Lower Bounds for Boxicity Abhijin Adiga Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore Outline Boxicity The boxicity of a graph is the minimum dimension k in which a given graph can be represented as an intersection graph of axis parallel k-dimensional boxes. Interval Graphs • An interval graph is a graph that can be represented as the intersection of closed intervals on the real line. • Interval graphs are precisely the class of graphs with boxicity at most 1. Equivalent Definition of Boxicity A graph G has box(G) ≤ k if and only if there exist k interval graphs I1, I2,..., Ik , such that E(G)= E(I1) ∩ E(I2) ∩···∩ E(Ik ). Significance: These problems can be looked at as combinatorial problems instead of as geometric problems. • Boxicity was introduced by Roberts in 1969. • Cozzens showed that computing the boxicity of a graph is NP-hard. Later Yannakakis showed that deciding whether boxicity or cubicity of a graph is at most 3 is NP-complete. Finally, Kratochvil showed that deciding whether the boxicity of a graph is at most 2 is NP-complete. • A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs • max-clique problem • max-independent set problem n • Boxicity of a graph is at most ⌊ 2 ⌋. n is the number of vertices in the graph. • Boxicity was introduced by Roberts in 1969. • Cozzens showed that computing the boxicity of a graph is NP-hard. Later Yannakakis showed that deciding whether boxicity or cubicity of a graph is at most 3 is NP-complete. Finally, Kratochvil showed that deciding whether the boxicity of a graph is at most 2 is NP-complete. • A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs • max-clique problem • max-independent set problem n • Boxicity of a graph is at most ⌊ 2 ⌋. n is the number of vertices in the graph. • Boxicity was introduced by Roberts in 1969. • Cozzens showed that computing the boxicity of a graph is NP-hard. Later Yannakakis showed that deciding whether boxicity or cubicity of a graph is at most 3 is NP-complete. Finally, Kratochvil showed that deciding whether the boxicity of a graph is at most 2 is NP-complete. • A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs • max-clique problem • max-independent set problem n • Boxicity of a graph is at most ⌊ 2 ⌋. n is the number of vertices in the graph. • Boxicity was introduced by Roberts in 1969. • Cozzens showed that computing the boxicity of a graph is NP-hard. Later Yannakakis showed that deciding whether boxicity or cubicity of a graph is at most 3 is NP-complete. Finally, Kratochvil showed that deciding whether the boxicity of a graph is at most 2 is NP-complete. • A number of NP-hard problems are either polynomial time solvable or have much better approximation ratio on low boxicity graphs • max-clique problem • max-independent set problem n • Boxicity of a graph is at most ⌊ 2 ⌋. n is the number of vertices in the graph. Bounds Upper Bounds n • box(G) ≤⌊ 2 ⌋ (Roberts 1969) • box(G) ≤ ∆ log n (Chandran et al 2008) • box(G) ≤ ∆2 (Chandran et al 2008) There have been several Lower Bound • Minimum interval supergraph of the given graph • Vertex-isoperimetric properties of the graph: subset neighborhoods Minimum Interval Supergraph A minimum interval supergraph of a graph G(V , E), is an interval supergraph of G with vertex set V and with least number of edges among all interval supergraphs of G. Let G(V , E) be any non-complete graph. Let Imin be a minimum interval supergraph of G. Then, Number of edges absent in G ||G|| box(G) ≥ = . Number of edges absent in Imin ||Imin|| Proof. Suppose G is the intersection of interval graphs {I1,..., Ik }, where k = box(G). Then, E(G)= E(I1) ∪ E(I2) ∪···∪ E(Ik ). Therefore, ||G|| ≤ ||I1|| + ||I2||··· + ||Ik || ≤ k||Imin||. ||G|| box(G) ≥ . ||Imin|| It is NP-complete to determine ||Imin||. So, now we proceed to use vertex-isoperimetric parameters of G to give an upperbound for ||Imin||. Vertex-Boundary and the Parameter bv (i) • Let X ⊆ V . Then, its vertex-boundary N(X , G)= {u ∈ V − X |∃v ∈ X , with uv ∈ E}. • bv (k, G) = min X ⊆V |N(X , G)|. |X |=k 1 5 In this example: • bv (1, G)= δ(G)=3 