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Multicoloring Unit Disk Graphs on Triangular Lattice Points Yuichiro MIYAMOTO Sophia University Tomomi MATSUI University of Tokyo Slide 2 Main purpose: Discuss perfectness & imperfectness of unit disk graphs on triangular lattice points Outline Definition Unit disk graph Multicoloring, weighted coloring Triangular lattice points Perfectness & imperfectness Approximation algorithms for multicoloring Maximum weight independent set Imperfection ratio Slide 3 Multicoloring problem Output multicoloring function c: V 2 N c(u)c(v)=, {u,v} E (Every adjacent pair of two vertices doesnt share a common color) 1 2 2 3 Input simple undirected graph G=(V,E) vertex weight function w: V Z + Objective val.= 6 w(v) {0,1}, v V Coloring problem {1} {4,5,6} {}{2,3} Objective minimize required number of colors |c(v)|=w(v), v V (Every vertex requires w(v) colors) 0 {2,3} Weight Assigned colors Constraints Slide 4 Unit disk graph T Given a set of unit disks (diameter = T) on a 2D plain, a unit disk graph is an undirected graph such that centers of two disks are adjacent if and only if the pair of disks has intersection. Slide 5 Unit disk graph d E (v,w): Euclidean distance between the pair v & w P: a set of finite points on a 2D plain T: a non-negative real threshold T We restrict centers of disks to triangular lattice points. unit disk graph (P,T) vertex set: P edge set: {{v,w}: v,w P,d E (v,w) T} Slide 6 Triangular lattice points (0,0) e1e1 e2e2 (1,0) This figure shows triangular lattice points. Slide 7 Weighted unit disk graph on triangular lattice points 3 3 5 5 0 1 9 0 4 4 2 1 0 0 0 4 2 1 6 0 1 0 0 1 NP-hard [ Miyamoto & Matsui (2004)] We deal with finite graphs. weight Height=4 1 2 3 4 Slide 8 We investigate polynomial time approximation algorithms for multicoloring unit disk graphs on triangular lattice points. It is important to find well-solvable cases to develop efficient approximation algorithms. Key property of this talk: graph perfectness. Slide 9 Multicoloring problem and perfect graph (G,w): weighted clique number of (G,w) (G,w): multicoloring number of (G,w) If graph G is perfect, then (G,w)= (G,w), for every w. An optimal multicoloring of (G,w) is obtained in (strongly) polynomial time. For weighted cases, the following theorem is known. Notation Theorem [Grtschel, Lovsz & Schrijver (1988)] Slide 10 An approximation algorithm We find perfect subgraphs. We propose a polynomial time approximation algorithm based on graph perfectness. We show a simple case. Slide 11 [Height=3, Threshold=1] perfect H: (vertex) induced subgraph When (H)=1 or 3, it is trivial. If (H)=2, then H contains no odd-cycle since height = 3 bipartite graph (H)=2 Given vertex weights, we proposed a simple polynomial time multicoloring algorithm. Proof (abstract) Slide 12 An approximation algorithm for multicoloring U.D.G. on T.L.P. when threshold=1 3 6 3 9 3 9 3 0 0 6 9 3 3 6 9 0 3 3 6 9 9 3 1 2 1 3 1 3 1 0 0 0 1 2 3 0 1 1 2 3 3 1 1 2 1 3 0 0 0 0 0 2 3 1 1 2 3 0 1 1 0 0 0 0 0 0 1 3 1 0 0 2 3 1 1 2 3 0 0 0 2 3 3 1 1 2 1 3 1 3 1 0 0 2 3 1 0 0 0 0 1 1 2 3 3 1 =+++ 6 0 6 9 3 3 Every layer is perfect from previous observation (slide). Every layer is optimally multicolorable in polynomial time. The union of multicolored layers implies feasible multicoloring. Multicoloring number of each layer = Weighted clique number of each layer 1/3(G,w) 1/3(G,w) layer1layer2layer3layer4 6 6 3 9 0 3 Proper weights The lines of 0 weights appear every 4 lines. Lines of 0 weights cover all the lines. Every non-zero weight of every layer is 1/3 of original graph. Similar to the shifting strategy [Hochbaum (1987)] Requied # of colors 4/3(G,w) Theorem For simplicity, w(v) is multiple of 3, for every v Slide 13 Approximation algorithm: known results When threshold = 1 & w(v) is not multiple of 3, 4/3(G,w)+4 [Miyamoto &Matsui (2004)] 4/3(G,w)+1/3 [McDiarmid & Reed (2000)] If there is a polynomial time approximation algorithm whose ratio < 4/3, then P=NP. [McDiarmid & Reed (2000)] hard to extend to the case threshold > 1. Our algorithm easy to extend to the case threshold > 1, if a perfect subgraph is known Slide 14 Perfect? Imperfect? 