Parameterized Hardness of Art Gallery Problems

Édouard Bonnet

and Tillmann Miltzow

1 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

edouard.bonnet@lamsade.dauphine.fr

2 Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

t.miltzow@gmail.com

Abstract

Given a simple polygonPonnvertices, two pointsx, yinPare said to be visible to each other if the line segment betweenxandyis contained inP. ThePoint Guard Art Galleryproblem asks for a minimum setSsuch that every point inP is visible from a point inS. TheVertex Guard Art Gallery problem asks for such a set S subset of the vertices of P. A point in the setS is referred to as a guard. For both variants, we rule out af(k)n^o(k/logk) algorithm, for any computable functionf, wherek:=|S|is the number of guards, unless the Exponential Time Hypothesis fails. These lower bounds almost match then^O(k) algorithms that exist for both problems.

1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

Keywords and phrases art gallery problem, computational geometry, parameterized complexity, ETH-based lower bound, geometric set cover/hitting set

Digital Object Identifier 10.4230/LIPIcs.ESA.2016.19

1 Introduction

Given a simple polygonP onnvertices, two pointsx, yinP are said to be visible to each other if the line segment between x andy is contained in P. The Point Guard Art Galleryproblem asks for a minimum set S such that every point inP is visible from a point inS. TheVertex Guard Art Galleryproblem asks for such a setSsubset of the vertices ofP. The setS is referred to as guards. In what follows,nrefers to the number of vertices ofP andk to the size of an optimal set of guards.

The art gallery problem is arguably one of the most well-known problems in discrete and computational geometry. Since its introduction by Viktor Klee in 1976, three books [32, 34, 14]

and two extensive surveys appeared [33, 8]. O’Rourke’s book from 1987 has over a thousand citations, and each year, top conferences publish new results on the topic. Many variants of the art gallery problem, based on different definitions of visibility, restricted classes of polygons, different shapes of guards, have been defined and analyzed. One of the first results is the elegant proof of Fisk thatbn/3cguards are always sufficient and sometimes necessary for a polygon withnvertices [12].

NP-hardness and APX-hardness have been shown for many variants of the art gallery problem and other related problems [11, 23, 4, 26]. Due to those negative results, most

∗ Supported by the ERC grant PARAMTIGHT: “Parameterized complexity and the search for tight complexity results”, no. 280152.

© Édouard Bonnet and Tillmann Miltzow;

licensed under Creative Commons License CC-BY

papers concentrate on finding approximation algorithms and restrictions that are polynomially tractable [15, 25, 24, 30, 26]. However, considering the recent lack of progress in this direction, the study of other approaches becomes interesting. One such approach is finding heuristics to solve large instances of the art gallery problem [8]. The fundamental drawback of this approach is the lack of performance guarantee.

In the last twenty-five years, a fruitful paradigm, parameterized complexity, has been gaining some popularity. The underlying idea is to study algorithmic problems with de- pendence on a natural parameter. If the dependence on the parameter is practical and the parameter is small for real-life instances, one gets algorithms that give optimal solutions with reasonable running times. For a gentle introduction to parameterized complexity, we recommend Niedermeier’s book [31]. For a thorough reading highlighting complexity classes, we suggest the book by Downey and Fellows [9]. For a recent book on the topic with an emphasize on algorithms, we advise to read the book by Cygan et al. [6]. An approach based on logic is given by Flum and Grohe [13]. Despite the recent successes of parameterized complexity, only very few results on the art gallery problem are known.

The first such result is the trivial algorithm for the vertex guard variant to check if a solution of sizekexists in a polygon withnvertices. The algorithm runs inO(n^k+2) time, by checking all possible subsets of sizekof the vertices. The secondnot so well-knownresult is the fact that one can find in timen^O(k) a set ofkguards for the point guard variant, if it exists [10], using tools from real algebraic geometry [2]. This was first observed by Sharir [10, Acknowledgment]. Despite the fact that the first algorithm is extremely basic and the second algorithm, even with remarkably sophisticated tools, uses almost no problem-specific insights, no better exact parameterized algorithms are known.

The Exponential Time Hypothesis (ETH) asserts that there is no 2^o(N)time algorithm forSatonN variables. The ETH is used to attain more precise conditional lower bounds than the mere NP-hardness. A simple reduction fromSet Coverby Eidenbenz et al. shows that there is non^o(k)algorithm for these problems, when we consider polygons with holes [11, Sec.4], unless the ETH fails. However, polygons with holes are very different from simple polygons. For instance, they have unbounded VC-dimension while simple polygons have bounded VC-dimension [35, 20, 22, 19]. Our contribution is to show that, even on simple polygons, one cannot expect a large improvement over then^O(k) algorithms. More precisely, we prove:

ITheorem 1(Parameterized hardness point guard). Assuming the ETH,Point Guard Art Galleryis not solvable in time f(k)n^o(k/logk), for any computable function f, even on simple polygons, wheren is the number of vertices of the polygon andk is the number of guards allowed.

