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Generalized vertex colorings and their applications to permutation graphs

Generalized vertex colorings and their applications to permutation graphs. T ınaz Ekim tinaz.ekim@epfl.ch EPFL - ROSE - Switzerland. T ınaz Ekim. Contents. Definitions Vertex coloring and applications Generalized vertex coloring s State of the art: complexity results

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Generalized vertex colorings and their applications to permutation graphs

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  1. Generalized vertex colorings and their applications to permutation graphs Tınaz Ekim tinaz.ekim@epfl.ch EPFL - ROSE - Switzerland

  2. Tınaz Ekim Contents • Definitions • Vertex coloring and applications • Generalized vertex colorings • State of the art: complexity results • Applications to permutation graphs • Car sorting • Robotics • Related results • Future research

  3. Tınaz Ekim Definitions (1) • Min Coloring: Partitioning the vertex set of a given graph into a minimum number of stable sets. The optimal value is (G) • Applications: • Telecommunications • Timetabling • Scheduling • etc… Stable set

  4. Tınaz Ekim Definitions (2) Given a graph G = (V,E), G is (p,k)-colorable if V can be partitioned into p cliques and k stable sets. • Min Coloring: (G) = min(k : G is (0,k)-colorable) • Min Cocoloring: z(G) = min (p+k : G is (p,k)-colorable) [Lesniak,77] • Min Split-coloring: S(G) = min (max(p,k) : G is (p,k)-colorable) = min (k : G is (k,k)-colorable) [Ekim, de Werra, 05]

  5. Tınaz Ekim Definitions and examples (3) G is a split graph if its vertex set can be partitioned into a stable set and a clique. S(G) = 2 z(G) = 2

  6. Tınaz Ekim Complexity results • Min Split-coloring and Min Cocoloring NP-hard • Min Split-coloring and Min Cocoloring P in cacti [Ekim, de Werra, 05], cographs [Demange, Ekim, de Werra, 05], in chordal graphs [Hell et al. 04]. • Min Cocoloring P in L(Bipartite), L(line-perfect graph), Min Split-coloring NP-hard in L(Bipartite) [Demange, Ekim, de Werra, 05] • Min Cocoloring NP-hard in permutation graphs [Wagner, 84]

  7. Tınaz Ekim clique = decreasing subsequence 2 7 3 6 stable set= increasing subsequence 4 1 5 Permutation graphs Given a permutation (N) where N=1, … ,n the permutation graph G=(V,E) corresponding to  is defined as follows: V= 1, … ,n and ijE iff i < j and (i) > (j) 5 1 3 7 6 2 4

  8. Tınaz Ekim Application 1: Sorting cars 4 6 436 346 1 2 3 4 5 6 4 3 6 1 5 2 3 5 152 12 5 1 2 Number of tracks needed to reorder  is (G()) = 3 In the modified structure, we only need S(G ()) = 2 tracks

  9. 7 2 Tınaz Ekim 3 6 4 1 5 Storage Area Storage Area 5 1 3 7 6 2 4

  10. 7 2 Tınaz Ekim 3 6 4 1 5 Storage Area Storage Area 5 1 3 7 6 2 4

  11. Tınaz Ekim NP-hardness results • Theorem [Demange, Ekim, de Werra, 05]: Let Gbe a class of graphs closed under addition of cliques without link to the rest of the graph and under addition of stable sets completely linked to the rest of the graph, then Min Cocoloring reduces to Min Split-coloring in G. • Corollary: Min Split-coloring is NP-hard in permutation graphs.

  12. Tınaz Ekim S(G’)  k  n k-p additional cliques are in any min split-coloring: S(G’)=max(k-p+p’,k’)  k p’  p and k’  k Also, since z(G)=p+k, p’+k’  p+k p’+k’ = p+k Proof Kn+1 Kn+1 G … G’ z(G)=p+k l=k-p cliques Kn+1

  13. Tınaz Ekim New version • The robot makes l+1 trips back and forth before unloading the items l-modal sequence. label Position   -1 Position label

  14. Tınaz Ekim Min l-modal • Partitioning a permuatation into a minimum number of l-modal subsequences is NP-hard even for l=1 [di Stefano, Krause, Lübbecke, Zimmermann, 05] • Deciding if  can be partitioned into 2 unimodal subsequences P O(m+nlogn)[Demange, Ekim, de Werra, 05]

  15. Tınaz Ekim Max l-modal subsequence How many items can the robot collect at most durin l+1 trips along the corridor? maximum l-modal subsequence • It can be found in time O(n logn) if l is fixed and in time O(n2 logn) if l is arbitrary [Demange, Ekim, de Werra, 05] • Hint: in permutation graphs, a maximum stable set can be found in time O(n logn)  Polynomial time approximation scheme for Min l-modal

  16. Tınaz Ekim π3 = 4 2 3 6 5 7 1 8 π2 = 5 7 6 3 4 2 8 1 π1 = 4 2 3 6 5 7 1 8 l=2, max l-modal subsequence = 6

  17. Tınaz Ekim DPTAS l-modal input: permutation , an integer p output: l-modal covering of  with diff. approx. ratio of (1-1/p) • while the current  has a maximum l-modal subseq. of size at least p(l+2) do color such an l-modal subseq. with a new color; • complete the solution by an exhaustive search on the remaining .

  18. Tınaz Ekim Thank you for your attention

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