Incremental Network Design Martin Savelsbergh Georgia Tech Aussois - - PowerPoint PPT Presentation

Incremental Network Design

Martin Savelsbergh Georgia Tech

Aussois 2016

Collaborators

• Thomas Kalinowski – University of Newcastle
• Dmytro Matsypura – University of Sydney

Outline

• Motivation
• Incremental Network Design with Maximum Flows

– IP formulations – Heuristics – Special case: unit arc capacities

• Open questions
• Other results
• Future research

Motivation

Some of my recent research projects:

• Optimizing coal export chain

– Transporting coal by rail from mines to terminals at a port

• Optimizing carbon capture and storage

– Transporting CO2 captured at power plants via a pipeline to subsurface aquifers

• Optimizing renewable energy integration

– Transporting electricity produced by renewable energy sources to consumers

Motivation

Network Network Use Expansions Over Time Planning Operational

Motivation

• Where should upgrades/expansions of a network take

place?

• When should upgrades/expansions of a network take

place?

• In what sequence should upgrades/expansions of a

network take place when budget and/or resources are constrained?

Incremental Network Design

• Network optimization problem

– Minimum spanning tree, shortest path, maximum flow, minimum cost flow, TSP, …

• Multiple periods

– Incur the cost / realize the benefit of an optimal solution to the network optimization problem in each period – Minimize the total cost / maximize the total benefit over the planning horizon

• Option to invest in capacity expansion

– Limited budget to invest in capacity expansion in each period (capacity expansions may decrease cost / increase benefit)

Incremental Network Design

build single edge per period at no cost

SIMPLEST VARIANT:

Questions of interest

• How difficult are incremental network design

problems?

• If they are difficult, are there constant factor

approximation algorithms?

• If they are difficult, what size instances can we solve

using integer programming techniques?

• If they are difficult, are there effective and efficient

heuristics?

Incremental Network Design with Maximum Flows

• Given:

– Graph D=(N, Ae U Ap),

• Existing arcs and Potential arcs

– Arc capacity ca – Source s – Sink t – Planning horizon T = | Ap | – Option to construct one potential arc each period

• Decide:

– Which arc to construct in each period so as to maximize the total flow over the planning horizon (i.e., sum of the s-t flows in each period)

Incremental Network Design with Maximum Flows

1-1-6 = 8 0-5-6 = 11 0-0-2-3 = 5 1-1-1-3 = 6 Which path to build first? Note: We assume that an arc built in a period can immediately be used in that period.

Incremental Network Design with Maximum Flows

COMPLEXITY

Incremental Network Design with Maximum Flows

Incremental Network Design with Maximum Flows

if yes-instance 3+6+…+3n+3n+…+3n

Incremental Network Design with Maximum Flows

INTEGER PROGRAMMING FORMULATIONS

arc a built in

• r before

period k

Integer Programming

Maximize flow out of source Flow balance Arc capacity Arc capacity & forcing Budget Consistency Initial condition Concern: Symmetry when multiple arcs need to be build to increase flow flow on arc a in period k Note: Assumes T = |Ap|+1 and that an arc built can be used only in subsequent periods

Integer Programming

An Alternative Formulation

Total flow: #arcs built to make a flow of f+k+1 possible #arcs built to make a flow of f+1 possible

Integer Programming

Maximize total flow Flow balance Arc capacity Arc capacity & forcing Consistency Concern: Size when F – f is large

Example

2 2 1 1 a b c d Solution:

Incremental Network Design with Maximum Flows

HEURISTICS

Heuristics

STEP 1: Determine the cardinality of a minimum set of potential arcs that have to be build to increase the maximum flow from f+k to at least f+k+1 (MinArcs) flow on arc a build arc a or not set of potential arcs already built capacity of arc a

Heuristics

STEP 2: Among the minimal sets of potential arcs that have to be build to increase the maximum flow from f+k to at least f+k+1 choose one that increases the flow the most (MaxVal)

Quickest-increment

restricted set of potential arcs Ap

Quickest-to-ultimate

Notation: r = F - f

Quickest-to-target

Remarks

• Heuristics are not polynomial in general
• Heuristics are polynomial if the maximum arc capacity
• f potential arcs is equal to 1 (MinArcs becomes a

minimum cost flow problem)

• If the maximization of the flow increment is omitted,

then quickest-increment can be implemented to run in polynomial time even with general arc capacities

Incremental Network Design with Maximum Flows

SPECIAL CASE: All capacities equal to 1

Quickest-to-ultimate

Quickest-to-ultimate: 0-0-…-0-0 | 1-1-…-1-1 | 2-2 Opt: 0 | 1-1-…-1-1 | 1-1-…-1 | 2

path of k potential arcs

Quickest-improvement

Quickest-improvement: 0-0-…-0-0 | 1-1-…-1-1 | 1-1-…-1-1 | 2 Opt: 0-0-…-0-0 | 1-1-…1-1 | 2-2-…-2

Incremental Network Design with Maximum Flows

SPECIAL CASE: All capacities equal to 1 and a bipartite graph

Special Case

Incremental Network Design with Maximum Cardinality Matchings

Observations

• Quickest-improvement does not always produce an
• ptimal solution
• Quickest-to-ultimate does not always produce an
• ptimal solution
• Quickest-to-ultimate is a 4/3-approximation algorithm

Quickest-increment is not optimal

Quickest-increment: (5,8)(7,10) (1,2)(3,4)(5,6)(9,10)(11,12)(13,14) Value: 2*5+6*6+7=53 Optimal: (1,2)(3,4)(5,6) (9,10)(11,12)(13,14) (5,8)(7,10) Value: 3*5+3*6+2*7=54

Quickest-to-ultimate is not optimal

Quickest-to-ultimate: (1,2)(3,4)(5,6)(7,8) (9,10)(11,12)(13,14)(15,16) Value: 4*6+4*7+2*8=68 Optimal: (1,2)(9,4) (3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16) Value: 2*6+7*7+8=69

Incremental Network Design with Maximum Flows

OPEN QUESTIONS

Open Questions

• Is incremental network design problem with maximum

flows and capacities equal to 1 solvable in polynomial time or NP-complete?

– Formulation 2 always gives integer solution – Formulation 2 is not TU

• Is the incremental network design problem with

maximum cardinality matchings solvable in polynomial time or NP-complete?

• What are the approximation ratios of quickest-to-

ultimate and quickest-improvement for the incremental network design problem with maximum cardinality matchings?

Incremental Network Design with Maximum Flows

OTHER RESULTS

Other results

• The Incremental Network Design Problem with Shortest

Paths is NP-hard

• The Incremental Network Design Problem with Minimum

Spanning Trees is solvable in O(max{n2,m log m}) time

Incremental Network Design with Maximum Flows

FUTURE RESEARCH

• Multi-commodity flow
• Investment budgets
• Multi-period build times
• Uncertainty

QUESTIONS ?

• M. Baxter, T. Elgindy, A.T. Ernst, T. Kalinowski, and

M.W.P. Savelsbergh. “Incremental Network Design with Shortest Paths”. EJOR 238, 675-684, 2014.

• T. Kalinowski, D. Matsypura, and M.W.P. Savelsbergh.

“The Incremental Network Design Problem with Maximum Flows”, EJOR 242, 51-62, 2015.

• C. Engel, T. Kalinowski, and M.W.P. Savelsbergh. “The

Incremental Network Design Problem with Minimum Spanning Trees”, arXiv:1306.1926 [math.CO].