ITheorem 2(Parameterized hardness vertex guard). Assuming the ETH, Vertex Guard Art Galleryis not solvable in timef(k)n^o(k/logk), for any computable functionf, even on simple polygons, where nis the number of vertices of the polygon and kis the number of guards allowed.

Our reductions are fromSubgraph Isomorphism. Therefore an algorithm solving the art gallery problem in time f(k)n^o(k/logk) would also improve current running times for Subgraph Isomorphismand for solving CSPs parameterized by treewidth, which are major open questions [28]. Our results imply, in particular, that both variants are W[1]-hard parameterized by the number of guards.

Finally, let us mention a sample of works on the parameterized complexity (with an emphasis on hardness) of other geometric problems. The complexity of some fundamental

a1 a2a3 a4 a5 a6

p1

p2

p3

p4

p5

p6

Figure 1Reduction fromHitting Seton interval graphs to a restricted version of the art gallery problem.

Figure 2Two instances of Hitting Set “magically” linked.

problems parameterized by the dimension d has been addressed [17]; it was shown that, assuming the ETH, algorithms running in timen^O(d)are essentially optimal (withnbeing the size of the instance). Extracting from a finite set of points ofR³the largest subset in convex position and whose convex-hull interior is empty is W[1]-hard [16]. More results on geometric covering or packing problems include the following papers [5, 27, 1, 29, 7]. We refer the interested reader to the extensive survey of Giannopoulos et al. [18].

Proof ideas. In order to achieve these results, we slightly extend some known hardness results of geometric set cover/hitting set problems and combine them with problem-specific insights of the art gallery problem. One of the first problem-specific insights is the ability to encodeHitting Set on interval graphs. The reader can refer to Figure 1 for the following description. Assume that we have some fixed pointsp₁, . . . , pn with increasing y-coordinates in the plane. We can build a pocket “far enough to the right” that can be seen only from {pi, . . . , pj} for any 1≤i < j≤n.

Let I₁, . . . , I_n benintervals with endpointsa₁, . . . , a_2n. Then, we construct 2n points p1, . . . , p2n representinga1, . . . , a2n. Further, we construct one pocket “far enough to the right” for each interval as described above. This way, we reduceHitting Seton interval graphs to a restricted version of the art gallery problem. This observation isnot so useful in itself since hitting set on interval graphs can be solved in polynomial time.

The situation changes rapidly if we consider Hitting Seton 2-track interval graphs, as described in Section 2. Unfortunately, we are not able to just “magically” link some specific pairs of points in the polygon of the art gallery instance. Therefore, we construct linker gadgets, which basically work as follows. We are given two set of points P andQ and a bijectionσbetweenP andQ. The linker gadget is built in a way that it can be covered by two points (p, q) ofP×Q, if and only ifq=σ(p). TheStructured 2-Track Hitting Setproblem will be specifically designed so that the linker gadget is the main remaining ingredient to show hardness.

Organization of the paper. In Section 2, we introduce some notations, discuss the encoding of the polygon, and give some useful ETH-based lower bounds. We show a lower bound for Structured 2-Track Hitting Setbased on the lower bound known forMulticolored Subgraph Isomorphism. Due to space limitation, this proof is only included in the arxiv version of the paper [3]. Then, we reduce from the particularly convenient Structured 2-Track Hitting Set. In Section 3, we show the lower bound for thePoint Guard Art Galleryproblem (Theorem 1). We design a linker gadget, show its correctness, and show how several linker gadgets can be combined consistently. In Section 4, we tackle theVertex Guard Art Galleryproblem (Theorem 2). We have to design a very different linker gadget, that has to be combined with other gadgets and ideas.

2 Preliminaries

For any two integersx6 y, we set [x, y] :={x, x+ 1, . . . , y−1, y}, and for any positive integer x, [x] := [1, x]. Given two points a, b in the plane, we define seg(a, b) as the line segment with endpointsa, b. Givennpointsv₁, . . . , vn∈R², we define a polygonal closed curvec by seg(v1, v2),. . . ,seg(vn−1, vn), seg(vn, v1). Ifcis not self intersecting, it partitions the plane into a closed bounded area and an unbounded area. The closed bounded area is a simple polygonon the vertices v₁, . . . , v_n. Note that we do not consider the boundary as the polygon but rather all the points bounded by the curvec as described above. Given two pointsa, bin a simple polygon P, we say thatasees b orais visiblefromb if seg(a, b) is contained inP. By this definition, it is possible to “see through” vertices of the polygon.

We say thatS is a set ofpoint guards ofP, if every point p∈ P is visible from a point ofS.

