Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Max-Point-Tolerance Graphs D. Catanzaro 1 , S. Chaplick 2 , S. Felsner 6 , B. Halldórsson 3 , M. Halldórsson 4 , T. Hixon 6 , J. Stacho 5 1 Computer Science Department, Université Libre de Bruxelles 2 Department of Applied Mathematics, Charles University 3 School of Science and Engineering, Reykjavik University 4 School Computer Science, Reykjavik University 5 Mathematics Institute, University of Warwick 6 Institut für Mathematik, TU Berlin Minisymposium on Geometric Representations of Graphs CanaDAM (June 11, 2013) 1
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Outline Background and Motivation 1 Properties and Characterizations of MPT graphs 2 Combinatorial Optimization Problems 3 2
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Outline Background and Motivation 1 Properties and Characterizations of MPT graphs 2 Combinatorial Optimization Problems 3 3
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Intersection Graphs Definition For a collection of sets S = { S 0 , ..., S n − 1 } the intersection graph of S has vertex set S and edge set { S i S j : i , j ∈ { 0 , ..., n − 1 } , i � = j , and S i ∩ S j � = ∅} Figure : http://upload.wikimedia.org/wikipedia/ commons/e/e9/Intersection_graph.gif 4
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Interval Intersection Graph Classes Interval : intersection graphs of intervals of R . Tolerance : interval graphs where each pair of intervals tolerate intersections up to min { t u , t v } without corresponding to edges. Max-Tolerance : interval graphs where each pair of intervals tolerate intersections up to max { t u , t v } without corresponding to edges. d d b 4 5 1 a c 3 5 4 c a 2 1 2 3 b 5
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Geometric Graph Classes (all in the plane) Rectangle : intersection graphs of axis-aligned rectangles. Right-Triangle : intersection graphs of axis-aligned right triangles. L : intersection graphs of axis-aligned L-shapes. Segment : intersection graphs of line segments. 2-DIR : intersection graphs of vertical and horizontal line segments. Semi-square: isosceles right-triangle = max-tolerance [M. Kaufmann, J. Kratochvil, K.A. Lehmann, A.R. Subramanian, SODA 2006] . v 1 v 1 v 1 v 2 v 2 v 2 v 1 v 3 v 3 v 3 v 2 v 4 v 4 v 4 v 4 v 6 v 5 v 5 v 5 v 5 v 3 G v 6 v 6 v 6 6
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Max Point Tolerance (MPT). Definition G = ( V , E ) is MPT when there exists pointed intervals { ( I v , p v ) } v ∈ V such that uv ∈ E iff { p u , p v } ⊆ I u ∩ I v . T T x x v 1 T 1 v 2 v 1 v 2 v k ... T 2 . . . . . v k ... T 1 T 2 T k T k [D. Catanzaro, B.V. Halldórsson, M. Labbé 2012]: At most n 2 maximal cliques; i.e., polytime algorithm for maximum clique. Weighted Clique Cover is NP-complete. 7
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems In this talk Graph Class relationships: MPT includes interval graphs, complete bipartite graphs, and outerplanar graphs. MPT is included in: L, right-triangle, and rectangle. Characterizations: (the following are equivalent) G is a max-point-tolerance graph. G is linear L = linear rectangle = linear right-triangle. G has a specific four point vertex ordering condition. G is a *special* intersection of two interval graphs. G is a *special* segment graph. Combinatorial Optimization Problems: Weighted Independent set can be solved in polytime on MPT graphs. Colouring is NP-complete for MPT graphs. 2-Approximation of Clique Cover. 8
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Outline Background and Motivation 1 Properties and Characterizations of MPT graphs 2 Combinatorial Optimization Problems 3 9
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Linear Ls c L r L -t L L -c L Figure : Anatomy of an L-shape in a linear L-system. Notice that linear L = linear rectangle = linear right-triangle. v 1 v 1 v 1 v 2 v 2 v 2 v 1 v 3 v 3 v 3 v 2 v 4 v 4 v 4 v 4 v 6 v 5 v 5 v 5 v 5 v 3 G v 6 v 6 v 6 10
