 1





4 Create list {(η, `)}Toll on edges 1, . . . , ηMaximally induce `


1 2 3 4

ℓ1

ℓ2∞

Length

Edges

Un

use

d

Use

d

Use

d

Use

d

List


Induce ℓ

Solution

ℓ0

Figure: Example: Placingtolls


Series Parallel in P: Extension to SP

Induce length LSeries comb. G1(S)G2: Induce L1(≤ L) in G1 and L− L1 in G2.Parallel comb. G1(P)G2: Induce L in G1 and G2.

Monotonicity LemmaIn a SP graph G with maximum used path length `max,We can induce length L ⇐⇒ L ≥ `max

L is induced optimally with support T⇒ ` ≤ L can be induced optimally with support t ≤ T



MakeList: Bottom-up List creation: Dynamic Programming1) Start from PL 2) Combine list in series/parallel 3) Recur till root

3

5

6

1

1

2

8

10

A B

C

P

S

A B C

Edge Length1 𝟓2 𝟔3 ∞

Edge Length0 𝟏2 ∞

Edge Length1 𝟖2 𝟏𝟎3 ∞

P

A(S)B C

Edge Length1 𝟔2 𝟕3 ∞


[A(S)B](P)C


MAKELIST

MINTB

Figure: Step1: Example Run of MINTB AlgorithmSoumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 17 / 21


Placetoll: Top-down Induction of length: Traceback1) Start from root. 2) Branch in series/parallel. 3) Recur till PL.

P

S

A B C

Edge Length1 𝟓2 𝟔3 ∞

Edge Length0 𝟏2 ∞


P

A(S)B C

Edge Length1 𝟔2 𝟕3 ∞


8 8 7 1 8

[A(S)B](P)C


8

3+4

5+2

6+1

1

1

2+6

8

10

A B

C

PLA

CET

OLL

MINTB Solution

Figure: Step2: Example Run of MINTB AlgorithmSoumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 18 / 21


TheoremThe MINTB on a SP graph with |E | = m, is solved optimally in timeO(m3).



TheoremThe MINTB on a SP graph with |E | = m, is solved optimally in timeO(m3).

Presence of Braess Structure:The Monotonicity Lemma breaks.Edges to induce length 3 = 3.Edges to induce length 4 = 2.

A 𝐷

B

C

0

Figure: Counter


Summary

MINTBon

General Graphs and

Multicommodity Network

NP Hard

Figure: Complexity Diagram Prior to the WorkSoumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 20 / 21

Summary

MINTBon

General Graphs

APX Hard

General Graphs with Used edges

NP Hard

Series-Parallel Graphs

P

• Reduction From Vertex Cover Problem• APX hardness of 1.1377• Exploits Unused Edges in an Optimal Flow

• Reduction from Partition Problem• Exploits Braess Structure

• Absence of Braess Structure• Monotonicity Property• Tree Decomposition

• Dynamic Programming • Bottom Up List Creation• Top Down Toll Placement

Complexity Results

Algorithm


Future Directions

Is the APX hardness result tight?YES Find matching approximation algorithms.NO Give tighter APX hardness results.

Design Practical Algorithms:Improved heuristics with performance guarantee.Faster algorithms for large-scale traffic networks.

What happens while Taxing Sub-networks?


Future Directions

Is the APX hardness result tight?YES Find matching approximation algorithms.NO Give tighter APX hardness results.

Design Practical Algorithms:Improved heuristics with performance guarantee.Faster algorithms for large-scale traffic networks.

What happens while Taxing Sub-networks?

Questions?


New Complexity Results and Algorithms for ... - Soumya Basu · Soumya Basu, UT Austin New Complexity Results and Algorithms for Minimum Tollbooth Problem 15 / 21. Series Parallel

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