We say thatS is a set ofvertex guards ofP, if additionallyS is a subset of the vertices of P. ThePoint Guard Art Galleryproblem and theVertex Guard Art Gallery problem are formally defined as follows.

Point Guard Art Gallery

Input: The vertices of a simple polygon P in the plane and a natural number k.

Question: Does there exist a set ofkpoint guards forP? Vertex Guard Art Gallery

Input: A simple polygon P onnvertices in the plane and a natural numberk.

Question: Does there exist a set ofkvertex guards for P?

For any two distinct pointsv andwin the plane we denote by ray(v, w) the ray starting atv and passing throughw, and by`(v, w) the supporting line passing through v andw.

For any pointxin a polygonP, V_P(x), or simplyV(x), denotes the visibility region ofx withinP, that is the set of all the pointsy∈ Pseen by x. We say that two verticesvandw of a polygonP are neighborsorconsecutive if vw is an edge ofP. A sub-polygon P⁰ of a simple polygonP is defined by anyl distinct consecutive verticesv1, v2, . . . , vl ofP (that is, for everyi∈[l−1],viandvi+1 are neighbors inP) such thatv1vl does not cross any edge ofP. In particular,P⁰ is a simple polygon.

We assume that the vertices of the polygon are either given by integers or by rational numbers. We also assume that the output is given either by integers or by rational numbers.

The instances we generate as a result of Theorem 1 and Theorem 2 have rational coordinates.

We can represent them by specifying the nominator and denominator. The number of bits is bounded byO(logn) in both cases. We can transform the coordinates to integers by multiplying every coordinate with the least common multiple of all denominators. However, this leads to integers usingO(nlogn) bits.

ETH-based lower bounds. The Exponential Time Hypothesis (ETH) is a conjecture by Impagliazzo et al. [21] asserting that there is no 2^o(n)-time algorithm for3-SATon instances withnvariables.

TheMulticolored Subgraph Isomorphismproblem can be defined in the following equivalent way. One is given a graph withnvertices partitioned intolcolor classesV₁, . . . , V_l such that onlykof the ₂^l

setsEij=E(Vi, Vj) are non empty. The goal is to pick one vertex in each color class so that the selected vertices inducek edges. Observe thatl corresponds to the number of vertices of the pattern graph. The technique of color coding and a result of Marx imply that:

I Theorem 3 ([28]). Unless the ETH fails, Multicolored Subgraph Isomorphism cannot be solved in timef(k)n^o(k/logk)where kis the number of edges of the solution andf any computable function.

Naturally, this result still holds when restricted to connected input graphs. In that case, k>l−1.

In the 2-Track Hitting Setproblem, the input consists of an integerk, two totally ordered ground setsA andB of the same cardinality, and two setsS_A ofA-intervals, and SB of B-intervals. In addition, the elements ofAandB are in one-to-one correspondence φ:A→B and each pair (a, φ(a)) is called a 2-element. The goal is to find, if possible, a set S ofk 2-elements such that the first projection ofS is a hitting set ofSA, and the second projection ofS is a hitting set ofSB.

Structured 2-Track Hitting Setis the same problem with color classes over the 2-elements, and a restriction on the one-to-one mappingφ. Given two integers kandt,Ais partitioned into (C₁, C₂, . . . , Ck) whereCj={a^j₁, a^j₂, . . . , a^j_t}for eachj∈[k]. Ais ordered:

a¹₁, a¹₂, . . . , a¹_t, a²₁, a²₂, . . . , a²_t, . . . , a^k₁, a^k₂, . . . , a^k_t. We defineC_j⁰ := φ(Cj) and b^j_i := φ(a^j_i) for all i ∈ [t] and j ∈ [k]. We now impose that φ is such that, for eachj ∈ [k], the setC_j⁰ is aB-interval. That is, B is ordered: C_σ(1)⁰ , C_σ(2)⁰ , . . . , C_σ(k)⁰ for some permutation on [k], σ ∈ Sk. For each j ∈ [k], the order of the elements within C_j⁰ can be described by a permutationσ_j∈S_tsuch that the ordering ofC_j⁰ is: b^j_σ

j(1), b^j_σ

j(2), . . . , b^j_σ

j(t). In what follows, it will be convenient to see an instance of Structured 2-Track Hitting Setas a tuple I = (k∈N, t∈N, σ∈S_k, σ₁∈S_t, . . . , σ_k ∈S_t,S_A,S_B), where we recall that S_A is a set ofA-intervals andS_B is a set of B-intervals. We denote by [a^j_i, a^j_i0⁰] (resp. [b^j_i, b^j_i0⁰]) all the elementsa∈A (resp. b∈B) such thata^j_i ≤Aa≤Aa^j_i0⁰ (resp. b^j_i ≤Bb≤B b^j_i0⁰).