1 2 3 4 5 6 H T [Height 2, Threshold 1] perfect Perfect (already shown) [Height 3, Threshold 1] perfect Perfect (trivial) Which is the remainder? Perfect? Imperfect? Perfect? Imperfect? Perfect? Imperfect? Perfect? Imperfect? Slide 15 Main result 1 2 3 4 5 6 H T perfect imperfect height 3, threshold 1 perfect height 4, threshold 1, Main theorem The boundary is monotone. We show an abstract of the proof of the main theorem. Slide 16 First, we show the perfectness 1 2 3 4 5 6 H T perfect already shown Slide 17 The comparability graph is perfect. G=(V,E) is a comparability graph If there is an orientation F of E such that (a,b) F, (b,c) F (a,c) F. (transitivity) Comparability graph Definition comparability graph Theorem The complement of a comparability graph is perfect. The complement of a perfect graph is perfect. Theorem Slide 18 If every pair of non-adjacent vertices is connected by right headed arrow, then the transitivity holds. Proof abstract Slide 19 Hight = 3 Perfect 1 2 3 4 5 6 H T From previous proof, threshold is large co-compalability graph perfect graph Co-comparability Perfectness co-comparability perfectness Slide 20 Perfectness of U.D.G. on T.L.P. 1 2 3 4 5 6 H T co-comparability perfectness Next, we show the inverse implication. not co-comparability graph In a similar way, we can show other cases. Slide 21 Odd-hole imperfect Odd-hole: induced subgraph C 2k+3, k=1,2, If G contains an odd-hole, then G is imperfect. Theorem Slide 22 1 The graph contains C 9 as an induced subgraph. Slide 23 Imperfectness (case 1) 1 2 3 4 5 6 H T perfect imperfect Graphs of height 4 are induced subgraphs of height 5 Slide 24 Imperfectness 1 2 3 4 5 6 H T perfect imperfect case 2 case 3 case 4case 5 case 6 In the following, we show other cases. Slide 25 The graph contains C 7 as an induced subgraph. case 2 Slide 26 Imperfectness (case 2) 1 2 3 4 5 6 H T perfect case 3 case 4case 5 case 6 imperfect Slide 27 2 The graph contains C 5 as an induced subgraph. case 3 Slide 28 Imperfectness (case 3) 1 2 3 4 5 6 H T perfect case 4case 5 case 6 imperfect Slide 29 3 case 4 Slide 30 Imperfectness (case 4) 1 2 3 4 5 6 H T perfect case 5 case 6 imperfect Slide 31 3 case 5 Slide 32 Imperfectness (case 5) 1 2 3 4 5 6 H T perfect case 6 imperfect Slide 33 H-3 H-1 case 6 Slide 34 Imperfectness (case 6) 1 2 3 4 5 6 H T perfect Imperfect By the induction, the proof is completed. Before we describe our approximation algorithms, we discuss the square lattice case. Slide 35 Unit disk graphs on square lattice points H T 1 2 3 4 perfect imperfect The boundary is not monotone. Slide 36 An approximation algorithm (again) 3 6 3 9 3 9 3 0 0 6 9 3 3 6 9 0 3 3 6 9 9 3 1 2 1 3 1 3 1 0 0 0 1 2 3 0 1 1 2 3 3 1 1 2 1 3 0 0 0 0 0 2 3 1 1 2 3 0 1 1 0 0 0 0 0 0 1 3 1 0 0 2 3 1 1 2 3 0 0 0 2 3 3 1 1 2 1 3 1 3 1 0 0 2 3 1 0 0 0 0 1 1 2 3 3 1 =+++ 6 0 6 9 3 3 layer1layer2layer3layer4 6 6 3 9 0 3 arbitrary weight 0 0 0 0 0 Key: This induced subgraph is optimally multicolorable. The decomposition into 4 layers implies 4/3-approximation algorithm arbitrary weight 0 0 0 0 0 arbitrary weight 3 1 This component is optimally multicolorable. If lines of weight 0 are removed, these components are independently multiclorable. Slide 37 This component is optimally multicolorable. If lines of weight 0 are removed, these components are independently multiclorable. Approximation algorithm (general threshold) For given threshold T, the following graph is perfect (from our main theorem). arbitrary weight 0 arbitrary weight 0 arbitrary weight 0 -approx. When T > 1, Theorem Slide 38 Table of approximation ratios T ratio When threshold=2, our (5/3)-approx. (7/3)-approx.[Feder & Shende (2000)] T 1 ratio4/35/37/429/5211/6 ratio =(T ) not monotone Slide 39 Other results Maximum weight stable set problem Imperfection ratio Slide 40 Maximum weight stable set problem Our main theorem implies polynomial time approximation algorithms for the problem. Details are omitted. ratio: Slide 41 Table of approximation ratios ratio T T 1 3/43/54/71/25/91/26/11 ratio = (T ) Slide 42 Imperfection ratio f (G,w): fractional weighted coloring number 1 U.D.G. on T.L.P. of threshold T imp( ) Our main theorem implies the following. Definition Corollary Slide 43 Thanks for your attention.