Max-Point-Tolerance Graphs D. Catanzaro 1 , S. Chaplick 2 , S. - - PowerPoint PPT Presentation

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Max-Point-Tolerance Graphs

• D. Catanzaro1, S. Chaplick2, S. Felsner6, B. Halldórsson3,
• M. Halldórsson4, T. Hixon6, J. Stacho5

1Computer Science Department, Université Libre de Bruxelles 2Department of Applied Mathematics, Charles University 3School of Science and Engineering, Reykjavik University 4School Computer Science, Reykjavik University 5Mathematics Institute, University of Warwick 6Institut für Mathematik, TU Berlin

Minisymposium on Geometric Representations of Graphs CanaDAM (June 11, 2013)

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Outline

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Background and Motivation

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Properties and Characterizations of MPT graphs

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Combinatorial Optimization Problems

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Outline

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Background and Motivation

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Properties and Characterizations of MPT graphs

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Combinatorial Optimization Problems

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Intersection Graphs

Definition For a collection of sets S = {S0, ..., Sn−1} the intersection graph

• f S has vertex set S and edge set

{SiSj : i, j ∈ {0, ..., n − 1}, i = j, and Si ∩ Sj = ∅}

Figure : http://upload.wikimedia.org/wikipedia/ commons/e/e9/Intersection_graph.gif

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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{tu, tv} without corresponding to edges. Max-Tolerance: interval graphs where each pair of intervals tolerate intersections up to max{tu, tv} without corresponding to edges.

d 4 5 b 1 2 3 a c a c b d 1 2 4 5 3

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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].

G v1 v2 v3 v4 v1 v2 v4 v6 v6v5 v3 v5 v1 v2 v4 v6 v3 v5 v1 v2 v4 v6 v3 v5

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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 {(Iv, pv)}v∈V such that uv ∈ E iff {pu, pv} ⊆ Iu ∩ Iv.

Tk x v1 v2 vk T1 T2 Tk ... ... . . . . . T1 T2 v1 v2 vk x T T

[D. Catanzaro, B.V. Halldórsson, M. Labbé 2012]: At most n2 maximal cliques; i.e., polytime algorithm for maximum clique. Weighted Clique Cover is NP-complete.

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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.

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Outline

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Background and Motivation

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Properties and Characterizations of MPT graphs

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Combinatorial Optimization Problems

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Linear Ls

L

• tL

rL cL

• cL

Figure : Anatomy of an L-shape in a linear L-system.

Notice that linear L = linear rectangle = linear right-triangle.

G v1 v2 v3 v4 v1 v2 v4 v6 v6v5 v3 v5 v1 v2 v4 v6 v3 v5 v1 v2 v4 v6 v3 v5

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

MPT = linear L

• si
• pi

ei si pi

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

• f adjacent vertices as pointed-intervals and the corresponding linear

Ls.

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Some simple linear L-systems

v1 v2 v3 v6 v4 v5 v1 v2 v3 v6 v4 v5 v7 C6 v1 v3 v5 v6 v4 v2 v1 v3 v5 v6 v4 v2 v7 C7 v1 vb v2 u2 u1 ua Ka,b ... ... ... ... ... v2 v1 vb u1 ua u2 . . . v1 v2 vt Kt v1 vt v2 ... v3

Note: Outerplanar graphs precisely the contact linear L-graphs.

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Interval Graphs are ”Anchored” linear Ls

Ii Ij Li Lj Ii Ij Li Lj

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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.

v w u

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

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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 H1 = (V, E1) and H2 = (V, E2) such that E = E1 ∩ E2 and the vertices of G can be ordered by < so that for every u < v < w if uw ∈ E1 then uv ∈ E1 and if uw ∈ E2 then wv ∈ E2.

v1 v2 v3 v4 v5 v6 v7 G v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 v1 v2 v3 v4 v5 v6 v7 H1 H2 v1 v2 v3 v4 v5 v6 v7

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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.

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Background and Motivation Properties and Characterizations of MPT graphs Combinatorial Optimization Problems

Outline

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Background and Motivation

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Properties and Characterizations of MPT graphs

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Combinatorial Optimization Problems

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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 La,b denote the subset of {La, ..., Lb} which :

• ccurs strictly to the left of the line x = min{ra−1, rb+1}; and

includes no neighbours of va−1 (i.e., occurs strictly below the line y = a − 1); and includes no neighbours of vb+1. Optimal solution with Li right-dominant is: Opt(L1,i−1) ∪ {Li} ∪ Opt(Li+1,n). So, the table we need has O(n2) entries each of which takes O(n) to compute; i.e., O(n3) total.

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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].

1 2 3 4 5 2 3 4 5 1 2 3 4 5 1 1 2 3 4 5 1 2 3 4 5 Cut vertices (1,2,3,4,5) appear on the other side

• f the representation

in the order (5,3,2,4,1) This gadget realizes the permutation (5,3,2,4,1) (the construction works for any permutation) Here k= 5 Cut Curcular-Arc Graph

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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.

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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!