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Running time analyses of Benson type algorithmswith an application to multiobjectivecombinatorial optimization problems
Fritz Bokler, Petra Mutzel
Chair of Algorithm Engineering, TU Dortmund, Germany
September 12, 2014
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Outline
1 Model of Computation
2 Multiobjective Linear Optimization
3 Benson’s Algorithm
4 Multiobjective Combinatorial Optimization
5 Experiments
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Enumeration Complexity: Motivation
MOCO Intractability
Long known: Size of Pareto-frontier might be exponential
No polynomial time algorithm
Important: In the worst case!
And in practice?
Smoothed Analysis: E (|XE |) ∈ O(n2dΦd) [Brunsch, Roglin]Φ is a perturbation parameter
Algorithm wanted: Fast if YN is small, not too slow if YN islarge
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Enumeration Complexity: Introduction
Definition (Enumeration Problem)
Consist of
Set of instances I ⊆ Σ∗
Mapping C : I 7→ 2Σ∗
x ∈ I , y ∈ C(x) : |y | ∈ O(poly(|x |))
Goal: Enumerate C(x) for instance x ∈ I .
Observation
|C(x)| ∈ Ω(2|x |) possible
Polynomial Total Time (PTT), a.k.a.: Output PolynomialTime
Enumerate C(x) in time poly(|x |, |C(x)|) 4 of 38
Enumeration Complexity: Delay
Delay [Johnson, Yannakakis, Papadimitriou 1988]
The k-th delay is the time between outputting the k-th and(k + 1)-th element.
0-th delay: Time before outputting the first element,
|C (x)|-th delay: Time after outputting the last element untiltermination
Polynomial Time Delay (PTD)
Enumerate C(x) such that every delay is at most poly(|x |)
Incremental Polynomial Time (IncP)
Enumerate C(x) such that the k-th delay is at most poly(|x |, k)
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Enumeration Complexity: MOO Negative Results
Multiobjective Linear Programming
By [Khachiyan, Boros, Borys, Elbassioni, Gurvich 08]:P 6= NP⇒ cannot enumerate all Pareto-optimal basic feasiblesolutions of MOLP in PTT
Multiobjective Shortest Path
P 6= NP⇒ MO-SP /∈ PTT even for d = 2 and on outerplanargraphs
And the decision variant?
There exist MOCO P, such that PDec ∈ NPC, but P ∈ IncP
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Outline
1 Model of Computation
2 Multiobjective Linear Optimization
3 Benson’s Algorithm
4 Multiobjective Combinatorial Optimization
5 Experiments
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Multiobjective Linear Program
Definition (MOLP)
Matrices A ∈ Rm×n, C ∈ Rd×n and b ∈ Rm
pmin Cx
Ax ≥ b
Possible Goals: Finding Pareto-optimal basic feasible solutions,finding a representation of the Pareto-surface
pminCx | Ax ≥ b
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Multiobjective Linear Program
Definition (MOLP)
Matrices A ∈ Qm×n, C ∈ Qd×n and b ∈ Qm
pmin Cx
Ax ≥ b
Possible Goals: Finding Pareto-optimal basic feasible solutions,finding a representation of the Pareto-surface
pminCx | Ax ≥ b
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Is it really so hard?
Reminder
By [Khachiyan, Boros, Borys, Elbassioni, Gurvich 08]:P 6= NP⇒ cannot enumerate all Pareto-optimal basic feasiblesolutions of MOLP in PTT
More important in practice
Solutions x1, x2 with Cx1 = Cx2 are indistinguishable
Wanted: One solution for each nondominated extreme pointof P := Cx | Ax ≥ b+ Rd
≥
Our Question
Can we enumerate vertP in polynomial total time?
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Previous Work
Exact MOLP Solver
Multiobjective Simplex, e.g., [Evans, Steuer 1973, Zionts,Wallenius 1980, Steuer 1985, Armand 1993, ...]
Objective space methods, e.g., [Dauer, Liu 1990, Dauer, Saleh1990, Dauer 1993, Benson 1998, ...]
