On the Chromatic Art Gallery Problem

Transcription

1 CCCG 2014, Halifax, Nova Scotia, Auust 11 13, 2014 On the Chromatic Art Gallery Problem Sándor P. Fekete Stephan Friedrichs Michael Hemmer Joseph B. M. Mitchell Christiane Schmidt Abstract For a polyonal reion P with n vertices, a uard cover S is a set of points in P, such that any point in P can be seen from a point in S. In a colored uard cover, every element in a uard cover is assined a color, such that no two uards with the same color have overlappin visibility reions. The Chromatic Art Gallery Problem (CAGP) asks for the minimum number of colors for which a colored uard cover exists. We discuss the CAGP for the case of only two colors. We show that it is already NP-hard to decide whether two colors suffice for coverin a polyon with holes, even when arbitrary uard positions are allowed. For simple polyons with a discrete set of possible uard locations, we ive a polynomial-time alorithm for decidin whether a two-colorable uard set exists. This alorithm can be extended to optimize various additional objective functions for two-colorable uard sets, in particular minimizin the uard number, minimizin the maximum area of a visibility reion, and minimizin or maximizin the overlap between visibility reions. We also show results for a larer number of colors: computin the minimum number of colors in simple polyons with arbitrary uard positions is NP-hard for Θ(n) colors, but allows an O(lo(OP T )) approximation for the number of colors. 1 Introduction Consider a robot movin in a polyonal environment P. At any point p in P, the robot can naviate by referrin to a beacon that is directly visible from p. In order to ensure unique orientation, each beacon has a color ; the same color may be used for different beacons, if their visibility reions do not overlap. What is the minimum number χ G (P ) of colors for coverin P, and where should the beacons be placed? This is the Chromatic Art Gallery Problem (CAGP), which was first introduced by Erickson and LaValle [7]. Clearly, any feasible set of beacons for the CAGP must also be a feasible solution for the classical Department of Computer Science, TU Braunschwei, Germany. s.fekete@tu-bs.de, mhsaar@mail.com, c.schmidt@tu-bs.de. Department of Applied Mathematics and Statistics, Stony Brook University, USA. jsbm@ams.sunysb.edu. Fiure 1: An example polyon with n = 20 vertices. A minimum-cardinality uard cover with n/4 uards (shown in white) requires n/4 colors, while a minimumcolor uard cover (shown in black) has n/2 + 1 uards and requires only 3 colors. Art Gallery Problem (AGP). However, the number of uards and their positions for optimal AGP and CAGP solutions can be quite different, even in cases as simple as the one shown in Fiure 1. Related Work. The closely related AGP is NPhard [13], even for simple polyons. See [15, 16, 18] for three surveys with a wide variety of results. More recently, there has been work on developin practical optimization methods for computin optimal AGP solutions [12, 17, 10, 5]. The CAGP was first proposed by Erickson and LaValle, who presented a number of results, most notably upper and lower bounds on the number of colors for different classes of polyons [7, 8, 9]. In particular, they noted that the construction of Lee and Lin [13], which establishes NP-hardness of determinin a minimum-cardinality uard cover for a simple polyon P, can be used to prove NP-hardness of computin χ G (P ), as lon as all uards have to be picked from a specified candidate set [6, 9]. However, there is no straihtforward way to extend this construction for showin NP-hardness of the CAGP with arbitrary uard positions. An upcomin paper by Zambon et al. [19] discusses worst-case bounds, as well as exact methods for computin optimal solutions. Bärtschi and Suri introduced the Conflict-Free CAGP, in which they relax the chromatic requirements [2, 3]: Visibility reions of uards with the same color may overlap, as lon as there is always one

