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Multicoloring Unit Disk Graphs on Triangular Lattice Points Yuichiro MIYAMOTO Sophia University Tomomi MATSUI University of Tokyo

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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

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Multicoloring problem Output multicoloring function c: V 2 N c(u)c(v)=φ, {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

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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.

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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}

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Triangular lattice points (0,0) e1e1 e2e2 (1,0) This figure shows triangular lattice points.

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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

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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.

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Multicoloring problem and perfect graph ω(G,w): weighted clique number of (G,w) (G,w): multicoloring number of (G,w) If graph G is perfect, then ω(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 [Grötschel, Lovász & Schrijver (1988)]

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An approximation algorithm We find perfect subgraphs. We propose a polynomial time approximation algorithm based on graph perfectness. We show a simple case.

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[Height=3, Threshold=1] perfect H: (vertex) induced subgraph When ω(H)=1 or 3, it is trivial. If ω(H)=2, then H contains no odd-cycle since height = 3 bipartite graph χ(H)=2 Given vertex weights, we proposed a simple polynomial time multicoloring algorithm. Proof (abstract)

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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×ω(G,w) 1/3×χ(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×χ(G,w) Theorem For simplicity, w(v) is multiple of 3, for every v

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Approximation algorithm: known results When threshold = 1 & w(v) is not multiple of 3, 4/3ω(G,w)+4 [Miyamoto &Matsui (2004)] 4/3ω(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

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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?

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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.

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First, we show the perfectness 1 2 3 4 5 6 … H T perfect already shown

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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

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If every pair of non-adjacent vertices is connected by right headed arrow, then the transitivity holds. Proof abstract

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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

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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.

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Odd-hole imperfect Odd-hole: induced subgraph C 2k+3, k=1,2,… If G contains an odd-hole, then G is imperfect. Theorem

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1 The graph contains C 9 as an induced subgraph.

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Imperfectness (case 1) 1 2 3 4 5 6 … H T perfect imperfect Graphs of height 4 are induced subgraphs of height 5

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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.

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The graph contains C 7 as an induced subgraph. case 2

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Imperfectness (case 2) 1 2 3 4 5 6 … H T perfect case 3 case 4case 5 case 6 imperfect

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2 The graph contains C 5 as an induced subgraph. case 3

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Imperfectness (case 3) 1 2 3 4 5 6 … H T perfect case 4case 5 case 6 imperfect

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3 case 4

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Imperfectness (case 4) 1 2 3 4 5 6 … H T perfect case 5 case 6 imperfect

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3 case 5

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Imperfectness (case 5) 1 2 3 4 5 6 … H T perfect case 6 imperfect

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H-3 H-1 case 6

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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.

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Unit disk graphs on square lattice points H T 1 2 3 4 perfect imperfect The boundary is not monotone.

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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.

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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

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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

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Other results Maximum weight stable set problem Imperfection ratio

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Maximum weight stable set problem Our main theorem implies polynomial time approximation algorithms for the problem. Details are omitted. ratio:

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Table of approximation ratios ratio T T 1 3/43/54/71/25/91/26/11 ratio = (T )

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Imperfection ratio χ 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

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Thanks for your attention.

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