Rarely: Theoretical running time guarantees
Theoretical Running Time Guarantees
Okamoto and Uno 2007
Output: All efficient bfsAlgorithm based on reverse search [Avis, Fukuda 92]Polynomial delay for nondegenerate MOLPs
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 4
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Contribution
MOLP: Benson’s Algorithm and Dual Variant
For fixed number of objectives: Prove PTT for Benson’s andthe Dual Algorithm
Removing exponential overhead on both algorithms, provingIncP for the improved Dual Algorithm
Running time in # vertices and # facets of P
MOCO
Application for dual algorithm: Enumerating extreme pointsof MOCO Problems
Experiments to show practicability on multiobjectiveassignment and spanning tree problems
First experiments for d > 411 of 38
Outline
1 Model of Computation
2 Multiobjective Linear Optimization
3 Benson’s Algorithm
4 Multiobjective Combinatorial Optimization
5 Experiments
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Benson’s Algorithm
Very brief History
Proposed by [Benson 1998]
Further improvements, e.g., [Burton and Ozlen 2010, Hamel,Lohne, Rudloff 2013, Csirmaz 2013]
Major recap with geometric duality theory and a dual variant[Ehrgott, Lohne, Shao 2012]
Benson’s Algorithm
Starts with polyhedron containing PIn each iteration: Finds supporting hyperplane to PNecessary assumption: Exists y ∈ Rd : P ⊆ Rd
≥ + y
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Benson’s Algorithm
Very brief History
Proposed by [Benson 1998]
Further improvements, e.g., [Burton and Ozlen 2010, Hamel,Lohne, Rudloff 2013, Csirmaz 2013]
Major recap with geometric duality theory and a dual variant[Ehrgott, Lohne, Shao 2012]
Benson’s Algorithm
Starts with polyhedron containing PIn each iteration: Finds supporting hyperplane to PNecessary assumption: Exists y ∈ Rd : P ⊆ Rd
≥ + y
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Example
c1T x
c2T x
0
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Running Time: Subproblems
Subproblems
LP Oracles: Finding ideal point, finding point at boundary,finding new hyperplane
A special case of vertex enumeration
Vertex Enumeration
Have to find new extreme points of intermediate polyhedra
Employing one Double Description (DD) step[Motzkin, Raiffa, Thompson, Thrall 53, Fukuda, Prodon 96, ..]
P is of dimension d
Running time for one iteration in O(dHV 3)
Using upper bound theorem: dHO(d)
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LP Oracles
Example: P2(y)
minz ∈ R | x ∈ Rn,Ax ≥ b,Cx − 1z ≤ y
Solving with the Ellipsoid Method—Major Problem: How large cany become in the process of the algorithm?
Lemma (Encoding Length of Intermediate Extreme Points)
For each intermediate extreme point y : |yi | ∈ O(poly(n, L))
Theorem (Running Time: Benson’s Algorithm)
Let ve : # vertices of P, d fixed, running time:O(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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LP Oracles
Example: P2(y)
minz ∈ R | x ∈ Rn,Ax ≥ b,Cx − 1z ≤ y
Solving with the Ellipsoid Method—Major Problem: How large cany become in the process of the algorithm?
Lemma (Encoding Length of Intermediate Extreme Points)
For each intermediate extreme point y : |yi | ∈ O(poly(n, L))
Theorem (Running Time: Benson’s Algorithm)
Let ve : # vertices of P, d fixed, running time:O(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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LP Oracles
Example: P2(y)
minz ∈ R | x ∈ Rn,Ax ≥ b,Cx − 1z ≤ y
Solving with the Ellipsoid Method—Major Problem: How large cany become in the process of the algorithm?
Lemma (Encoding Length of Intermediate Extreme Points)
For each intermediate extreme point y : |yi | ∈ O(poly(n, L))
Theorem (Running Time: Benson’s Algorithm)
Let ve : # vertices of P, d fixed, running time:O(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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Finding Facets
Drawback
Example in [Ehrgott, Lohne, Shao 2012]: Supportinghyperplanes might support in face of dimension < d − 1
Exponentially many redundant inequalities
Redundant DD steps
Redundant intermediate extreme points
Nonredundancy: Speed up adjacency checks
Nice to have: Running time in number of extreme points andfacets of P
Can we find facets only?
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Finding Facets
Drawback
Example in [Ehrgott, Lohne, Shao 2012]: Supportinghyperplanes might support in face of dimension < d − 1
Exponentially many redundant inequalities
Redundant DD steps
Redundant intermediate extreme points
Nonredundancy: Speed up adjacency checks
Nice to have: Running time in number of extreme points andfacets of P
Can we find facets only?