2 26 th Canadian Conference on Computational Geometry, 2014 uniquely colored uard visible. They showed an upper bound of O(lo 2 n) on the worst-case number of uards, which was lowered to O(lo n) for simple polyons by Bärtschi et al. [1]. This is sinificantly smaller than the lower bound of Θ(n) for χ G (P ) established by Erickson and LaValle [8]. Another loosely related line of research is by Biro et al. [4], who consider beacons of a different kind: in order to et to a new location, a robot aims for a sequence of destinations for shortest eodesic paths, each from a finite set of beacons. Their results include a tiht worst-case bound of n 2 1 for the number of beacons and a proof of NP-hardness for findin a smallest set of beacons in a simple polyon. Our Results. We provide a number of positive and neative results on the CAGP. In particular, we show the followin: For a polyon P with holes, it is NP-hard to decide whether there is a k-colorable uard cover of P, even if k = 2 and arbitrary uard positions in P may be used. We also provide a proof for k 3 for lack of space. Because k = 1 requires a sinle uard, i.e., P must be a star-shaped polyon, we et a complete complexity analysis. For a simple polyon P and a iven discrete set L of c candidate uard locations, it can be decided in polynomial time whether there is a 2-colorable uard set S L. For a simple polyon P that has a 2-colorable uard set S C chosen from a iven discrete set L of l uard positions, we can find a smallest 2-colorable uard set in polynomial time. We can also compute 2-colorable subsets that optimize any from a variety of other objective functions, includin minimizin or maximizin the larest visibility reion, or minimizin or maximizin the overlap between reions. For a simple polyon P, it is NP-hard to compute a uard cover with the smallest number of colors, even when arbitrary uard positions may be used. The proof establishes this for χ G (P ) Θ(n). For a simple polyon with a iven discrete set L of l possible uard locations, there is a polynomialtime O(lo(χ G (P ))-approximation for the number χ G (P ) of colors. 2 Preliminaries Let P be a polyon. P is simple if its boundary is connected (and not self-intersectin). For p P, V(p) P denotes the visibility polyon of p, i.e., all points p that can be connected to p usin the line sement pp w 1 V w 2 w k+1 Fiure 2: Forcin one uard into the reion V (ray). P. For any S P, we denote by V(S) = S V(). A finite S P with V(S) = P is called a uard cover of P ; S is a uard. We say that covers all points in V(). Let S be a uard cover of P. We call G(S) the intersection raph of visbility polyons of uards S. For a colorin c : S {1,..., k} of the uards, (S, c) is a k-colorin of P, if it induces a k-colorin of G(S), i.e., no point in P sees two uards of the same color. χ G (P ) is the chromatic (uard) number of P, i.e., the minimal k, such that there is a k-colorin of P. (The index G in χ G (P ) as introduced in [8, 9] does not refer to a specific uard set, nor to a raph G.) Definition 2.1 (Chromatic Art Gallery Problem) For k N, the k-chromatic Art Gallery Problem (k- CAGP) is the followin decision problem: Given a polyon P, decide whether χ G (P ) k. 3 Polyons with Holes In this section, we show that the k-cagp is NP-hard for any fixed k 3, even when uards may be chosen arbitrarily in P Colorability Lemma 3.1 (Needle lemma) Consider k + 1 needles with end points W = {w 1,..., w k+1 } (tips of the needle spikes), such that there is some k+1 i=1 V(w i) with V(W ) V(), as in Fiure 2. Then a k-colorin must place a uard in V = {p P V(p) W 2}, e.. at. Proof. Suppose there is no uard in V. Then there must be a uard in each V(w i ), requirin a total of at least k + 1 uards; two of them share the same color. Because V(W ) V(), sees two uards with the same color, which is impossible in a k-colorin. We use a reduction from 3SAT for showin NPhardness of decidin whether χ G (P ) = 2. Throuhout this section, we associate colors with Boolean values; w. l. o.., blue corresponds to true and red to false. The of every adet is (accordin to Lemma 3.1) a uard at a specific position, colored in red or blue which entirely covers all tunnels servin as for the next adets.

