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

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

3. 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

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.

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}

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

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

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.

9. 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)]

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

11. [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)

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×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

13. 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

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?

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.

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

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

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

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

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.

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

22. 1 The graph contains C 9 as an induced subgraph.

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

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.

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

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

27. 2 The graph contains C 5 as an induced subgraph. case 3

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

29. 3 case 4

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

31. 3 case 5

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

33. H-3 H-1 case 6

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.

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

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.

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

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

39. Other results Maximum weight stable set problem Imperfection ratio

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

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

42. 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

43. Thanks for your attention.