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The Art Gallery Problem is ∃ℝ-complete
The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard...
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| Published in: | Journal of the ACM 2022-02, Vol.69 (1), p.1-70 |
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| creator | Abrahamsen, Mikkel Adamaszek, Anna Miltzow, Tillmann |
| description | The
Art Gallery Problem
(AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer
k
, the goal is to decide if there exists a set
G
of
k
guards
within 풫 such that every point
p
∈ 풫 is seen by at least one guard
g
∈
G
. Each guard corresponds to a point in the polygon 풫, and we say that a guard
g
sees
a point
p
if the line segment
pg
is contained in 풫.
We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set
S
⊆ [0, 1]
2
, there exists a polygon with corners at rational coordinates such that for every
p
∈ [0, 1]
2
, there is a set of guards of minimum cardinality containing
p
if and only if
p
∈
S
.
In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems. |
| doi_str_mv | 10.1145/3486220 |
| format | article |
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Art Gallery Problem
(AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer
k
, the goal is to decide if there exists a set
G
of
k
guards
within 풫 such that every point
p
∈ 풫 is seen by at least one guard
g
∈
G
. Each guard corresponds to a point in the polygon 풫, and we say that a guard
g
sees
a point
p
if the line segment
pg
is contained in 풫.
We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set
S
⊆ [0, 1]
2
, there exists a polygon with corners at rational coordinates such that for every
p
∈ [0, 1]
2
, there is a set of guards of minimum cardinality containing
p
if and only if
p
∈
S
.
In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.</description><identifier>ISSN: 0004-5411</identifier><identifier>EISSN: 1557-735X</identifier><identifier>DOI: 10.1145/3486220</identifier><language>eng</language><publisher>New York: Association for Computing Machinery</publisher><subject>Algebra ; Art galleries & museums ; Computational geometry ; Guards ; Hardness ; Mathematical analysis ; Polygons ; Polynomials ; Real numbers</subject><ispartof>Journal of the ACM, 2022-02, Vol.69 (1), p.1-70</ispartof><rights>Copyright Association for Computing Machinery Feb 2022</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c984-f42d66d6ba89ecc60160562155a6af6a61d2868817cdabadee7675ea8610cded3</citedby><cites>FETCH-LOGICAL-c984-f42d66d6ba89ecc60160562155a6af6a61d2868817cdabadee7675ea8610cded3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>315,786,790,27957,27958</link.rule.ids></links><search><creatorcontrib>Abrahamsen, Mikkel</creatorcontrib><creatorcontrib>Adamaszek, Anna</creatorcontrib><creatorcontrib>Miltzow, Tillmann</creatorcontrib><title>The Art Gallery Problem is ∃ℝ-complete</title><title>Journal of the ACM</title><description>The
Art Gallery Problem
(AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer
k
, the goal is to decide if there exists a set
G
of
k
guards
within 풫 such that every point
p
∈ 풫 is seen by at least one guard
g
∈
G
. Each guard corresponds to a point in the polygon 풫, and we say that a guard
g
sees
a point
p
if the line segment
pg
is contained in 풫.
We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set
S
⊆ [0, 1]
2
, there exists a polygon with corners at rational coordinates such that for every
p
∈ [0, 1]
2
, there is a set of guards of minimum cardinality containing
p
if and only if
p
∈
S
.
In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.</description><subject>Algebra</subject><subject>Art galleries & museums</subject><subject>Computational geometry</subject><subject>Guards</subject><subject>Hardness</subject><subject>Mathematical analysis</subject><subject>Polygons</subject><subject>Polynomials</subject><subject>Real numbers</subject><issn>0004-5411</issn><issn>1557-735X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNotkMFKAzEYhIMouFbxFRY8CEI0ySZ_0mMptgoFPezBW8gm_2LLrrsm20PPevAlfLk-iSvtaRj4mBmGkGvO7jmX6qGQBoRgJyTjSmmqC_V2SjLGmKRKcn5OLlLajJYJpjNyV75jPotDvnRNg3GXv8auarDN1ynf_3ztv3-p79q-wQEvyVntmoRXR52QcvFYzp_o6mX5PJ-tqJ8aSWspAkCAypkpeg-MA1Mgxi0OXA0OeBAGjOHaB1e5gKhBK3QGOPMBQzEhN4fYPnafW0yD3XTb-DE2WgECpkoaI0fq9kD52KUUsbZ9XLcu7ixn9v8He_yh-AMNOU6i</recordid><startdate>20220228</startdate><enddate>20220228</enddate><creator>Abrahamsen, Mikkel</creator><creator>Adamaszek, Anna</creator><creator>Miltzow, Tillmann</creator><general>Association for Computing Machinery</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20220228</creationdate><title>The Art Gallery Problem is ∃ℝ-complete</title><author>Abrahamsen, Mikkel ; Adamaszek, Anna ; Miltzow, Tillmann</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c984-f42d66d6ba89ecc60160562155a6af6a61d2868817cdabadee7675ea8610cded3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebra</topic><topic>Art galleries & museums</topic><topic>Computational geometry</topic><topic>Guards</topic><topic>Hardness</topic><topic>Mathematical analysis</topic><topic>Polygons</topic><topic>Polynomials</topic><topic>Real numbers</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Abrahamsen, Mikkel</creatorcontrib><creatorcontrib>Adamaszek, Anna</creatorcontrib><creatorcontrib>Miltzow, Tillmann</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of the ACM</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Abrahamsen, Mikkel</au><au>Adamaszek, Anna</au><au>Miltzow, Tillmann</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The Art Gallery Problem is ∃ℝ-complete</atitle><jtitle>Journal of the ACM</jtitle><date>2022-02-28</date><risdate>2022</risdate><volume>69</volume><issue>1</issue><spage>1</spage><epage>70</epage><pages>1-70</pages><issn>0004-5411</issn><eissn>1557-735X</eissn><abstract>The
Art Gallery Problem
(AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer
k
, the goal is to decide if there exists a set
G
of
k
guards
within 풫 such that every point
p
∈ 풫 is seen by at least one guard
g
∈
G
. Each guard corresponds to a point in the polygon 풫, and we say that a guard
g
sees
a point
p
if the line segment
pg
is contained in 풫.
We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set
S
⊆ [0, 1]
2
, there exists a polygon with corners at rational coordinates such that for every
p
∈ [0, 1]
2
, there is a set of guards of minimum cardinality containing
p
if and only if
p
∈
S
.
In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.</abstract><cop>New York</cop><pub>Association for Computing Machinery</pub><doi>10.1145/3486220</doi><tpages>70</tpages></addata></record> |
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| language | eng |
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| source | EBSCOhost Business Source Ultimate; Association for Computing Machinery:Jisc Collections:ACM OPEN Journals 2023-2025 (reading list) |
| subjects | Algebra Art galleries & museums Computational geometry Guards Hardness Mathematical analysis Polygons Polynomials Real numbers |
| title | The Art Gallery Problem is ∃ℝ-complete |
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