3 CCCG 2014, Halifax, Nova Scotia, Auust 11 13, 2014 w 2 x i x i 2 xi xi w 1 (a) Variable adet. (b) Inverter adet. (c) Crossin adet. V(w 1 ) V(w 2 ) indicated in ray. w 3 w inverter inverter w 1 w 2 2 w 1 inverter w 2 (d) Multiplexer adet. (e) Or adet, mixed. (f) Or adet, uniform. () And adet. Fiure 3: 3SAT to 2-CAGP reduction adets. Variable Gadet. This adet uses the construction in Fiure 3(a) to encode one decision variable x i. The needles enforce locatin two uards at the indicated positions. The color used for the left uard is interpreted as the value of x i : Blue means true, red means false. Inverter Gadet. The adet in Fiure 3(b) inverts colors. Its area is illuminated by one color; the uard forced to position must have the other, or a point in the lower riht corner can observe two uards of the same color. Crossin Gadet. Crossins of channels propaatin colors is achieved by the adet in Fiure 3(c). For any uard that sees w 1, V(w 1 ) V(). As V(w 1 ) intersects both the area and V( ), must have the same color as the correspondin ; the same holds for w 2. If both areas are covered by uards of the same color, then one uard of the opposite color is placed in V(w 1 ) V(w 2 ). Otherwise, we place two uards of different color outside of V(w 1 ) V(w 2 ), e.., at w 1 and w 2. Multiplexer Gadet. Multiplexin is achieved with the adet in Fiure 3(d). It uses an inverter adet, and forces a uard to position 2 which covers all tunnels. The adet is easily eneralized to an arbitrary number of tunnels. Or Gadet. The adet in Fiures 3(e) 3(f) is a binary or, allowin construction of a ternary one. We arue that there is a uard cover that colors the area blue, if and only if at least one area is blue. If two different colors are applied, a uard cover with blue exists: blue, 2 red, and 3 blue in Fiure 3(e). The same is true for blue/blue : red and 3 blue in Fiure 3(f). If the is red/red, the cannot be blue: By Lemma 3.1, the ray area must contain a uard, which can only be blue. V() V( 3 ), so 3 is red. And Gadet. This adet is similar to the multiplexer adet, see Fiure 3(). The uard forced to position can be colored if and only if all reions have the same color. Note that this adet forces all s to be identical, either true or false. The 3SAT Reduction. Any 3SAT instance S = C 1 C m with variables x 1,..., x n can be encoded as a 2-CAGP instance usin the adets and the overall layout depicted in Fiures 3 and 4. If S is satisfiable, settin every true variable to blue results in a valid 2-colorin. If there is a 2-colorin of the polyon, the final and adet s must be uniformly colored, w. l. o.. blue. Then the blue uards in the variable adets encode which variable is true and which is false in a valid truth assinment for S.

4 26 th Canadian Conference on Computational Geometry, 2014 H C 1 C 2 x 1 x1 x 1 v 3 x 2 x 3 x 4 x 5 C m x n and or crossin multiplexer variables Fiure 4: 3SAT reduction adet usae. v 4 k Fiure 5: 3- and (3 + k)-colorability. 3-colorability is shown in continuous lines, (3+k)-colorability is achieved by addin the structures drawn in dashed lines. The ray areas contain k or 3 forced uards. Theorem CAGP, i.e., decidin if a polyon is 2-colorable, is NP-hard. 3 v 3 k v 4 The proof is based on a number of structural lemmas, then proceeds by dynamic prorammin. Basic Lemmas. We state a number of topoloical properties of 2-colorable simple polyons. Note that these hold even if we do not have a discrete candidate set for uard locations. Lemma 4.2 Let P be a simple polyon. A 2- colorable uard set S for P cannot contain three uards, 2, 3 S, such that V( ) V( 2 ) V( 3 ). Proof. Trivial. Lemma 4.3 Let P be a simple polyon, and let S be a 2-colorable uard set of P. Let, 2 S be two different uards. Then any point p V( ) V( 2 ) must be in P, i.e., boundaries of visibility reions intersect only on the boundary of the polyon. Proof. Suppose p is in the interior of P and consider an infinitesimal neihborhood of p. This contains points that are in neither visibility reion, so they must be in a third visibility reion, say, of 3. Because visibility reions are closed sets, this must also intersect p, so the claim follows from Lemma and (3 + k)-colorability The NP-hardness of 3-CAGP follows from the NPhardness of decidin whether a planar raph H is 3- colorable-complete [11]. The idea is shown in continuous lines in Fiure 5: Each node of the raph H is turned into a convex reion of the polyon P, and needles force exactly one uard into each of them by Lemma 3.1. Those uards cover P ; it is easy to see that P is 3- colorable iff H is 3-colorable. For 1 k N, the construction can be eneralized to (3+k)-CAGP by addin the structures drawn in dashed lines to Fiure 5. Details are straihtforward. 