Taking inspiration from previous results, we show hardness of Structured 2-Track Hitting Setby a reduction fromMulticolored Subgraph Isomorphism. Due to lack of space, we do no include the proof of the following theorem. The interested reader can find this proof in the arxiv version of the paper [3].

ITheorem 4. Structured 2-Track Hitting SetisW[1]-hard, and not solvable in time f(k)|I|^o(k/logk) for any computable functionf, unless the ETH fails.

3 Parameterized hardness of the point guard variant

As exposed in the introduction, we give a reduction from theStructured 2-Track Hitting Setproblem. The main challenge is to design alinker gadget that groups together specific pairs of points in the polygon. The following introductory lemma inspires the linker gadgets for bothPoint Guard Art Gallery andVertex Guard Art Gallery.

a¹₁ a¹₂ a¹₃ a¹₄ a¹₅ a¹₆ C₁

a²₁ a²₂ a²₃ a²₄ a²₅ a²₆ C₂

a³₁ a³₂ a³₃ a³₄ a³₅ a³₆ C₃

a⁴₁ a⁴₂ a⁴₃ a⁴₄ a⁴₅ a⁴₆ C₄

A

b³₄ b³₂ b³₃ b³₆ b³₁ b³₅ C₃⁰

b¹₂ b¹₄ b¹₁ b¹₅ b¹₆ b¹₃ C₁⁰

b⁴₃ b⁴₆ b⁴₅ b⁴₂ b⁴₁ b¹₄ C₄⁰

b²₁ b²₅ b²₂ b²₄ b²₆ b²₃ C₂⁰

B σ1 σ

≤A:

≤B:

Figure 3An illustration of thek+ 1 permutationsσ∈Sk,σ1∈St, . . . ,σk∈Stof an instance ofStructured 2-Track Hitting Set, withk= 4 andt= 6.

I Lemma 5. The only minimum hitting sets of the set-system S = {Si = {1,2, . . . , i, i+ 1, i+ 2, . . . , n} | i∈[n]} ∪ {Si={1,2, . . . , i, i+ 1, i+ 2, . . . , n} |i∈[n]} are {i, i}, for each i∈[n].

Proof. First, for eachi∈[n], one may easily observe that{i, i}is a hitting set ofS. Now, because of the setsS_n andS_n one should pick one elementiand one element j for some i, j∈[n]. Ifi < j, then setSi is not hit, and ifi > j, thenSj is not hit. Therefore,ishould

be equal toj. J

ITheorem 1(Parameterized hardness point guard). Assuming the ETH,Point Guard Art Galleryis not solvable in time f(k)n^o(k/logk), for any computable function f, even on simple polygons, wheren is the number of vertices of the polygon andk is the number of guards allowed.

Proof. Given an instance I = (k ∈ N, t ∈ N, σ ∈ Sk, σ₁ ∈ St, . . . , σ_k ∈ St,SA,SB) of Structured 2-Track Hitting Set, we build a simple polygonP withO(kt+|SA|+|SB|) vertices, such that I is a YES-instance iffP can be guarded by 3kpoints.

Outline. We recall thatA’s order is: a¹₁, . . . , a¹_t, . . . , a^k₁, . . . , a^k_t andB’s order is determined byσ and the σj’s (see Figure 3). Let us focus on one color classj ∈[k] together with a permutationσ_j:A→B. The global strategy of the reduction is toallocate, 2tspecial points for this polygon. The pointsa^j₁, . . . , a^j_t on trackAare represented byα^j₁, . . . , α^j_t points inP. and the pointsσ_j(a^j₁), . . . , σ_j(a^j_t) on trackB are represented byβ₁^j, . . . , β^j_t in the polygon.

Placing a guard inα^j_i andβ_i^j shall correspond to picking the 2-element (a^j_i, σj(b^j_i)). The pointsα_i^j’s and β_i^j’s ordered by increasing y-coordinates will match the order of thea^j_i’s along the order≤A and then of theb^j_i’s along ≤B. Then, far in the horizontal direction, we will place pockets to encode eachA-interval ofSA, and eachB-interval ofSB.

The first critical issue will be tolink pointα^j_i to pointβ^j_i. Indeed, in theStructured 2-Track Hitting Setproblem, one selects 2-elements (one per color class), so we should prevent one from placing two guards inα^j_i andβ^j_i0 withi6=i⁰. The so-calledpoint linker gadget will realize the intervals as described in Lemma 5.

The second critical issue is to enforce these positions. For this purpose, we will need to introduce acopy α^j_i of eachα^j_i. In each part of the gallery encoding a color classj∈[k], the only way of guarding all the pockets with only three guards will be to place them inα_i^j,α^j_i, andβ_i^j for somei∈[t] (see Figure 5). Hence, 3kguards will be necessary and sufficient to

guard the wholeP iff there is a solution to the instance of Structured 2-Track Hitting Set.