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems MPT = linear L s i p i e i -s i -p i Figure : Illustrating the equivalence between MPT representations and linear L-systems. From left-to-right: the L-shape corresponding to a pointed-interval, two examples of non-adjacent vertices as pointed-intervals and the corresponding linear Ls, and one example of adjacent vertices as pointed-intervals and the corresponding linear Ls. 11
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Some simple linear L-systems u 1 ... v 1 v 1 v 1 u 2 v 3 v 3 v 2 ... K t C 6 C 7 K a,b v 2 v 2 u a ... v 3 v 5 v 5 v 1 u 1 v 1 v 1 v 1 ... v 1 v 6 v 2 v 7 v 4 v 4 u 2 v 2 v 2 v t v 2 v 3 v t v 2 v 5 v 6 v 6 . v 6 . . ... ... v b v 3 v 4 v 4 v 5 v 7 u a v b Note: Outerplanar graphs precisely the contact linear L-graphs. 12
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Interval Graphs are ”Anchored” linear Ls I i I i I j I j L i L i L j L j 13
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Vertex Orders Theorem (Olariu 1991,Ramalingam and Pandu Rangan 1988, Raychaudhuri 1987) G = ( V , E ) is an interval graphs iff V can be ordered by < so that for every u < v < w, if uw ∈ E, then uv ∈ E. u v w Theorem G = ( V , E ) is an MPT graph iff V can be ordered by < so that for every u < v < w < x, if uw , vx ∈ E, then vw ∈ E u v w x 14
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems MPT graphs as the intersection of two Interval Graphs Theorem There are two interval graphs H 1 = ( V , E 1 ) and H 2 = ( V , E 2 ) such that E = E 1 ∩ E 2 and the vertices of G can be ordered by < so that for every u < v < w if uw ∈ E 1 then uv ∈ E 1 and if uw ∈ E 2 then wv ∈ E 2 . v 1 v 1 v 2 v 2 v 1 v 1 v 1 v 3 v 3 v 2 v 2 v 2 v 4 v 4 v 3 v 3 v 3 v 5 v 5 v 4 v 4 v 4 v 6 v 6 v 5 v 5 v 5 v 6 v 7 v 6 v 7 v 6 v 7 v 7 v 7 H 2 H 1 G 15
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems MPT graph as Segment Graphs Theorem G = ( V , E ) is an MPT graph iff each vertex can be represented by a line segment tangent to a parabola. 16
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Outline Background and Motivation 1 Properties and Characterizations of MPT graphs 2 Combinatorial Optimization Problems 3 17
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Weighted Independent Set idea: right-dominant Ls. L L For a , b ∈ { 1 , ...., n } and a ≤ b , let L a , b denote the subset of { L a , ..., L b } which : occurs strictly to the left of the line x = min { r a − 1 , r b + 1 } ; and includes no neighbours of v a − 1 (i.e., occurs strictly below the line y = a − 1); and includes no neighbours of v b + 1 . Optimal solution with L i right-dominant is: Opt( L 1 , i − 1 ) ∪ { L i } ∪ Opt( L i + 1 , n ). So, the table we need has O ( n 2 ) entries each of which takes O ( n ) to compute; i.e., O ( n 3 ) total. 18
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Colouring is NP-complete From the Hardness of coloring circular arc graphs. [Garey, Johnson, Miller, Papadimitriou; 1980]. This gadget realizes the (the construction works permutation for any permutation) (5,3,2,4,1) 1 1 1 4 4 4 2 2 2 3 3 3 5 5 5 1 2 3 4 Cut 5 Cut vertices (1,2,3,4,5) appear on the other side of the representation in the order (5,3,2,4,1) Curcular-Arc Graph 19 5 3 2 4 Here k= 5 1
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems 2-Approx For Clique Cover Clique Cover Problem : Partition the graph into a minimum number of cliques. Algorithm: Choose a greedy independent set following an MPT-order. Build part of the clique cover from this independent set. Remove this partial clique cover. The remainder is an interval graph. Construct a clique cover of the remaining interval graph. 20
Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems Open Problems Recognition of MPT graphs. k-colouring. Other combinatorial optimization problems. Thank you for your attention! 21
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