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Finding Facets
Drawback
Example in [Ehrgott, Lohne, Shao 2012]: Supportinghyperplanes might support in face of dimension < d − 1
Exponentially many redundant inequalities
Redundant DD steps
Redundant intermediate extreme points
Nonredundancy: Speed up adjacency checks
Nice to have: Running time in number of extreme points andfacets of P
Can we find facets only?
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Improved Algorithm: Finding Facets
Theorem (Finding Facets)
By solving a lexicographic variant of hyperplane finding LP, weensure that for a point y at the boundary of P the hyperplanesupports P in a facet containing y .
Theorem (Running Time w.r.t. vertices and facets)
Let ve : # vertices, vf : # facets of P, d fixedImproved version’s running time:O((ve + vf ) poly(n,m, L) + vf
Θ(d))
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Improved Algorithm: Finding Facets
Theorem (Finding Facets)
By solving a lexicographic variant of hyperplane finding LP, weensure that for a point y at the boundary of P the hyperplanesupports P in a facet containing y .
Theorem (Running Time w.r.t. vertices and facets)
Let ve : # vertices, vf : # facets of P, d fixedImproved version’s running time:O((ve + vf ) poly(n,m, L) + vf
Θ(d))
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A Dual Variant of Benson’s Algorithm
History
Geometric duality in MOLP from [Heyde and Lohne 2008]
Algorithm proposed in [Ehrgott, Lohne, Shao 2007/2012]
Dual Variant of Benson’s Algorithm
Pretty much the same strategy
Different polyhedron: D,“dual” to P
Theorem (Running Time)
Let ve : # vertices of P, d fixedO(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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A Dual Variant of Benson’s Algorithm
History
Geometric duality in MOLP from [Heyde and Lohne 2008]
Algorithm proposed in [Ehrgott, Lohne, Shao 2007/2012]
Dual Variant of Benson’s Algorithm
Pretty much the same strategy
Different polyhedron: D,“dual” to P
Theorem (Running Time)
Let ve : # vertices of P, d fixedO(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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A Dual Variant of Benson’s Algorithm
History
Geometric duality in MOLP from [Heyde and Lohne 2008]
Algorithm proposed in [Ehrgott, Lohne, Shao 2007/2012]
Dual Variant of Benson’s Algorithm
Pretty much the same strategy
Different polyhedron: D,“dual” to P
Theorem (Running Time)
Let ve : # vertices of P, d fixedO(ve
Θ(d) poly(n,m, L) + veΘ(d2))
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Encoding Lengths
LP Oracle
P1(`) : min`TCx | x ∈ Rn,Ax ≥ b
Again: How large can ` become?
Lemma (Encoding Length of Intermediate dual ExtremePoints)
For each intermediate extreme point v : |vi | ∈ O(poly(n, L))
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Improved Algorithm
Theorem (Finding Facets)
By solving a lexicographic variant of the single objective LP, weensure that for an optimal point y of D the hyperplane supports Din a facet containing y .
Theorem (Running Time w.r.t. vertices and facets of P)
Let ve : # vertices, vf : # facets of P, d fixedImproved version’s running time:O((ve + vf ) poly(n,m, L) + ve
Θ(d))
Theorem (Running Time w.r.t. vertices P)
Let ve # vertices of P, d fixedImproved version’s running time: O(ve
Θ(d) poly(n,m, L) + veΘ(d))
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Improved Algorithm
Theorem (Finding Facets)
By solving a lexicographic variant of the single objective LP, weensure that for an optimal point y of D the hyperplane supports Din a facet containing y .
Theorem (Running Time w.r.t. vertices and facets of P)
Let ve : # vertices, vf : # facets of P, d fixedImproved version’s running time:O((ve + vf ) poly(n,m, L) + ve
Θ(d))
Theorem (Running Time w.r.t. vertices P)
Let ve # vertices of P, d fixedImproved version’s running time: O(ve
Θ(d) poly(n,m, L) + veΘ(d))
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And considering the delay?
Original Algorithm
Number of iterations: number of faces/facets
Might take many iterations until new extreme point is found
Theorem (Delay of the Dual Algorithm)
Let’s fix d , k is number of extreme points so far,using the improved algorithm we get a delay of:
O(kΘ(d) poly(n,m, L))
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Outline
1 Model of Computation
2 Multiobjective Linear Optimization
3 Benson’s Algorithm
4 Multiobjective Combinatorial Optimization
5 Experiments
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Multiobjective Combinatorial Optimization
Definition (Instance of a MOCO Problem)
An instance of a MOCO problem consists of
a finite base set A,
a set of feasible solutions S ⊆ 2A and
a cost function c : A 7→ Rd .