We now get into the details of the reduction. We will introduce several characteristic lengths and compare them; whenl1l2 means thatl1 should be thought as really small compared to l₂, and l₁ ≈ l₂ means that l₁ and l₂ are roughly of the same order. The motivation is to guide the intuition of the reader without bothering her/him too much about the details. At the end of the construction, we will specify more concretely how those lengths are chosen.

Construction. We start with an explicit specification of the coordinates. The description will be dependent on some parametersx, y, L, D, F that we will specify later. The valuex represents the offset between elements with respect to the x-coordinate and likewise the valuey represents the offset between elements with respect to they-coordinate. Drepresents the vertical distance between different color classes andLrepresents the horizontal distance between all theα⁰sand theβ⁰s, see also Figure 6. The valueF will become relevant later and describes the distance of the points to the pockets to the far right. The crucial point of the construction is that the order of theα’s corresponds exactly to the order of thea’s along trackAand the same relation holds between theβ’s andb’s.

We recall that we want the pointsα^j_i’s andβ_i^j’s ordered by increasingy-coordinates, to match the order of thea^j_i’s andb^j_i’s along≤Aand≤B, with first all the elements ofAand then all the elements ofB. Starting from somey-coordinatey₁ (which is the one given to pointα¹₁), they-coordinates of theα^j_i’s are regularly spaced out by an offsety; that is, the y-coordinate ofα^j_i isy1+ (i+ (j−1)t)y. Between they-coordinate of the last element inA (i.e.,a^k_t whosey-coordinate isy₁+ (kt−1)y) and the first element inB, there is a large offset L, such that they-coordinate ofβ_i^j is y1+ (kt−1)y+L+ (ord(b^j_i)−1)y (for anyj ∈[k]

andi∈[t]) where ord(b^j_i) is the rank of b^j_i along the order≤_B.

For each color class j∈[k], letxj :=x1+ (j−1)D for somex-coordinatex1and valueD, andy_j:=y₁+ (j−1)ty. The allocated pointsα^j₁, α^j₂, α^j₃, . . . , α^j_t are on a line at coordinates:

(xj, yj),(xj+x, yj+y),(xj+ 2x, yj+ 2y), . . . ,(xj+ (t−1)x, yj+ (t−1)y), for some value x. We place, to the left of those points, a rectangular pocket P_j,r of width, say, y and length, say¹,txsuch that the uppermost longer side of the rectangular pocket lies on the line

`(α^j₁, α^j_t) (see Figure 4). They-coordinates ofβ₁^j, β^j₂, β₃^j, . . . , β^j_t have already been defined.

We set, for eachi∈[t], thex-coordinate ofβ_i^j toxj+ (i−1)x, so thatβ_i^j andα^j_i share the same x-coordinate. One can check that it is consistent with the previous paragraph. We also observe that, by the choice of they-coordinate for theβ_i^j’s, we have both encoded the permutationsσj’s and permutationσ(see Figure 6 or Figure 4). This finishes the description of the coordinates.

Now, we will give a description how, we can encode intervals by on trackA andB by small pockets and, we describe, where to place them. From hereon, for a vertexv and two points pand p⁰, we informally call triangular pocket rooted at vertex v and supported by ray(v, p)and ray(v, p⁰) a sub-polygonw, v, w⁰ (a triangle) such that ray(v, w) passes through p, ray(v, w⁰) passes throughp⁰, whilewandw⁰are close tov(sufficiently close not to interfere with the rest of the construction). We say thatv is theroot of the triangular pocket, that we often denote byP(v). We also say that the pocket P(v)points towardspandp⁰. It is easy to see that each point that seesv also sees the entire triangular pocketP(v).

1 The exact width and length of this pocket are not relevant; the reader may just think ofPj,r as a thin pocket which forces to place a guard on a thin strip whose uppermost boundary is`(α^j₁, α^j_t)

For eachA-intervalI_q= [a^j_i, a^j_i0⁰]∈ SA we construct one triangular pocketP(zA,q) rooted at vertex z_A,q and supported by ray(z_A,q, α^j_i) and ray(z_A,q, α^j_i0⁰). The placement of this triangular pocket is very far to the right. Thex-coordinate ofzA,q equalsxk+ (t−1)x+F, for some large valueF to be specified later. The y-coordinate shall be between y1 and y_k+ (kt−1)y. We place those|SA|pockets along the y-axis, and space them out by some small distances. To guarantee that we have enough room to place all those pockets, swill be chosen sufficiently small (sy).

We will show later, for appropriate valuesyxDF, the only α^j_i00⁰⁰ seeing vertex zA,q should be the points such thata^j_i ≤Aa^j_i00⁰⁰ ≤Aa^j_i0⁰ (see Figure 6).