2 3 6 7 (always) • bv (2, G) = 2 4 8 Counting the Number of Edges in an Interval Graph Let I be an interval graph. Consider an interval representation with distinct end points. • For each vertex, count the number of intervals containing the right end-point of its interval. • Sum this up over all vertices to get ||I ||. Observation: The ”count” for the i th right end-point is equal to the size of the vertex-boundary of the set of vertices to the left of this end-point. We denote it as |N(Xi , I )|. n−1 Therefore, ||I || = |N(Xi , I )|. Xi=1 Counting the Number of Edges in an Interval Graph Let I be an interval graph. Consider an interval representation with distinct end points. • For each vertex, count the number of intervals containing the right end-point of its interval. • Sum this up over all vertices to get ||I ||. Observation: The ”count” for the i th right end-point is equal to the size of the vertex-boundary of the set of vertices to the left of this end-point. We denote it as |N(Xi , I )|. n−1 Therefore, ||I || = |N(Xi , I )|. Xi=1 Counting the Number of Edges in an Interval Graph Let I be an interval graph. Consider an interval representation with distinct end points. • For each vertex, count the number of intervals containing the right end-point of its interval. • Sum this up over all vertices to get ||I ||. Observation: The ”count” for the i th right end-point is equal to the size of the vertex-boundary of the set of vertices to the left of this end-point. We denote it as |N(Xi , I )|. n−1 Therefore, ||I || = |N(Xi , I )|. Xi=1 An Upper Bound for ||Imin|| Let Imin be the minimum interval supergraph of G. |N(Xi , Imin)|≥|N(Xi , G)|≥ bv (i, G). n−1 n−1 Therefore, ||Imin|| = |N(Xi , Imin)|≥ bv (i, G) Xi=1 Xi=1 Revisiting the lower bound: ||G|| ||G|| box(G) ≥ ≥ n n−1 min ||I || 2 − i=1 bv (i, G) P An Upper Bound for ||Imin|| Let Imin be the minimum interval supergraph of G. |N(Xi , Imin)|≥|N(Xi , G)|≥ bv (i, G). n−1 n−1 Therefore, ||Imin|| = |N(Xi , Imin)|≥ bv (i, G) Xi=1 Xi=1 Revisiting the lower bound: ||G|| ||G|| box(G) ≥ ≥ n n−1 min ||I || 2 − i=1 bv (i, G) P An Upper Bound for ||Imin|| Let Imin be the minimum interval supergraph of G. |N(Xi , Imin)|≥|N(Xi , G)|≥ bv (i, G). n−1 n−1 Therefore, ||Imin|| = |N(Xi , Imin)|≥ bv (i, G) Xi=1 Xi=1 Revisiting the lower bound: ||G|| ||G|| box(G) ≥ ≥ n n−1 min ||I || 2 − i=1 bv (i, G) P Proceeding further ... n n n n −1 −1 −1 − bv (i, G) = n − i − bv (i, G) 2 Xi=1 Xi=1 Xi=1 n−1 = n − i − bv (i, G). Xi=1 Therefore, ||G|| ||G|| box(G) ≥ n n−1 ≥ n−1 2 − i=1 bv (i, G) i=1 n − i − bv (i, G) P P Proceeding further ... n n n n −1 −1 −1 − bv (i, G) = n − i − bv (i, G) 2 Xi=1 Xi=1 Xi=1 n−1 = n − i − bv (i, G). Xi=1 Therefore, ||G|| ||G|| box(G) ≥ n n−1 ≥ n−1 2 − i=1 bv (i, G) i=1 n − i − bv (i, G) P P Strong Neighbourhood and the Parameter cv (i) • Let X ⊆ V . Then, its strong neighbourhood NS (X , G)= {u ∈ V − X |uv ∈ E, ∀v ∈ X }. • Let cv (k, G) = max X ⊆V |NS (X , G)|. |X |=k 1 5 In this example: • cv (1, G) = ∆(G)=3 2 3 6 7 (always). • cv (2, G) = 2 and cv (i, G) = 0 for i > 2. 4 8 Lower Bound revisited Since, cv (i, G)= n − i − bv (i, G), we have the following result: Theorem: Let G be a non-complete graph with n vertices. Then, G G || || box( ) ≥ n−1 . cv (i, G) Xi=1 Some Observations regarding cv • cv (i) ≥ cv (i + 1), i.e. cv is a monotonically non-increasing function of i. • What is cv (∆ + 1)? Immediate Applications: n • Boxicity of an (n − k − 1)-regular graph is at least 2k . What about a lower bound for a general graph? n • box(Cn) ≥ 3 . • Boxicity of an (n − k − 1)-regular graph whose complement is n C4 free is at least 4 . n • Boxicity of an (n − k − 1)-regular co-planar graph is at least 8 . Immediate Applications: n • Boxicity of an (n − k − 1)-regular graph is at least 2k . What about a lower bound for a general graph? n • box(Cn) ≥ 3 . • Boxicity of an (n − k − 1)-regular graph whose complement is n C4 free is at least 4 . n • Boxicity of an (n − k − 1)-regular co-planar graph is at least 8 . Immediate Applications: n • Boxicity of an (n − k − 1)-regular graph is at least 2k .