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Subcoloring and Hypocoloring Interval Graphs R. Gandhi1 and B. Greening1 and S. Pemmaraju2 and R. Raman3 Subcoloring and Hypocoloring Interval Graphs WG 2009 1Rutgers University, Camden. 2University of Iowa, USA. 3Max-Planck-Institut für Informatik, Germany June 24, 2009 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 2 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 3 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring S2 Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint S1 cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 4 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint S1 cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 4 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring S2 Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 4 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring S2 Let G = (V , E) be a graph. A Subcoloring of G is a partition V1, ··· , Vk of V , such that each Vi is a union of disjoint S1 cliques. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 4 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subchromatic Number S2 The subchromatic number of a graph G, χs (G) is the smallest k for which such a partition S1 V1, ··· , Vk exists. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 5 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Example χs (Kn) = 1 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 6 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) S2 χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S1 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions χs(G) ≤ χ(G) S2 χs (G) ≤ χ(G) Infact, χs (G) ≤ min{χ(G), χ(G)} S1 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 7 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Approximation Algorithms Several Optimization problems on graphs are easier when the underlying graph is a clique, or a disjoint union of cliques. If χs (G) is small, obtaining a good solution to each subcolor class, and picking the best gives a good approximation. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 8 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x 1 1 1 0 0 Given l ≤ Ax ≤ u, Aij ∈ {0, 1}, x ≥ 0 x2 0 1 0 1 Find the largest system of inequalities that has a feasible solution. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 9 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x 1 1 1 0 0 Given l ≤ Ax ≤ u, Aij ∈ {0, 1}, x ≥ 0 x2 0 1 0 1 Find the largest system of inequalities that has a feasible solution. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 9 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x1 1 1 0 0 A is a clique-vertex incidence matrix if there is a graph G s.t. x ∈ C iff A = 1. x2 0 1 0 1 i j ij Consecutive-ones ⇒ Interval Graphs. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 10 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x1 1 1 0 0 A is a clique-vertex incidence matrix if there is a graph G s.t. x ∈ C iff A = 1. x2 0 1 0 1 i j ij Consecutive-ones ⇒ Interval Graphs. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 10 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x1 1 1 0 0 Computing a subcoloring gives a x2 0 1 0 1 partition of the inequalities that are easier to handle. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 11 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications: Maximum Feasible subsystems C1 C2 C3 C4 x1 1 1 0 0 Computing a subcoloring gives a x2 0 1 0 1 partition of the inequalities that are easier to handle. x3 1 1 1 0 S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 11 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Applications Item Pricing: Highway Problem. Unspittable Flow on a path. Distributed Computing: Cluster graph. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 12 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Results Graph U.B L.B n n n General + O( 2 ) 2 log n+1 [Albertson et al 89] log2 n/4 log2 n 2 [Broersma et al. 03] √ Perfect p2n + 1/4 2n − 1 [Erdös et al. 91] Chordal log n log n [Broersma et al. 03] S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 13 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Results F -free coloring: Color so that no color class has an induced graph isomorphic to F . Pk -free coloring: Each color class does not contain an induced Pk (path with k vertices). Subcoloring = P3-free coloring. [Fiala et al. 01] Deciding if G has an F -free coloring with r ≥ 2 colors is NP-hard for triangle-free planar graphs of max. degree 4. [Stacho 08] Deciding of a chordal graph has a subcoloring with r ≥ 3 colors is NP-hard. For r = 2, poly. time. [Hoàng Le 01] P4-free coloring of comparability, co-comparability graphs is NP-hard. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 14 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Interval Graphs a b c d G = (V , E) is an interval graph if the vertices can be e represented as an intersection e graph of intervals. a b c d S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 15 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subchromatic Number Interval Graphs Pick the middle clique and recurse. χs (G) ≤ blog2(n + 1)c S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 16 Introduction Applications Results Approximation Algorithm Hypocoloring Questions K1,k -free graphs K1,3 [Albertson et al. 89] Interval graphs without an induced K1,k have subchromatic number at most k − 1. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 17 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Interval Graphs Dynamic Programming [Broersma et al.] Proceed in the order of the left endpoints of the intervals. For a color class C, either Ii+1 can be added to the last clique (if it does not intersect Ii+1 the previous clique), or Ii+1 can form a new clique in C. Hence we only need max(Cq−1), min(Cq), max(Cq) to decide the two cases. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 18 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Interval Graphs Dynamic Programming [Broersma et al.] Proceed in the order of the left endpoints of the intervals. For a color class C, either Ii+1 can be added to the last clique (if it does not intersect Ii+1 the previous clique), or Ii+1 can form a new clique in C. Hence we only need max(Cq−1), min(Cq), max(Cq) to decide the two cases. S. Pemmaraju et al. Subcoloring and Hypocoloring Interval Graphs 18 Introduction Applications Results Approximation Algorithm Hypocoloring Questions Subcoloring Interval Graphs Dynamic Programming [Broersma et al.] State specified by a (3r + 1)-tuple [k; union0, inter, union] 0 Ii+1 where union , inter, union are r-tuples. Total states O(n3r+1). Compute Boolean value B[k; union0, inter, union] S.
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