SLIDE 1
Approximation Algorithms
A 2-approximation algorithm for TSP
c(TSP) c(MST) Double tree algorithm:
- 1. T ← MST
- 2. Double all edges in T
- 3. Etour ← Eurler tour
- 4. H ← vertices in order of appearance in Etour
DM865 (10 ECTS) Heuristikker og Approximationsalgoritmer - - PowerPoint PPT Presentation
DM865 (10 ECTS) Heuristikker og Approximationsalgoritmer - - PowerPoint PPT Presentation
DM865 (10 ECTS) Heuristikker og Approximationsalgoritmer [Heuristics and Approximation Algorithms] dm865.github.io Spring semester Lene Monrad Favrholdt Marco Chiarandini lektorer, IMADA Approximation Algorithms A 2 -approximation
SLIDE 2
SLIDE 3
Approximation Algorithms
A 2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP)
SLIDE 4
Approximation Algorithms
A 2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP) c(H) ≤ 2 · c(MST) ≤ 2 · c(TSP)
SLIDE 5
Approximation Algorithms
A 3/2-approximation algorithm for TSP
c(TSP) c(MST) Christofide’s algorithm:
- 1. T ← MST
- 2. M ← minimum perfect matching of odd degree vertices in T
- 3. Etour ← Euler tour in the subgraph (V , E(T) ∪ M)
- 4. H ← vertices in order of appearance in the Etour
SLIDE 6
Approximation Algorithms
A 3/2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP)
SLIDE 7
Approximation Algorithms
A 3/2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP) c(H) ≤ c(MST) + c(M)
SLIDE 8
Approximation Algorithms
A 3/2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP) c(H) ≤ c(MST) + c(M) ≤ c(TSP) + 1 2c(TSP)
SLIDE 9
Approximation Algorithms
A 3/2-approximation algorithm for TSP
c(TSP) c(MST) c(MST) ≤ c(TSP) c(H) ≤ c(MST) + c(M) ≤ c(TSP) + 1 2c(TSP) = 3 2 · c(TSP)
SLIDE 10
Approximation Algorithms
Theorem (2015)
For α < 185
184, there does not exist an α-approximation algorithm for the TSP.
SLIDE 11
Local Search
SLIDE 12
Local Search
SLIDE 13
Local Search
SLIDE 14
Local Search
SLIDE 15
Local Search
SLIDE 16
Metaheuristics
Accepting worsening changes
SLIDE 17
Metaheuristics
Accepting worsening changes
SLIDE 18
Metaheuristics
Accepting worsening changes
SLIDE 19
Metaheuristics
Accepting worsening changes
SLIDE 20
Metaheuristics
Accepting worsening changes
SLIDE 21
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 22
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 23
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 24
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 25
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 26
Metaheuristics
Accepting worsening changes Trying different changes
SLIDE 27
Contents
Apporx Algorithms Local Search + Metaheuristics Set Cover Satisfiability Traveling Salesman Scheduling Knapsack Bin packing
SLIDE 28
Course Formalities
Prerequisites: ✧ Programming (DM502, DM503, DM550) ✧ Algorithms and Datastructures (DM507) ✦ Complexity and Computability (DM508, DM553) ✦ Linear and Integer Programming (DM559, DM545, DM554) Credits: 10 ECTS Language: English or Danish Classes: intro: 2h × 24; training: 2h × 24 Material: slides + text book + articles + starting code
SLIDE 29
Assessment (10 ECTS)
- Two practical project assignments passed/failed with internal censor by the teacher
(include programming in Python)
- Oral exam based on:
- the theoretical part
- two practical assignments
Grading by the Danish 7-mark scale with external examiner. Exam aids allowed.
SLIDE 30