Given a simple polygon P on n vertices, two points x,y in P are said to be visible to each other if the line segment between x and y is contained in P. The Point Guard Art Gallery problem asks for a minimum set S such that every point in P is visible from a point in S.
The Vertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the set S is referred to as a guard.   For both variants, we rule out a f(k)*n^{o(k/log k)} algorithm, for any computable function f, where k := |S| is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match the n^{O(k)} algorithms that exist for both problems

Bonnet, Édouard

Miltzow, Tillmann

SZTAKI Publication Repository

EuroCG 2016, Lugano, Switzerland, March 30–April 1, 2016Parameterized Hardness of Art Gallery ProblemsE´douard Bonnet ∗ Tillmann Miltzow †AbstractGiven a simple polygon P on n vertices, two pointsx, y in P are said to be visible to each other if theline segment between x and y is contained in P. Thepoint guard art gallery problem asks for a minimumset S such that every point in P is visible from a pointin S. The vertex guard art gallery problem asks forsuch a set S subset of the vertices of P. The set Sis referred to as guards. We show W [1]-hardness ofboth variants, when parameterized by the number kof guards. We even rule out any no(k/ log k) algorithmunder the exponential time hypothesis.1 IntroductionGiven a simple polygon P on n vertices, two pointsx, y in P are said to be visible to each other if theline segment between x and y is contained in P. Thepoint-guard art gallery problem asks for a minimumset S of points called guards such that every point inP is visible from a point in S. The vertex guard artgallery problem asks for such a set of guards S subsetof the vertices of P.One of the first combinatorial results is the elegantproof of Fisk that bn/3c guards are always sufficientand sometimes necessary for a polygon with n ver-tices [8]. On the algorithmic side, very few variantsare solvable in polynomial time [5, 11], but most re-sults are on approximating the minimum number ofguards [3, 4, 6, 9]. On the lower bound side the paperof Eidenbenz et al. showed for most relevant variantsNP-hardness and inapproximability [7]. In particular,their reduction from Set-Cover implies that the artgallery is W [2]-hard on polygons with holes and thatthere is no no(k) algorithm, to determine if k guardsare sufficient for a given gallery with n vertices, underthe exponential time hypothesis [7, Sec.4]. However,polygons with holes are very different to simple poly-gons as they have unbounded VC-dimension [12]. Inparticular none of these reductions rule out a fixed pa-rameter tractable algorithm (i.e., whose running timeis O(f(k)nc) where f is any computable function andc is a constant) for simple polygons (see [2] for an∗Institute for Computer Science and Control,Hungarian Academy of Sciences (MTA SZTAKI),edouard.bonnet@lamsade.dauphine.fr†Institute for Computer Science and Control, HungarianAcademy of Sciences (MTA SZTAKI), t.miltzow@gmail.comintroduction to parameterized complexity.).Obviously, the vertex guard variant can be solvedin time O(nk+2) by trying out all possible subsets ofsize k of the vertices and checking if one of those sub-sets sees the whole polygon. Not obvious at all is thealgorithm running in time nO(k) for the point guardvariant using standard tools from real algebraic geom-etry [1]. Despite the fact that the first algorithm isextremely basic and the second algorithm, even withremarkably sophisticated tools, uses almost no prob-lem specific insights, no better exact parameterizedalgorithms are known.We present the first conditional lower bounds forthe parameterized art gallery problem for simple poly-gons:Theorem 1 (Point guard hardness) PointGuard Art Gallery parameterized by the numberof guards k is W [1]-hard, and is not solvable in timeno(k/ log k), under the ETH.Theorem 2 (Vertex guard hardness) VertexGuard Art Gallery is W [1]-hard, and is notsolvable in time no(k/ log k), under the ETH.2 PreliminariesFor any two integers x < y, we set [x, y] := {x, x +1, . . . , y − 1, y}, and for any positive