slide1
Download
Skip this Video
Download Presentation
The Ferry Cover Problem

Loading in 2 Seconds...

play fullscreen
1 / 134

The Ferry Cover Problem - PowerPoint PPT Presentation


  • 117 Views
  • Uploaded on

The Ferry Cover Problem. Michael Lampis - Valia Mitsou National Technical University of Athens. Wolf. Goat. Cabbage. Guard. Boat. Previous Work. “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D (in latin). We propose a generalization of Alcuin’s puzzle.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' The Ferry Cover Problem' - shania


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

The Ferry Cover Problem

Michael Lampis - Valia Mitsou

National Technical University of Athens

previous work
Previous Work
  • “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D (in latin).
  • We propose a generalization of Alcuin’s puzzle
our generalization1
Our generalization
  • We seek to transport n items, given their incompatibility graph.
  • Objective: Minimize the size of the boat
  • We call this the Ferry Cover Problem
the ferry cover problem
The Ferry Cover Problem

Lemma:

OPTVC (G) ≤ OPTFC (G) ≤ OPTVC (G) + 1

Graphs are divided into two categories:

  • Type-0, ifOPTFC (G) = OPTVC (G)
  • Type-1, if OPTFC (G) = OPTVC (G) + 1
hardness and approximation results
Hardness and Approximation Results
  • Ferry Cover is NP and APX-hard (like Vertex Cover [Håstad1997]).
  • A ρ-approximation algorithm for Vertex Cover yields a (ρ+1/ OPTFC)-approximation algorithm for Ferry Cover.
ferry cover on trees
Ferry Cover on Trees

Lemma:

For trees with OPTVC (G) > 1 (i.e. not stars)

OPTFC (G) = OPTVC (G) (Type-0)

slide61

u

w

v

slide62

u

v

w

slide63

u

v

w

slide64

u

v

w

slide65

u

v

w

slide66

u

v

w

slide67

u

v

w

slide68

u

v

w

slide69

u

v

w

slide70

u

v

w

slide71

u

v

w

slide72

u

v

w

slide73

u

v

w

slide74

u

v

w

slide75

u

v

w

slide76

u

v

w

slide77

u

v

w

slide78

v

w

u

slide79

v

w

u

slide80

v

w

u

slide81

v

w

u

slide82

v

w

u

slide83

v

w

u

slide84

v

w

u

slide85

v

w

u

slide86

u

w

v

slide87

u

w

v

slide88

u

w

v

ferry cover on trees1
Ferry Cover on Trees
  • For a star with three or more leaves

OPTFC (G) = OPTVC (G)+1 = 2 (Type-1)

  • For any other tree

OPTFC (G) = OPTVC (G) (Type-0)

Fact:

The Vertex Cover Problem can be solved in Polynomial time on trees.

ferry cover on trees2
Ferry Cover on Trees

Theorem:

The Ferry Cover Problem can be solved in polynomial time on trees.

trip constrained ferry cover
Trip Constrained Ferry Cover
  • Variation of Ferry Cover: we are also given a trip constraint. We seek to minimize the size of the boat s.t. there is a solution within this constraint.
  • Definition: FCi→ determine the minimum boat size s.t. there is a solution with at most 2i+1 trips (i round-trips).
slide93
FC1
  • An interesting special case: only one round-trip allowed.
  • FC1 is NP-hard.
  • 2-approximation for general graphs.
  • A (4/3+ε)-approximation for trees.
h colorings

1

2

3

H-Colorings
  • A traditional 3-coloring of graph G:

Vertices of color 2 are connected with vertices of colors 1 or 3

H :

h colorings1
H-Colorings
  • A constrained 3-coloring of graph G:

Vertices of color 2 are only connected with vertices of color 3

1

H :

2

3

h colorings2

1

2

3

H-Colorings
  • A loose 3-coloring of graph G:

Vertices of color 2 can be connected with any vertex.

H :

fc 1 as an f 1 coloring problem
FC1 as an F1-Coloring problem

Vertices are partitioned into 3 groups:

  • Those loaded and unloaded on the first trip
  • Those remaining on the boat for all three trips
  • Those loaded and unloaded on the third trip

1

2

F1

3

fc 1 as an f 1 coloring problem1
FC1 as an F1-Coloring problem
  • Boat size is |V2|+ max{|V1|, |V3|}
  • FC1 is equivalent to finding an F1-coloring that minimizes the above function.

1

2

F1

3

fc 1 is np hard1
FC1 is NP-hard

Via a reduction from NAE3SAT

Sketch:

  • Given a NAE3SAT formula φ with m clauses, create a new formula φ’
  • From φ’ create a graph G
  • G has an F1-coloring of cost 7m iff φ is satisfiable.
reduction step 1
Reduction: Step 1

In NAESAT

For example:

Then:

reduction step 2
Reduction: Step 2
  • For every clause construct a triangle.
  • For every variable construct a complete bipartite graph.
  • Connect each triangle vertex to one corresponding bipartite vertex.
step 3 if is satisfiable
Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1

step 3 if is satisfiable1
Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1

step 3 if is satisfiable2
Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1

step 3 if is satisfiable3
Step 3: If φ’ issatisfiable

false

true

true

Cost = 7m = 2m + 2m + 3m

false

...

...

step 3 if cost 7m
Step 3: If cost=7m
  • Observe that 7m is the minimum possible cost.
  • It is possible to show that a coloring of this cost is a coloring of the previous form.
  • Therefore, φ is satisfiable.
  • Bonus: This reduction is also gap-preserving. Therefore, FC1 is APX-hard.
2 approximation
2-approximation
  • The boat arrives to the destination bank twice.
  • Therefore, its size must be at least n/2
  • A boat of size n is a 2-approximation!
4 3 approximation for fc 1 on trees boat size 2n 3
(4/3+ε)– approximation for FC1 on trees (boat size 2n/3)

Fact: For a tree G OPTVC(G) ≤ n/2 (because tree is a bipartite graph)

ALGORITHM

  • Load a vertex cover of size 2n/3.
  • Unload n/3 vertices that form an Independent Set and return.
  • Load the remaining vertices and transfer all of them to the destination.
results for the trip constrained ferry cover problem

i:

0

1

2

n-2

n-1

2n-1

Results for the Trip Constrained Ferry Cover Problem

NP-hard

≡FC

?

Trivial

NP-hard

further work
Further Work
  • Is it NP-hard to determine whether a graph G is Type-0 or Type-1?
  • Is FC equivalent to FCn?
  • Is FCi for 1 < i < n-1 polynomially solved?
  • Can we have an efficient approximation of FC1 in the general case?
ad