Implicit cost function for x ∈ S : c(x) :=∑
ω∈x c(ω)
Definition (MOCO Problem)
For each non-dominated y ∈ Y := c(S): find a solution x ∈ Ssuch that c(x) = y
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Multiobjective Combinatorial Optimization
Definition (Instance of a MOCO Problem)
An instance of a MOCO problem consists of
a finite base set A,
a set of feasible solutions S ⊆ 2A and
a cost function c : A 7→ Rd .
Implicit cost function for x ∈ S : c(x) :=∑
ω∈x c(ω)
Definition (MOCO Problem)
For each non-dominated y ∈ Y := c(S): find a solution x ∈ Ssuch that c(x) = y
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Finding Extreme Points of the Pareto-Frontier
Previous work for d > 2
Przybylski, Gandibleux, Ehrgott 2010 (PGE10)
Huge improvements for d = 3Experiments for d = 3No performance guarantees
Ozpeynirci and Koksalan 2010 (OK10)
Experiments for d ∈ 3, 4No performance guarantees
Theoretical Running Time Guarantees
Okamoto and Uno 2007
Polynomial delay for finding supported spanning treesNo experiments
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Finding Extreme Points of the Pareto-Frontier
Previous work for d > 2
Przybylski, Gandibleux, Ehrgott 2010 (PGE10)
Huge improvements for d = 3Experiments for d = 3No performance guarantees
Ozpeynirci and Koksalan 2010 (OK10)
Experiments for d ∈ 3, 4No performance guarantees
Theoretical Running Time Guarantees
Okamoto and Uno 2007
Polynomial delay for finding supported spanning treesNo experiments
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Why Dual of Benson’s Algorithm?
Relation MOCO and MOLP
For instance of MOCO exists MOLP s.t. YX = vertPP is always bounded from below
We do not have to construct the MOLP explicitly
We only have to solve oracle problems:
of the single objective version (P1)of a lexicographic version (lex-P)
of the problem
Lexicographic Optimization
Usually not much harder than single objective optimization
e.g., Matroids, Shortest Path, Assignments, ...
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Why Dual of Benson’s Algorithm?
Relation MOCO and MOLP
For instance of MOCO exists MOLP s.t. YX = vertPP is always bounded from below
We do not have to construct the MOLP explicitly
We only have to solve oracle problems:
of the single objective version (P1)of a lexicographic version (lex-P)
of the problem
Lexicographic Optimization
Usually not much harder than single objective optimization
e.g., Matroids, Shortest Path, Assignments, ...
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Results
Theorem (Dual Algorithm for MOCO)
For MOCO O, fixed d and P1 ∈ P: YX of O can be enumerated inpolynomial total time
Theorem (Improved Dual Algorithm for MOCO)
For MOCO O, fixed d and lex-P1 ∈ P: YX of O can beenumerated in incremental polynomial time
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Outline
1 Model of Computation
2 Multiobjective Linear Optimization
3 Benson’s Algorithm
4 Multiobjective Combinatorial Optimization
5 Experiments
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Experiment Overview
Setup
Implemented combinatorial dual algorithm in C++
DD Method:
For d ∈ 3, 4: DD algorithm with adjacency informationFor d > 4: CDD library by K. Fukuda
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MO Assignment Problem Experiments
General
Benchmark problem used in many experiments before
Complete bipartite graph, U ∪ W = V , |U| = |W |We draw weights i.i.d. from 0, . . . , 20For each number of nodes: 10 independent instances
For d = 3: Instances from PGE10
lex-P Solver
Using Hungarian algorithm to find lexicographic minimalassignments
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MO Assignment Problem Experiments
General
Benchmark problem used in many experiments before
Complete bipartite graph, U ∪ W = V , |U| = |W |We draw weights i.i.d. from 0, . . . , 20For each number of nodes: 10 independent instances
For d = 3: Instances from PGE10
lex-P Solver
Using Hungarian algorithm to find lexicographic minimalassignments
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Comparison - Outline
PGE10
CPU: P4 EE 3.73 GHz,Mem: 4 GB
Does not filternon-extreme points
Numerical problems forlarger n
OK10
CPU: Pentium M 1.6 Ghz,Mem: 256MB
Not same instances,similarly generated
Our implementation
CPU: Core i7-3770 3.4 GHz, Mem: 16 GB
Did not find numerical problems
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Comparison: d = 3
20 40 60 80 100
02
46
810
Number of Nodes
Mea
n R
unni
ng T
ime
[s]
PGE10OK10Dual Benson
Summary
PGE10 faster
OK10 for n = 60: 418.84s
Does OK10 suffer fromCPU only?