Similarly, we represent each intervalIq ∈ SB by a triangular pocket rooted atzB,q. These pockets are placed at thex-coordinatex_k+ (t−1)x+F and spaced out by distance salong they-axis betweeny-coordinatesy1+ (kt−1)y+Landy1+ 2(kt−1)y+L. TheB-interval Iq = [b^j_i, b^j_i0⁰] is represented by the triangular pocketP(zB,q) rooted at vertexzB,q supported by ray(z_B,q, σ_j(a^j_i)) and ray(z_B,q, σ_j(a^j_i0⁰)). Note that σ_j(a^j_i) is the point on track B that corresponds toβ^j_i. The different values (s,x,y,D,L, andF) introduced so far compare in the following way: syxDF, and xLF, see Figure 6.

Now, we describe how welink each pointα^j_i to its associateβ_i^j. For eachj∈[k], let us mentally draw ray(α^j_t, β₁^j) and consider points slightly to the left of this ray at a distance, say,L⁰from pointα^j_t. Let us callR^j_leftthat informal region of points. Any point inR^j_leftsees, from right to left, in the orderα^j₁,α^j₂ up to α^j_t, and then,β₁^j, β₂^j up toβ_t^j. This observation relies on the fact thatyxL. So, from the distance, the pointsβ^j₁, . . . , β_t^j look almost flat. It makes the following construction possible. InR^j_left, for eachi∈[t−1], we place a triangular pocketP(c^j_i) rooted at vertexc^j_i and supported by ray(c^j_i, α^j_i+1) and ray(c^j_i, β^j_i).

We place also a triangular pocketP(c^j_t) rooted atc^j_t supported by ray(c^j_i, β₁^j) and ray(c^j_i, β_t^j).

We place verticesc^j_i andc^j_i+1at the samey-coordinate and spaced out by distancexalong the x-axis (see Figure 4). Similarly, let us informally refer to the region slightly to the right of ray(α₁^j, β_t^j) at a distance L⁰ from point α^j₁, asR^j_right. Any pointR^j_right sees, from right to left, in this orderβ^j₁,β^j₂ up toβ_t^j, and then, α^j₁, α^j₂ up toα_t^j. Therefore, one can place inR^j_left, for each i ∈ [t−1], a triangular pocket P(d^j_i) rooted atd^j_i supported by ray(d^j_i, β_i+1^j ) and ray(c^j_i, α^j_i). We place also a triangular pocketP(d^j_t) rooted atd^j_t supported by ray(d^j_i, α^j₁) and ray(c^j_i, α^j_t). Again, thosetpockets are placed at the samey-coordinate and spaced out horizontally byx (see Figure 4). We denote byPj,α,β the set of pockets {P(c^j₁), . . . ,P(c^j_t),P(d^j₁), . . . ,P(d^j_t)}and informally call it theweak point linker (or simply, weak linker) ofα^j₁, . . . , α^j_t andβ₁^j, . . . , β^j_t. We may call the pockets ofR^j_left (resp. R^j_right)left pockets (resp. right pockets).

As we will show later, if one wants to guard with only two points all the pockets of P_j,α,β={P(c^j₁), . . . ,P(c^j_t),P(d^j₁), . . . ,P(d^j_t)} and one first decides to put a guard on point α^j_i (for somei ∈[t]), then one is not forced to put the other guard on point β_i^j but only on an area whose uppermost point isβ^j_i (see the shaded areas below theb^j_i’s in Figure 4).

Now, if the pointsβ^j₁, . . . , β_t^j would all lie on a common line`, we could shrink the shaded area of eachβ<sub>i</sub><sup>j</sup> (Figure 4) down to the single pointβ<sub>i</sub><sup>j</sup> by adding a thin rectangular pocket on ` (similarly to what we have for α^j₁, . . . , α^j_t). Naturally, we need that β₁^j, . . . , β_t^j are not on a common line to be able to encode the permutation σj. The remedy we pursue is the following. For eachj ∈[k], we allocate t pointsα^j₁, α^j₂, . . . , α^j_t on a horizontal line, spaced out by distance x, say, ≈ ^D₂ to the right and ≈L above of β_t^j. We place a thin horizontal rectangular pocketP_j,r of the same dimension asPj,r such that the lowermost longer side ofPj,r is on the line`(α^j₁, α^j_t). We add the 2tpockets corresponding to a weak

α1α2α3α₄α₅α₆ β1β2β₃β4β5β6

d1

c1c2c3c4c5c6 d2d3d4d5d6

Figure 4Weak point linker gadget.

