Multicoloring Unit Disk Graphs on Triangular Lattice Points

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 Assigned colors Weight Input：simple undirected graph G=(V,E)vertex weight function w: V →Z+ {4,5,6} Output：multicoloring function c: V → 2N 3 {1} Objective： minimize required number of colors {2,3} 1 2 Constraints： |c(v)|=w(v), ∀v∈V (Every vertex requires w(v) colors) 2 0 c(u)∩c(v)=φ, ∀ {u,v}∈E (Every adjacent pair of two vertices doesn’t share a common color) {2,3} {} Objective val.= 6 w(v)∈{0,1}, ∀ v∈V →　Coloring problem

4. Unit disk graph 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. T

5. Unit disk graph P: a set of finite points on a 2D plain T: a non-negative real threshold unit disk graph (P,T) vertex set: P edge set: {{v,w}: v,w∈P,dE(v,w)≦T} T dE(v,w): Euclidean distance between the pair v & w We restrict centers of disks to triangular lattice points.

6. Triangular lattice points This figure shows triangular lattice points. e2 e1 (1,0) (0,0)

7. Weighted unit disk graph on triangular lattice points 3 weight We deal with finite graphs. 2 0 4 0 2 1 3 Height=4 1 4 5 0 1 4 0 1 5 9 2 0 6 0 3 1 4 0 1 0 NP-hard [Miyamoto & Matsui (2004)]

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 Notation ω(G,w): weighted clique number of (G,w) (G,w): multicoloring number of (G,w) For weighted cases, the following theorem is known. Theorem [Grötschel, Lovász & Schrijver (1988)] 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.

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 Proof (abstract) 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.

12. 6 0 6 9 3 3 Theorem Requied # of colors ≦ 4/3×χ(G,w) An approximation algorithm for multicoloring U.D.G. on T.L.P. when threshold=1 For simplicity, w(v) is multiple of 3, for every v 3 6 3 9 3 9 3 0 0 6 9 3 3 6 9 0 3 3 3 3 6 9 9 3 1 2 1 3 1 3 1 0 0 0 0 0 1 2 3 0 1 1 1 1 2 3 3 1 1 2 1 3 0 0 0 0 0 2 3 1 1 2 3 0 1 1 1 1 0 0 0 0 1 2 1 3 1 3 1 0 0 2 3 1 0 0 0 0 1 1 1 1 2 3 3 1 0 0 0 0 1 3 1 0 0 2 3 1 1 2 3 0 0 0 0 0 2 3 3 1 6 0 6 9 3 3 = + + + layer1 layer2 layer3 layer4 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. Every layer is perfect from previous observation (slide). Every layer is optimally multicolorable in polynomial time. →The union of multicolored layers implies feasible multicoloring. Similar to the shifting strategy [Hochbaum (1987)] Multicoloring number of each layer = Weighted clique number of each layer ≦ 1/3×ω(G,w) ≦ 1/3×χ(G,w)

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? T H Perfect (trivial) ←Perfect (already shown) Perfect? Imperfect? Imperfect? Perfect? Imperfect? Imperfect? Imperfect? Perfect? Perfect? Perfect? [Height ＝3, Threshold ＝1]  perfect [Height ≦2, Threshold ≧1]  perfect Which is the remainder?

15. Main result Main theorem height ≦ 3, threshold ≧ 1  perfect height ≧ 4, threshold ≧ 1, T H perfect We show an abstract of the proof of the main theorem. The boundary is monotone. imperfect

16. First, we show the perfectness T H already shown perfect

17. Comparability graph Definition（comparability graph） 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) Theorem The comparability graph is perfect. Theorem The complement of a perfect graph is perfect. ↓ The complement of a comparability graph is perfect.

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

19. Hight = 3  Perfect T H co-comparability  perfectness Co-comparability  Perfectness From previous proof, threshold is large  co-compalability graph  perfect graph

20. Perfectness of U.D.G. on T.L.P. T H co-comparability  perfectness not co-comparability graph In a similar way, we can show other cases. Next, we show the inverse implication.

21. Odd-hole → imperfect Theorem If G contains an odd-hole, then G is imperfect. Odd-hole: induced subgraph C2k+3, k=1,2,…

22. The graph contains C9 as an induced subgraph. 1

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

24. Imperfectness T H perfect case 2 case 3 imperfect case 4 case 5 case 6 In the following, we show other cases.

25. The graph contains C7 as an induced subgraph. case 2

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

27. The graph contains C5 as an induced subgraph. case 3 2

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

29. case 4 3

30. Imperfectness (case 4) T H perfect case 5 imperfect case 6

31. case 5 3

32. Imperfectness (case 5) T H perfect imperfect case 6

33. ・・・・・・ ・・・・・・ ・・・・・・ ・・・・・・ case 6 H-1 ・・・・・・ ・・・・・・ ・・・・・・ ・・・・・・ ・・・・・・ ・・・・・・ H-3

34. Imperfectness (case 6) T H 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 T H The boundary is not monotone. perfect imperfect

36. 6 0 6 9 3 3 An approximation algorithm (again) 3 6 3 9 3 9 3 0 0 6 9 3 3 6 9 0 3 3 3 3 6 9 9 3 1 2 1 3 1 3 1 0 0 0 0 0 1 2 3 0 1 1 1 1 2 3 3 1 1 2 1 3 0 0 0 0 0 2 3 1 1 2 3 0 1 1 1 1 0 0 0 0 1 2 1 3 1 3 1 0 0 2 3 1 0 0 0 0 1 1 1 1 2 3 3 1 0 0 0 0 1 3 1 0 0 2 3 1 1 2 3 0 0 0 0 0 2 3 3 1 If lines of weight 0 are removed, these components are independently multiclorable. 6 0 6 9 3 3 = + + + This component is optimally multicolorable. layer1 layer2 layer3 layer4 arbitrary weight 3 Key: This induced subgraph is optimally multicolorable.  The decomposition into 4 layers implies 4/3-approximation algorithm 1 0 0 0 0 0 arbitrary weight 0 0 0 0 0 arbitrary weight

37. Approximation algorithm (general threshold) For given threshold T, the following graph is perfect (from our main theorem). arbitrary weight 0 This component is optimally multicolorable. Theorem arbitrary weight If lines of weight 0 are removed, these components are independently multiclorable. When T > 1, -approx. 0 arbitrary weight 0

38. Table of approximation ratios ratio not monotone ratio = (T∞) T When threshold=2, our (5/3)-approx.　＜　(7/3)-approx.[Feder & Shende (2000)]

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. ratio: Details are omitted.

41. Table of approximation ratios ratio ratio = (T∞) T

42. Imperfection ratio Definition χf(G,w): fractional weighted coloring number Our main theorem implies the following. Corollary U.D.G. on T.L.P. of threshold T imp( ) 1 ≦ ≦

43. Thanksforyourattention.