D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola...

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Advantage in the discrete Voronoi game

Daniel Gerbner

Renyi Institute

Joint work with Viola Meszaros, Domotor Palvolgyi, AlexeyPokrovskiy and Gunter Rote

Daniel Gerbner Advantage in the discrete Voronoi game

Advantage in the discrete Voronoi game Daniel Gerbner

What is the Voronoi game?

Applet by Jens Anuth.

Two players, First and Second claim points alternating for trounds.

At end area is divided, each point goes to closest claimed.

Competitive facility location problem

Two chains of supermarkets build shops in a city.

The customers always go to the nearest shop.

What is the discrete Voronoi game?

Same on a graph!

Players can claim only vertices and vertices are divided at end.

So the above game ended in a draw.

Theorem (Kiyomi, Saitoh, Uehara)

Game on path is a draw unless odd vertices and t = 1.

Moreover, even then First wins with only one.

Same on a graph!

Game on path is a draw unless odd vertices and t = 1.Moreover, even then First wins with only one.

What percentage can each player get?

Definition

VR(G , t) =# Closer to First + 1

2# Tied

Goal: Bound VR(G , t) for certain graph(family)

VR(path, t) ≥ 12 and → 1

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Definition

2# Tied

VR(star , t)→ 1

Can the second player win?

Questions

VR(G , t) < ε?

What if t = 1?

VR(T , t) < 12 for a tree?

ClaimVR(T , 1) ≥ 1

2 for trees.

Proof.First takes center of tree.

(A vertex which cuts the tree intoconnected components of order at most n/2.)

Questions

VR(G , t) < ε?

What if t = 1?

2 for trees.

Proof.First takes center of tree. (A vertex which cuts the tree intoconnected components of order at most n/2.)

Questions

VR(G , t) < ε?

What if t = 1?

2 for trees.

Questions

VR(G , t) < ε?

What if t = 1?

2 for trees.

Questions

VR(G , t) < ε?

What if t = 1?

2 for trees.

Questions

VR(G , t) < ε?

What if t = 1?

2 for trees.

For any t ≥ 2 there exists a tree T with VG (T , t) < 0.34:

2 4 8kN

xlegshead

Basic idea: most of the legs are controlled by the player whoclaims the last vertex there; the head is controlled by the playerwho claims h.

Optimal for t = 2: VG (T , 2) > 1/3 for any tree T .

2 4 8kN

xlegshead

2 4 8kN

xlegshead

2 4 8kN

xlegshead

Strategy stealing

TheoremFor every graph and t we have

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.

First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game v

then plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy

(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).

Cannot make last pick v ′ of Second’s strategybut v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategy

but v is there.If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

Proof of Lower bound.First starts with best pick in one round game vthen plays Second’s strategy(ignoring own first pick).Cannot make last pick v ′ of Second’s strategybut v is there.

If he could now change v to v ′, he could notcontrol more than 1− VR(G , 1) new vertices,as a one round game with v and v ′ shows.

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

VR(G , 1)

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

< VR(G , 1)/2

> 1− VR(G , 1)/2

Strategy stealing

2VR(G , 1) ≤ VR(G , t) ≤ 1

2(VR(G , 1) + 1).

< VR(G , 1)/2

> 1− VR(G , 1)/2

> 1− VR(G , 1)

TheoremFor every graph and t we have 1

2VR(G , 1) ≤ VR(G , t).

Corollary

For every tree and t we have VR(T , t) ≥ 14 .

TheoremFor every tree we have VR(T , 2) > 1

Where is truth for t > 2 between 14 and 1

2VR(G , 1) ≤ VR(G , t).

Corollary

2VR(G , 1) ≤ VR(G , t).

Corollary

2VR(G , 1) ≤ VR(G , t).

Corollary

VR(G , t) < ε

TheoremFor all t and ε there is G with VR(G , t) < ε.

Proof for t = 1 and semicontinuous case.Play on d-dimensional simplex with weights on vertices.

VR(G , t) < ε

Proof for t = 1 and semicontinuous case.

Play on d-dimensional simplex with weights on vertices.

VR(G , t) < ε

�Daniel Gerbner Advantage in the discrete Voronoi game

Summary

For treesif t = 1 then VR(T , 1) ≥ 1

2 sharpif t = 2 then VR(T , 2) > 1

3 sharpif t ≥ 3 then 1

4 ≤ VR(T , t) and VR(T , t) < 13 + ε possible.

For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?

Equivalent problem: Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?

SummaryFor treesif t = 1 then VR(T , 1) ≥ 1

For general graphsVR(G , t) < ε possible for all t.

What if First makes t moves and Second makes 1?Is VR(G , t : 1) < ε?

For general graphsVR(G , t) < ε possible for all t.What if First makes t moves and Second makes 1?

Is VR(G , t : 1) < ε?

Equivalent problem:

Is there a finite function family, F , such thatfor any f1, . . . , ft ∈ F there is a g ∈ F such thatg > max(f1, . . . , ft) on (1− ε) fraction of inputs?

Thank you for your attention!

D aniel Gerbner - 上海交通大学数学系D aniel Gerbner R enyi Institute Joint work with Viola...

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