4 Simple Polyons 4.1 Two Colors In the followin, we consider the CAGP for a simple polyon P and a iven discrete set L of l candidate locations, from which a 2-colorable uard set S L must be chosen. Overall, we will prove the followin main result. Theorem 4.1 For a simple polyon P with n vertices and l discrete uard locations L, it can be decided in polynomial time whether there is a 2-colorable uard set. Lemma 4.4 Let P be a simple polyon, and let S be a 2-colorable uard set. Then G(S) must be a tree. Proof. Consider a cycle in G(S), which corresponds to a sequence of visibility reions. By Lemma 4.3, all of their boundary intersections must be on the boundary of P. Furthermore, these intersections must be pairwise disjoint by Lemma 4.2, so the intersection between overlappin visibility reions is non-deenerate. Thus, there is a closed path throuh the interior of all involved visibility reions (and thus throuh the interior of P ) that separates two of these boundary points. Therefore, the boundary of P cannot be connected, a contradiction. Lemma 4.5 Let P be a simple polyon and let S be a 2-colorable uard set. Consider a trianulation T of the overlay of visibility reions V(), S. Then the dual raph of T is a tree. Proof. By Lemma 4.3, boundaries of visibility reions only intersect on the boundary of P, so all vertices of cells in the overlay lie on the boundary. It follows that any chord in a cell separates P into two pieces, provin the claim.

5 CCCG 2014, Halifax, Nova Scotia, Auust 11 13, 2014 v P o(e) e V() P i (e) e Fiure 6: Illustratin Definition 4.6. P i(e ) P o(e) e e v 3 e P i(e ) Fiure 7: Solvin a subproblem (e, Γ) by combinin solutions for subproblems (e, Γ ) and (e, Γ ). Shadow Vertices and Labeled Edes. Next we consider some basic structures for our alorithm. By Lemma 4.3, we know that the vertices of subreions in the overlay of visibility reions of active uards must lie on the boundary. This allows us to compute a 2- colorable uard set by buildin a trianulation of the overlay of the visibility reions of active uards in the polyon, one trianle at a time, based on considerin the limited set of possible chords between overlay vertices. Refer to Fiure 6 for the followin definitions. Definition 4.6 Let P be a simple polyon, and let L be a discrete set of l possible uard locations. Then we define the followin. (a) For a (not necessarily active) uard position L, a shadow vertex v is a vertex of the boundary of V(), or a polyon vertex. Any shadow vertex lies on the boundary of P. We write V for the set of all O(nl) shadow vertices. Similarly, a shadow ede is a line sement between two shadow vertices that separates V() from P \ V() for some uard. Fiure 8: Different possible positions for v 3 for combinin a solution for subproblem (e, Γ) from solutions for subproblems ((, v 3 ), Γ ) and ((v 3, ), Γ ). It suffices if one of these combinations yields a feasible combination for (e, Γ). (b) A shadow chord e = (, ) is a directed sement between two shadow vertices, and. e subdivides P into two subpolyons, the inner polyon P i (e) P that lies riht of e, and the outer polyon P o (e) P that lies left of e. (c) A subproblem (e, Γ) is defined by a shadow chord toether with a subset Γ L of one or two uards, and possibly 2. It asks for a subset of uards S L and a trianulation of P i (e), such that S is a 2-colorable uard cover of P i (e), in which the set of uards coverin the trianle to the riht of e is precisely Γ. If such a set exists, we say that (e, Γ) is solvable. Dynamic Prorammin. Now the idea for our dynamic-prorammin alorithm is to consider buildin a valid trianulation of a 2-colorable visibility overlay (whose dual must be a tree by Lemma 4.5), based