Guarding Polyominoes under k-hop Visibility or Minimum k ...

Guarding Polyominoes under k-hop Visibility or Minimum k-Dominating Sets in Grid GraphsChristiane Schmidt ICCG 2021, Yazd/Online, February 18, 2021

Research funded by Sweden’s Innovation Agency VINNOVA within the project NEtworK Optimization for CarSharing Integration into a Multimodal TRANSportation System (EkoCS-Trans)

Click to edit Master title styleAgenda

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• Motivation and Formal Problem Definition • NP-Completeness • Art Gallery Theorems • Open Problems

Click to edit Master title styleMotivation• Serve a city with carsharing (CS) stations: - Demand in granularity of square cells - Customers willing to walk a certain distance - Same distance bound for the complete city - City ➜ Polyomino - Walking only within the polyomino

• Goal: Place as few CS stations as possible to serve the complete city

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Image source: Gunnar Flötteröd, Waterborne Urban Mobility, Final Project Report

Click to edit Master title styleMotivation• So, what can a station serve? - All unit squares of the polyomino reachable

when walking when walking inside the polyomino for at most the given walking range - Walking range k - “Visibility”: We can look around corners for k ≥2

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k=5

Click to edit Master title styleFormal Definition• Polyomino: connected polygon P in plane,

formed by joining m unit squares on the square lattice • Dual graph GP is a grid graph • Unit square v∈P k-hop visible to unit square u∈P, if shortest path from u to v

in GP has length at most k. Minimum k-hop Guarding Problem (MkGP) Given: Polyomino P, range k Find: Minimum cardinality unit-square guard cover in P under k-hop visibility.

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k=3

Click to edit Master title styleAlternative FormulationMinimum k-dominating Set Problem (MkDSP) Given: Graph G Find: Minimum cardinality Dk ⊆ V(G), each graph vertex connected to vertex in Dk with a path of length at most k. MkDSP is NP-complete in general graphs. ➡ We want to solve MkDSP in grid graphs

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NP-Completeness

Click to edit Master title styleTheorem 1: MkGP is NP-complete for k=2 in polyominoes with holes. • Proof by reduction from PLANAR 3 SAT • Given a set of guards it can be verified in polynomial time whether each unit square of the polyomino is covered

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Variable gadget with two corridor gadgets: • In case variable appears in clause • In case negated variable appears in clause

“true” “false”Corridor propagates variable value

Click to edit Master title style

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Corridor bend:


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Clause gadget

FFF: 3 witnesses - pairwise disjoint visibility regions

3 guards (necessary and sufficient)

FFT:


FTT:


TTT:

No guard can see all three squares



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Theorem 1: MkGP is NP-complete for k=2 in polyominoes with holes. Equivalent formulation: Minimum 2-dominating Set Problem is NP-complete in grid graphs.

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Click to edit Master title styleTheorem 2: MkGP is NP-complete for k=1 in polyominoes with holes. • Proof by reduction from PLANAR 3 SAT • Given a set of guards it can be verified in polynomial time whether each unit square of the polyomino is covered

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Variable gadget with two corridor gadgets: • In case variable appears in clause • In case negated variable appears in clause

“true” “false”Corridor propagates variable value

Corridor bending works as before


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Clause gadget

FFF: 3 witnesses - pairwise disjoint visibility regions


FFT:


FTT:


TTT:

2 witnesses pairwise disjoint visibility regions



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Art Gallery Theorems

Click to edit Master title styleTheorem 3: There exist simple polyominoes with m unit squares that require guards to cover their interior under k-hop visibility.

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m not divisible by k+1 ➜ we add x=(m mod (k+1)) unit squares to the right of the shaft

k=1

k=6

Click to edit Master title styleTheorem 4: guards are always sufficient and sometimes necessary to cover a polyomino with m unit squares under k-hop visibility for k∈{1,2}.

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always sufficient

Proof: k=1: • Use dual grid graph GP • Compute a maximum matching M in the dual grid graph GP • Every vertex in GP that is not matched is adjacent to matched vertices only • For each matching edge {v,w} unmatched vertices only adjacent to v or w • For each matching edge: place guard at unit square adjacent to

unmatched vertices (if any)


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Proof: k=2: • Again: Compute a maximum matching M in the dual grid graph GP • Graph GM based on M:

- Vertex for each matching edge - Vertex for each unmatched vertex - {v,v’}∈V(GM) if:

- For v=v{u,u’}, v’=v’{w,w’}: if at least one of the edges {u,w}, {u,w’}, {u’,w} or {u’,w’} ∈E(GP)

- For v=v{u,u’}, v’=v’w: if at least one of {u,w} or {u,w’} ∈E(GP) • Compute maximum matching M’ in GM


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Proof: k=2: Case distinction

u' u

w w'

u'uw

w'

y

y'

y''

u'u

w w'

y y'

y''

e={v{u,u’}, v{w,w’}} ∈M’ not adjacent to any other unmatched vertex Single guard covers 4 unit squares

e={v{u,u’}, v{w,w’}} ∈M’ adjacent to unmatched vertices, all of which represent one vertex from GP.

W.l.o.g. all unmatched vertices adjacent to v{u,u’}

{u,w}∈ E(GP) or {u,w’}∈ E(GP), but {u’,w}∉E(GP) and {u’,w’}∉E(GP) Single guard on u covers u, u’, w, w’ and all unmatched vertices adjacent to covered v{u,u’} Single guard covers at least 5 unit squares

{u,w}∈ E(GP) and {u’,w’}∈ E(GP) (or {u’,w}∈ E(GP) and {u’,w’}∈ E(GP)) Single guard on u (or u’) covers u, u’, w, w’ and all unmatched vertices adjacent to covered v{u,u’} Single guard covers at least 5 unit squares


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u'uw

w' x

x'

u'u

w

u'uwy

y'

Two guards cover at least 6 unit squares

Single guard covers 3 unit squares

Single guard covers at least 4 unit squares

w u'u

y

x

x'

w u'u

y

x

x'



w u'u

y

z

z'

x x'

u'

u

x

x'w


Two guards cover at least 6 unit squares

➜Each guard covers at least 3 unit squares ➜ ⌊m/(k+1)⌋=⌊m/3⌋ guards always sufficient

Open Problems

Click to edit Master title styleOpen Problems

• Computational complexity in simple polyominoes (grid graphs without holes) • Upper bounds on the number of guards for k≥3 • Approximation algorithms (k=1: trivial 5/2-approximation)

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THANKS

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