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.