integer x, [x] :=[1, x]. The Exponential Time Hypothesis (ETH) is aconjecture by Impagliazzo et al. [10] asserting thatthere is no 2o(n)-time algorithm for 3-SAT on in-stances with n variables.Polygons and visibility. For any two distinct pointsv and w in the plane, we denote by seg(v, w) the seg-ment whose two endpoints are v and w, by ray(v, w)the ray starting at v and passing through w, by `(v, w)the supporting line passing through v and w.A polygon is simple if it is not self-crossing and hasno holes. For any point x in a polygon P, VP(x), orsimply V (x), denotes the visibility region of x withinP, that is the set of all the points y ∈ P such thatsegment seg(x, y) is entirely contained in P. We saythat two vertices v and w of a polygon P are neighborsor consecutive if vw is an edge of P. A subpolygon P ′of a simple polygon P is defined by any l distinct con-secutive vertices v1, v2, . . . , vl of P (that is, for everyThis is an extended abstract of a presentation given at EuroCG 2016. It has been made public for the benefit of the community and should be considered apreprint rather than a formally reviewed paper. Thus, this work is expected to appear in a conference with formal proceedings and/or in a journal.32nd European Workshop on Computational Geometry, 2016i ∈ [l − 1], vi and vi+1 are neighbors in P) such thatv1vl does not cross any edge of P.Given a vertex v and two points p and p′, we calltriangular pocket rooted at vertex v and supported byray(v, p) and ray(v, p′) a sub-polygon w, v, w′ suchthat ray(v, w) passes through p, ray(v, w′) passesthrough p′. We say that v is the root of the trian-gular pocket that we denote P(v). We also say thatthe pocket P(v) points towards p and p′.Structured 2-Track Hitting Set. We introduce anew problem which will constitute a handy startingpoint to show Theorem 1 and 2. In the 2-Track Hit-ting Set problem, the input consists of an integer k,two sets A and B of the same cardinality totally or-dered by ≤A and ≤B , and two sets SA of A-intervals(that is a set of consecutive elements of A according to≤A), and SB of B-intervals. In addition, the elementsof A and B are in one-to-one correspondence φ : A→B and each pair (a, φ(a)) is called a 2-element. Thegoal is to find a set S of k 2-elements such that the firstprojection of S is a hitting set of A, and the secondprojection of S is a hitting set of B. Structured2-Track Hitting Set is the same problem withcolor classes over the 2-elements, and a restriction onthe one-to-one mapping φ. A is partitioned into kclasses (C1, C2, . . . , Ck) where Cj = {aj1, aj2, . . . , ajt}for each j ∈ [k], where |A| = tk, and is or-dered: a11, a12, . . . , a1t , a21, a22, . . . , a2t , . . . , ak1 , ak2 , . . . , akt .We define C ′j := φ(Cj) and bji := φ(aji ) for all i ∈ [t]and j ∈ [k]. We now impose that φ is such that, foreach j ∈ [k], the t elements of C ′j are consecutive along≤B . That is, B is ordered: C ′σ(1), C ′σ(2), . . . , C ′σ(k) forsome permutation on [k], σ ∈ Sk. For each j ∈ [k],the order of the elements within C ′j can be describedby a permutation σj ∈ St such that the ordering of C ′jis: bjσj(1), bjσj(2), . . . , bjσj(t). Due to space limitations,we omit the proof of the following theorem.Theorem 3 Structured 2-Track Hitting Setis W [1]-hard, and not solvable in time |I|o(k/ log k),unless the ETH fails.3 Point GuardOverview of the reduction. Given an instance I =(k ∈ N, t ∈ N, σ ∈ Sk, σ1 ∈ St, . . . , σk ∈ St,SA,SB),we build a simple polygon P with O(kt+ |SA|+ |SB |)vertices, such that I is a YES-instance iff P can beguarded by 3k points.The global strategy of the reduction is to allocate,for each color class j ∈ [k], 2t special points in thepolygon αj1, . . . , αjt and βj1, . . . , βjt . Placing a guard inαji (resp. βji ) shall correspond to picking a 2-elementwhose first (resp. second) component is aji (resp. bji ).z1z2z3z4p1p2p3p4p5p6Figure 1: Interval gadgets encoding {p1, p2, p3},{p2, p3, p4, p5}, {p4, p5}, and {p4, p5, p6}.The points αji ’s and βji ’s ordered by increasing y-coordinates will match the order of the aji ’s along ≤Aand then of the bji ’s along ≤B . Then, far