on shadow chords and their active uards. As each new ede in such a trianulation must be a shadow chord that forms a trianle with a pair of existin edes, we can choose such a pair that is feasible, and recurse. More formally, we consider the Boolean function B(e, Γ) := 1, if the subproblem defined by (e, Γ) is solvable. We initialize B(e, Γ) := 1 for all combinations of e and Γ for which e is a counterclockwise polyon ede, i.e, for which P i (e) = e. Then the update rule is as follows. B(e, Γ) := 1, if and only if there is a shadow vertex v 3 in P i (e) with e = (, v 3 ) and e = (v 3, ) with uard sets Γ and Γ, such that the subproblems (e, Γ ) and (e, Γ ) are both solvable in a way that stays valid when we cover trianle (, v 3, ) by uard set Γ. (It is straihtforward to see the necessary relationship

6 26 th Canadian Conference on Computational Geometry, 2014 between Γ, Γ and Γ : when crossin a shadow ede, the coverin uard set must chane by the correspondin uard; when crossin any other shadow chord, it must stay the same.) The recursion terminates with a feasible result, if and only if there is a clockwise polyon ede e with B(e, Γ) = 1, i.e, a subproblem (e, Γ) for which P i (e) = P, i.e., B(e Γ ) := 1. Conversely, it is straihtforward to see that any feasible result implies that a feasible solution will be found. Additional Objective Functions. It is not hard to see that the dynamic-prorammin approach can be extended to computin not just any 2-colorable uard set, but also one that is optimal with respect to an objective functions. In the followin, we state this for some of the more interestin ones; of course others exist. Corollary 4.7 For a simple polyon P with n vertices and a set L of l candidate locations for uards, we can compute in polyomial time a 2-colorable uard set that is optimal with respect to one of the followin objectives. Minimize/maximize the number of uards. Minimize/maximize the larest/smallest visibility reion. Minimize/maximize the averae area of visibility reions. Minimize/maximize the larest/smallest overlap between visibility reions. Minimize/maximize the total overlap between visibility reions. This can be done by replacin the Boolean function B and its update rule by an appropriate objective function; for averae values, subproblems can be extended appropriately. 4.2 Many Colors NP-hardness. Our proof of NP-hardness of the eneral CAGP for simple polyons is based on a reduction from computin a minimum-cardinality set of points for coverin a iven set of lines; this auxiliary problem is easily seen to be NP-hard by eometric duality applied to a result of Meiddo and Tamir [14], who showed that it is NP-hard to determine the minimum number of lines to cover a set of points in the plane. Theorem 4.8 It is NP-hard to determine the chromatic number of a simple polyon, even for arbitrary uard positions. Fiure 9: NP-hardness of the eneral CAGP for simple polyons: A minimum-color uard cover of the spike box (red) corresponds to a minimum-cardinality point cover of a set of lines (blue). Proof. Refer to Fiure 9. For a iven set of lines, construct a spike box, which is formed by a square that contains all intersection points of the lines, and has two narrow niche extensions for each line, one at either intersection with the square. Note that the visibility reions of any two points in the spike box overlap; thus, minimizin the number of colors is equivalent to minimizin the number of uards. Now any uard cover corresponds to a point cover of the lines; conversely, any line cover can be converted to a uard cover of the spike box: if there is a uard placed in a niche, replace it by one inside the square. Approximation. On the positive side, we can establish an O(lo(χ G (P )))-approximation alorithm for the CAGP in simple polyons with a discrete set of candidate locations. Theorem 4.9 For a simple polyon P with a set L of l candidate uard locations, we can find an O(χ G (P ) lo(χ G (P )))-colorable uard set in polynomial time. The alorithm uses reedy set cover. In each iteration, we use dynamic prorammin to find an independet set of uard positions that covers a maximum number of uncovered cells in the overlay of all visibility polyons V() for L.