Reimplementation needed
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Comparison: d = 3
20 40 60 80 100
02
46
810
Number of Nodes
Mea
n R
unni
ng T
ime
[s]
PGE10OK10Dual Benson
Summary
PGE10 faster
OK10 for n = 60: 418.84s
Does OK10 suffer fromCPU only?
Reimplementation needed
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Comparison: d = 3
20 40 60 80 100
02
46
810
Number of Nodes
Mea
n R
unni
ng T
ime
[s]
PGE10OK10Dual Benson
Summary
PGE10 faster
OK10 for n = 60: 418.84s
Does OK10 suffer fromCPU only?
Reimplementation needed
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Comparison: d = 3
20 40 60 80 100
02
46
810
Number of Nodes
Mea
n R
unni
ng T
ime
[s]
PGE10OK10Dual Benson
Summary
PGE10 faster
OK10 for n = 60: 418.84s
Does OK10 suffer fromCPU only?
Reimplementation needed
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Assignment Instances, d = 4
20 30 40 50 60 70 80 90
01000
2000
3000
4000
Number of Nodes
Run
ning
Tim
e [s
]
0 5000 10000 15000
01000
2000
3000
4000
Number of Extreme Points
Run
ning
Tim
e [s
]
n = 20: Mean running times: OK10: 18.58s, Dual Benson: 0.12s33 of 38
Assignment Instances, d = 5
16 20 24 28 32 36
01000
2000
3000
4000
Number of Nodes
Run
ning
Tim
e [s
]
0 1000 2000 3000 4000
01000
2000
3000
4000
Number of Extreme Points
Run
ning
Tim
e [s
]
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MO Spanning Tree Problem Experiments
Input Graphs
Using graphs similar to [Johnson, Minkoff, Phillips 2000]
MST on points in the plane, parameter to adjust density
Instances here: mean density of 2
First objective function: Euclidean distance
Other objective functions: i.i.d. from 1, . . . , 10030 instances per number of nodes
lex-P Solver
Using Kruskal’s algorithm with lexicographic sorting, UNION FINDdata structure, path compression
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MO Spanning Tree Problem Experiments
Input Graphs
Using graphs similar to [Johnson, Minkoff, Phillips 2000]
MST on points in the plane, parameter to adjust density
Instances here: mean density of 2
First objective function: Euclidean distance
Other objective functions: i.i.d. from 1, . . . , 10030 instances per number of nodes
lex-P Solver
Using Kruskal’s algorithm with lexicographic sorting, UNION FINDdata structure, path compression
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Spanning Tree Instances, d = 3
30 50 70 90 110 130 150
05
1015
2025
30
Number of Nodes
Run
ning
Tim
e [s
]
0 2000 4000 6000 8000
05
1015
2025
30
Number of Extreme Points
Run
ning
Tim
e [s
]
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Spanning Tree Instances, d = 5
8 10 12 14 16 18 20 22 24
05000
15000
25000
Number of Nodes
Run
ning
Tim
e [s
]
0 2000 4000 6000 8000
05000
15000
25000
Number of Extreme Points
Run
ning
Tim
e [s
]
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Summary
General
Enumerating (parts of) the Pareto-frontier is an enumerationproblem
MOLP
Fixed d : Benson’s Algorithm and Dual Variant run in PTT
Improvements through finding of facets, Dual Variant in IncP,Speedup when using suggested improvements
MOCO
Fixed d : If P1 ∈ P (lex-P1 ∈ P): Finding of YX in PTT(IncP)
Dual algorithm is practically capable of finding YX for MOCOs38 of 38
Sorting Gap
200 400 600 800
05
1015
20
Spanning Tree Instances, d=3, a=.75
Number of Edges
Kru
skal
: Cum
ulat
ed R
unni
ng T
ime
[s]
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Hungarian Algorithm for Lex-AP
C ∈ Rd×n
lexmin Cx
i ∈ [n] :∑j∈[n]
xij = 1
j ∈ [n] :∑i∈[n]
xij = 1
xij ≥ 0
lexmax2n∑i=0
Ui
(i , j) ∈ V ×W : Ui + Uj ≤lex Cij
U ∈ Rd×2n
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