in the hori-zontal direction, we will place pockets to encode eachA-interval of SA, and each B-interval of SB (see Fig-ure 1).The critical issue will be to link point αji to pointβji . Indeed, in the Structured 2-Track HittingSet problem, one selects 2-elements (one per colorclass), so we should prevent one from placing twoguards in αji and βji′ with i 6= i′. Due to a techni-cality, we will introduce a copy αji of each αji . In eachpart of the gallery encoding a color class j ∈ [k], theonly way of guarding all the pockets with only threeguards is to place them in αji , αji , and βji for somei ∈ [t]. Hence, 3k guards will be necessary and suffi-cient to guard the whole P iff there is a solution to theinstance of Structured 2-Track Hitting Set.We now sketch the construction.Allocated points and interval gadgets. The posi-tion of the αji ’s and βji can be seen on Figure 2 andFigure 4. It is such that the ordering of the αji ’s(resp. βji ) by increasing y-coordinate matches the or-der ≤A on the aji ’s (resp. ≤B on the bji ’s). Also,αji and βji shares the same x-coordinate for eachj ∈ [k], i ∈ [t]. There is a quite large gap D alongthe x-axis between a point αjt and αj+1t .For each A-interval Iq = [aji , aj′i′ ] ∈ SA, we put, ata very large distance F to the right of the αji ’s, onetriangular pocket P(zA,q) rooted at vertex zA,q andsupported by ray(zA,q, αji ) and ray(zA,q, αj′i′ ). Thisway, the only αj′′i′′ seeing vertex zA,q are all the pointssuch that aji ≤A aj′′i′′ ≤A aj′i′ (see Figure 1 and Fig-ure 4). We do the same for the B-intervals.Weak linkers. We now describe how we link eachpoint αji to its associate βji . See Figure 2 for a de-scription of the following weak linker gadget.For each j ∈ [k], let us mentally draw ray(αjt , βj1)and consider points slightly to the left of this rayand quite far. Let us call Rjleft that informal re-gion of points. Any point in Rjleft sees, from rightto left, in this order αj1, αj2 up to αjt , and then,EuroCG 2016, Lugano, Switzerland, March 30–April 1, 2016α1α2α3α4α5α6β1β2β3β4β5β6d1c1 d2c2 d3c3 d4c4 d5c5 d6c6Figure 2: Weak point linker gadget.βj1, βj2 up to βjt . In Rjleft, for each i ∈ [t − 1], weplace a triangular pocket P(cji ) rooted at vertex cjiand supported by ray(cji , αji+1) and ray(cji , βji ). Weplace also a triangular pocket P(cjt ) rooted at cjt sup-ported by ray(cji , βj1) and ray(cji , βjt ). We place mir-roring pockets P (dj1), . . . ,P(djt ) in the region slightlyto the right of ray(αj1, βjt ) and quite far. Finally, weadd a thin rectangular pocket Pj,r to the left of thepoints αj1, . . . , αjt such that the uppermost longer sideof the rectangular pocket lies on the line `(αj1, αjt )(see Figure 2). We denote by Pj,α,β the set of pockets{P(cj1), . . . ,P(cjt ),P(dj1), . . . ,P(djt )} and call it weaklinker.If one wants to guard Pj,α,β with only two pointsand place the first guard on αji , one is not forced toplace the second guard on βji , as we would desire, butanywhere on an area whose uppermost point is βji (seethe shaded areas below the bji ’s in Figure 2).Linkers. For each j ∈ [k], we allocate t pointsαj1, . . . , αjt on a horizontal line above and to the rightof βjt at a quite large distance. We add two weaklinkers Pj,α,α and Pj,α,β , one linking αj1, . . . , αjt andαj1, . . . , αjt , the other linking αj1, . . . , αjt and βj1, . . . , βjt(see Figure 3). We also add a thin horizontal pocketwhose lowermost side is in the same line as the pointsαj1, . . . , αjt . Pockets of Pj,α,α and the two thin rect-angular pockets force to put guards on αji and αji (fora same i ∈ t), if we have only two guards to spare.Now, pockets of Pj,α,β forces to place the third guardbelow βji while pockets of Pj,α,β forces to place thethird guard above βji (again if we have only threeguards to spare). So, the only solution is to place thethird guard exactly on βji . The k linkers are placedaccordingly to Figure 4.Lemma 4 ∀j ∈ [k], ∀i ∈ [t], the three associatepoints αji1 , αji2, βji3 guard entirely Pj iff i1 = i2 = i3.Figure 3: Point linker gadget: a triangle of (three)weak point linkers.F. . .F. . .DP1,α,β P2,α,β P3,α,βP1,α,α P2,α,α