Figure 5Point linker gadget: a triangle of (three) weak point linkers.

linker Pj,α,α betweenα^j₁, . . . , α^j_t and α^j₁, . . . , α^j_t as well as the 2t pockets of a weak linker Pj,α,β betweenα^j₁, . . . , α^j_t andβ^j₁, . . . , β_t^j as pictured in Figure 5. We denote byPj the union Pj,r∪ Pj,r∪ Pj,α,β∪ Pj,α,α∪ Pj,α,β of all the pockets involved in the encoding of color class j. Now, say, one wants to guard all the pockets ofP_j with only three points, and chooses to put a guard onα^j_i (for somei∈[t]). Because of the pockets ofPj,α,α∪Pj,r, one is forced to place a second guard precisely onα^j_i. Now, because of the weak linkerPj,α,βthe third guard should be on a region whose uppermost point isβ_i^j, while, because ofPj,α,β the third guard should be on a region whose lowermost point is β_i^j. The conclusion is that the third guard should be put precisely onβ_i^j. Thistriangleof weak linkers is called thelinker of color class j. Theklinkers are placed accordingly to Figure 6. This ends the construction.

Specification of the distances. We can specify the coordinates of positions of all the vertices by fractions of integers. These integers are polynomially bounded inn. If we want to get integer coordinates, we can transform the rational coordinates to integer coordinates by multiplying all of them with the least common multiple of all the denominators, which is not polynomially bounded anymore. The length of the integers in binary is still polynomially bounded.

We can safely set s to one, as it is the smallest length, we specified. We will put|S_a| pockets on track Aand|Sb| pockets on trackB. It is sufficient to have an opening space of one between them. Thus, the space on the right side ofP, for all pockets of trackAis bounded by 2|Sa|. Thus settingy to|Sa|+|Sb|secures us that we have plenty of space to

. . .F . . .F

D

P1,α,β P2,α,β P3,α,β

P1,α,α P2,α,α P3,α,α

P1,α,β

P2,α,β

P3,α,β

L

P1,r P2,r P3,r

P1,r P2,r P3,r

trackA trackB

α

α β

Figure 6The overall picture of the reduction withk= 3.

place all the pockets. We specifyF = (|Sa|+|Sb|)Dk=yDk. We have to show that this is large enough to guarantee that the pockets on trackAdistinguish the picked points only by they-coordinate. Letpandq be two points among the α^j_i. Their vertical distance is upper bounded byDkand their horizontal distance is lower bounded by y. Thus the slope of`=`(p, q) is at least _Dk^y . At the right side ofP the line `will be at leastF_Dk^y above the pockets of trackA. NoteF_Dk^y =yDk_Dk^y > y²>|Sa|²>2|Sa|. The same argument shows thatF is sufficiently large for trackB.

The remaining lengths x, L, L⁰, and D can be specified in a similar fashion. For the construction of the pockets, lets∈ S_a be anA-interval with endpointsaandb, represented by some pointspandqand assume the opening verticesvandwof the triangular pocket are already specified. Then the two lines`(p, v) and`(q, w) will meet at some pointxto the right ofv andw. It is easy to see thatxhas rational coordinates and the integers to represent them can be expressed by the coordinates ofp, q, v,andw. This way, all the pockets can be explicitly constructed using rational coordinates as claimed above.

Soundness. We now show that the reduction is correct. The following lemma is the main argument for the easier implication: ifI is a YES-instance, then the gallery that we build can be guarded with3k points.

ILemma 6. ∀j∈[k],∀i∈[t], the three associate pointsα^j_i,α^j_i,β_i^j guard entirely Pj. Proof. The rectangular pocketsP_j,r andP_j,r are entirely seen by respectivelyα^j_i andα^j_i. The pockets P(c^j₁),P(c^j₂), . . .P(c^j_i−1) andP(d^j_i),P(d^j_i+1), . . .P(d^j_t) are all entirely seen by α^j_i, while the pocketsP(c^j_i),P(c^j_i+1), . . .P(c^j_t) andP(d^j₁),P(d^j₂), . . .P(d^j_i−1) are all entirely seen byβ_i^j. This means thatα^j_i andβ_i^j jointly see all the pockets ofPj,α,β. Similarly,α^j_i and α^j_i jointly see all the pockets ofP_j,α,α, andα^j_i andβ_i^j jointly see all the pockets ofP_j,α,β. Therefore,α_i^j,α^j_i,β_i^j jointly see all the pockets ofPj. J

Assume that I is a YES-instance and let {(a¹_s1, b¹_s1), . . . ,(a^k_sk, b^k_sk)} be a solution. We claim thatG={α¹_s

1, α¹_s

1, β¹_s

1, . . . , α^k_s

k, α^k_s

k, β_s^k

k}guard the whole polygonP. By Lemma 6,

∀j∈[k],Pj is guarded. For eachA-interval (resp.B-interval) inSA (resp.SB) there is at least one 2-element (a^j_sj, b^j_sj) such thata^j_sj ∈ SA (resp. b^j_sj ∈ SB). Thus, the corresponding pocket is guarded byα^j_sj (resp.β^j_sj). The rest of the polygonP (which is not part of pockets)

is guarded by, for instance, {α¹_s1, . . . , α^k_sk}. So, Gis indeed a solution and it contains 3k points.