7 CCCG 2014, Halifax, Nova Scotia, Auust 11 13, Conclusion A number of open problems remain. These include the complexity of the CAGP for simple polyons with χ G (P ) = 2 without fixed a discrete set of uard locations as well as the complexity for χ G (P ) = 3 with a discrete candidate set. Note that the latter remains open for any fixed χ G (P ) = k, as the NP-hardness proof by Erickson and LaValle [9] requires lare χ G (P ). Amon the other open questions for fixed uard locations is the complexity of determinin the chromatic number of a iven uard set in a simple polyon; the claim by Erickson and LaValle stated in [9] that this problem has a polynomial solution is still unproven, as the correspondin conflict raph does not have to be chordal: this can be seen from the n/2 black locations at the spikes in Fiure 1. Acknowledements We thank Cid de Souza, Pedro de Rezende, Maurício Zambon and Bruno Espinosa Crepaldi for helpful and pleasant discussions. J. Mitchell is partially supported by NSF (CCF ) and the US-Israel Binational Science Foundation (project ). References [1] A. Bärtschi, S. K. Ghosh, M. Mihalák, T. Tschaer, and P. Widmayer. Improved bounds for the conflictfree chromatic art allery problem. In Proceedins of the 30th Symposium on Computational Geometry (SoCG), paes , [2] A. Bärtschi and S. Suri. Conflict-free chromatic art allery coverae. In Proceedins of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 14 of LIPIcs, paes , [3] A. Bärtschi and S. Suri. Conflict-free chromatic art allery coverae. Alorithmica, 68(1): , [4] M. Biro, J. Iwerks, I. Kostitsyna, and J. S. B. Mitchell. Beacon-based alorithms for eometric routin. In Proceedins of the 13th International Symposium on Alorithms and Data Structures (WADS), volume 8037 of LNCS, paes Spriner, [5] D. Borrmann, P. J. de Rezende, C. C. de Souza, S. P. Fekete, A. Kröller, A. Nüchter, and C. Schmidt. Point uards and point clouds: Solvin eneral art allery problems. In Proceedins of the 29th Symposium on Computational Geometry (SoCG), paes ACM, [6] L. Erickson. Visibility analysis of landmark-based naviation. PhD thesis, University of Illinois at Urbana- Champain, [7] L. H. Erickson and S. M. LaValle. A chromatic art allery problem. Technical report, University of Illinois at Urbana-Champain, [8] L. H. Erickson and S. M. LaValle. An art allery approach to ensurin that landmarks are distinuishable. In Robotics: Science and Systems, [9] L. H. Erickson and S. M. LaValle. How many landmark colors are needed to avoid confusion in a polyon? In Proceedins of the IEEE International Conference on Robotics and Automation (ICRA), paes IEEE, [10] S. P. Fekete, S. Friedrichs, A. Kröller, and C. Schmidt. Facets for art allery problems. In Proceedins of the 19th International Conference on Computin and Combinatorics (COCOON), volume 7936 of LNCS, paes Spriner, [11] M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP- Completeness. W. H. Freeman and Company, [12] A. Kröller, T. Baumartner, S. P. Fekete, and C. Schmidt. Exact solutions and bounds for eneral art allery problems. Journal of Experimental Alorithms, 17(1), [13] D. T. Lee and A. K. Lin. Computational Complexity of Art Gallery Problems. IEEE Transactions on Information Theory, 32(2): , [14] N. Meiddo and A. Tamir. On the complexity of locatin linear facilities in the plane. Operations Research Letters, 1, [15] J. O Rourke. Art Gallery Theorems and Alorithms. Oxford University Press, New York, NY, [16] T. C. Shermer. Recent results in art alleries. In Proceedins of the IEEE, volume 80, paes , [17] D. Tozoni, P. J. de Rezende, and C. C. de Souza. The quest for optimal solutions for the art allery problem: A practical iterative alorithm. In Proceedins of the 13th Symposium on Experimental Alorithms (SEA), volume 7933 of LNCS, paes Spriner, [18] J. Urrutia. Art allery and illumination problems. In J.-R. Sack and J. Urrutia, editors, Handbook on Computational Geometry, paes Elsevier, [19] M. J. O. Zambon, P. J. de Rezende, and C. C. de Souza. An exact alorithm for the chromatic art allery problem. In Proceedins of the 14th Symposium on Experimental Alorithms (SEA), (To appear).