P3,α,αP1,α,βP2,α,βP3,α,βLP1,rP2,rP3,rP1,rP2,rP3,rtrack 1track 2ααβFigure 4: The overall picture of the reduction withk = 3.4 Vertex GuardThe reduction is again from Structured 2-TrackHitting Set.Vertex linkers. For each j ∈ [k], permutation σjis encoded by a sub-polygon Pj that we call vertexlinker, or simply linker (see Figure 5). We regularlyset t consecutive vertices αj1, αj2, . . . , αjt in this order,along the x-axis. Opposite to this segment, we placet vertices βjσj(1), βjσj(2), . . . , βjσj(t) in this order, alongthe x-axis, too. The βjσj(1), . . . , βjσj(t), contrary toαj1, . . . , αjt , are not consecutive. We put reflex verticesin between the vertices βjσj(1), . . . , βjσj(t)to ensure thatthe only way of seeing entirely the walls djej and xjyjby taking two vertices αji and βji′ is that i = i′.Lemma 5 For any j ∈ [k], the sub-polygon Pj is seenentirely by {αjv, βjw} iff v = w.What we should now prevent is that one puts aguard in a reflex vertex of the linker.Filter gadget. The only way to see all the pock-ets of the filter gadget Fj (see Figure 6) with twoguards is to place them on ci and di for the same i.In the overall construction, the ci’s are in fact ver-tices βjσj(1), . . . , βjσj(t). Thus, if one wants to guard32nd European Workshop on Computational Geometry, 2016α1 α2 α3 α4 α5 α6β4 β2 β5 β3 β6 β1r↓defacr↑ xyhgbp1p2p3p4 q3q4q5q6Figure 5: Vertex linker gadget. We omitted the su-perscript j in all the labels. Here, σj(1) = 4, σj(2) =2, σj(3) = 5, σj(4) = 3, σj(5) = 6, σj(6) = 1.all the pockets of Fj and Pj with only three guards,one should place them at vertices αji , βji , and djσj(i).d1d2d3d4d5d6x1x2x3x4x5x6c1 c2 c3 c4 c5 c6y1y2y3y4y5Figure 6: The filter gadget Fj .Overall construction. Permutation σ is encoded inthe way depicted on Figure 7 by limiting the visibil-ity of the vertices βjσj(1), . . . , βjσj(t)to only one filtergadget, namely Fj . Finally, as for the point guardvariant, for each A- and B-interval, we place a tri-angular pocket seeing the corresponding vertices (seeTrack 1 and 2 of Figure 7).AcknowledgmentsBoth authors are supported by the ERC grantPARAMTIGHT: ”Parameterized complexity and thesearch for tight complexity results”, no. 280152. We thankMeirav Zehavi and Saeed Merhabi for useful discussionsduring a preliminary stage of the project.Track 1Track 2P1P2P3P4P5F5F3F1F4F2Figure 7: Overall picture of the reduction with k = 5.References[1] S. Basu, R. Pollack, and M.-F. Roy. Algorithms inreal algebraic geometry. AMC, 10:12, 2011.[2] M. Cygan, F. V. Fomin, L. Kowalik, D. Lokshtanov,D. Marx, M. Pilipczuk, M. Pilipczuk, and S. Saurabh.Parameterized Algorithms. Springer, 2015.[3] A. Deshpande. A pseudo-polynomial time O(log2 n)-approximation algorithm for art gallery problems.Master’s thesis, Department of Mechanical Engineer-ing, Department of Electrical Engineering and Com-puter Science, MIT, 2006.[4] A. Deshpande, T. Kim, E. D. Demaine, andS. E. Sarma. A pseudopolynomial time O(logn)-approximation algorithm for art gallery problems. InWADS 2007, pages 163–174, 2007.[5] S. Durocher and S. Mehrabi. Guarding orthogonal artgalleries using sliding cameras: algorithmic and hard-ness results. In MFCS 2013, pages 314–324. Springer,2013.[6] A. Efrat and S. Har-Peled. Guarding galleries andterrains. Inf. Process. Lett., 100(6):238–245, 2006.[7] S. Eidenbenz, C. Stamm, and P. Widmayer. Inap-proximability results for guarding polygons and ter-rains. Algorithmica, 31(1):79–113, 2001.[8] S. Fisk. A short proof of chva´tal’s watchman theorem.J. Comb. Theory, Ser. B, 24(3):374, 1978.[9] S. K. Ghosh. Approximation algorithms for artgallery problems in polygons. Discrete Applied Math-ematics, 158(6):718–722, 2010.[10] R. Impagliazzo and R. Paturi. Complexity of k-sat. InComputational Complexity, 1999. Proceedings. Four-teenth Annual IEEE Conference on, pages 237–240.IEEE, 1999.[11] R. Motwani, A. 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Parameterized Hardness of Art Gallery Problems

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Parameterized Hardness of Art Gallery Problems

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