Assume now that there is no solution to the instance I of Structured 2-Track Hitting Set. We show that there is no set of 3kpoints guardingP. We observe that no point of P sees inside two triangular pockets one being inPj,α,γ and the other inPj⁰,α,γ⁰

withj 6=j⁰ andγ, γ⁰ ∈ {β, α}. Further,V(r(Pj,α,β∪ Pj,α,α))∩V(r(Pj⁰,α,β∪ Pj⁰,α,α)) =∅ whenj6=j⁰, wherer maps a set of triangular pockets to the set of their root. Also, for each j∈[k], seeing entirelyP_j,α,β andP_j,α,αrequires at least 3 points. This means that for each j∈[k], one should place three guards inV(r(Pj,α,β∪ Pj,α,α)). Furthermore, one can observe among those three points one should guard a triangular pocket Pj⁰,r and another should guardPj⁰⁰,r. Let us try to guard entirelyP1 and two rectangular pockets Pj⁰,r andPj⁰⁰,r, with only three guards. Let call`1 (resp. `⁰₁) the line corresponding to the extension of the uppermost (resp. lowermost) longer side ofP1,r (resp. P1,r). The only points ofP that can see a rectangular pocketPj⁰,r and at leastt pockets ofP1,α,αare on`1: more specifically, they are the points α<sup>1</sup><sub>1</sub>, . . . , α<sup>1</sup><sub>t</sub>. The only points that can see a rectangular pocket P<sub>j</sub><sup>00</sup><sub>,r</sub> and at leastt pockets ofP1,α,αare on `⁰₁: they are the pointsα¹₁, . . . , α¹_t. As P1,α,αhas 2t pockets, one has to take a pointα¹_i andα¹_i0. By the same argument argument as in Lemma 5, ishould be equal to i⁰ (otherwise,i < i⁰ and the left pocket pointing towardsα¹_i0−1 andα¹_i0

is not seen, ori > i⁰ and the right pocket pointing towardsα¹_i+1 andα¹_i is not seen). We now denote bys₁ this shared value. Now, to see the left pocketP(c¹_s

1) and the right pocket P(d¹_s1−1) (that should still be seen), the third guard should be to the left of`(c<sup>1</sup><sub>s1</sub>, β<sub>s</sub><sup>1</sup><sub>1</sub>) and to the right of`(d¹_s

1−1, β_s¹

1) (see shaded area of Figure 4). That is, the third guard should be on a region in whichβ_s¹₁ is the uppermost point. The same argument with the pockets of P_1,α,β implies that the third guard should also be on a region in which β_s¹

1 is the lowermost point. Thus, the position of the third guard has to be point β_s¹₁. Therefore, one should put guards on pointsα¹_s1,α¹_s1, andβ_s¹₁, for someα1∈[t].

As none of those three points see any pocket P_j,α,β withj >1 (we already mentioned that no pocket ofPj,α,β andPj,α,αwithj >1 can be seen by those points), we can repeat the argument for the second color class; and so forth up to color classk. Thus, a potential solution with 3kguards should be of the form{α¹_s1, α¹_s

1, β_s¹

1, . . . , α^k_s

k, α^k_s

k, β_s^k

k}. As there is no solution toI, there should be a set inSA∪ SBthat is not hit by{(a¹_s1, b¹_s1), . . . ,(a^k_sk, b^k_sk)}.

By construction, the pocket associated to this set is not entirely seen. J

4 Parameterized hardness of the vertex guard variant

We now turn to the vertex guard variant and show the same hardness result. Again, we reduce fromStructured 2-Track Hitting Setand our main task is to design alinker gadget. Though,linking pairs of vertices turns out to be very different fromlinking pairs of points. Therefore, we have to come up with fresh ideas to carry out the reduction. In a nutshell, the principal ingredient is tolink pairs of convex vertices by introducing reflex vertices at strategic places. As placing guards on those reflex vertices is not supposed to happen in theStructured 2-Track Hitting Setinstance, we design a so-calledfilter gadget to prevent any solution from doing so.

ITheorem 2(Parameterized hardness vertex guard). Assuming the ETH,Vertex Guard Art Gallery is not solvable in timef(k)n^o(k/logk), for any computable functionf, even on simple polygons, wherenis the number of vertices of the polygon and k is the number of guards allowed.

Proof. From an instanceI= (k∈N, t∈N, σ∈Sk, σ1∈St, . . . , σk ∈St,SA,SB), we build