Distance-preserving Subgraphs of Interval Graphs Tata Institute of Fundamental Research, Mumbai Distance-preserving Subgraphs of Interval Graphs Kshitij Gajjar* Jaikumar Radhakrishnan ESA 2017, Vienna

Tata Institute of Fundamental Research, Mumbai Source: http://www.tifr.res.in

Preserving Distance Between Terminals 𝐺 Given 𝐺: undirected, unweighted (𝑛 vertices) 𝑇: terminals (𝑘 vertices) Typically, 𝑘≪𝑛 Objective 𝐻: subgraph of 𝐺 containing the terminals 𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇 Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻 Terminals Non-terminals 𝑣

Preserving Distance Between Terminals 𝐺 Given 𝐺: undirected, unweighted (𝑛 vertices) 𝑇: terminals (𝑘 vertices) Typically, 𝑘≪𝑛 Objective 𝐻: subgraph of 𝐺 containing the terminals 𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇 Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻 Terminals Non-terminals 𝑣

Preserving Distance Between Terminals 𝐺 Given 𝐺: undirected, unweighted (𝑛 vertices) 𝑇: terminals (𝑘 vertices) Typically, 𝑘≪𝑛 Objective 𝐻: subgraph of 𝐺 containing the terminals 𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇 Then 𝐻 is a distance-preserving subgraph of 𝐺 𝑢 𝐻 Terminals Non-terminals 𝑣

Definition Branching vertex A vertex 𝑣 is called a branching vertex if deg 𝑣 ≥3. 𝑣 𝑣 𝑣 deg 𝑣 =1 Not a branching vertex deg 𝑣 =2 Not a branching vertex deg 𝑣 =4 Branching vertex

Preserving Distance Between Terminals 𝐺 𝑑 𝐺 𝑢,𝑣 = 𝑑 𝐻 𝑢,𝑣 ∀ 𝑢, 𝑣∈𝑇 Solution Trivial: 𝐻=𝐺 Ideal: 𝐻=𝑇 Optimize Minimize the number of branching vertices in 𝐻 𝐻 Terminals Non-terminals

Related Work ⇒ Graph homeomorphisms [Fortune, Hopcroft, Wyllie 1980] Graph compression [Feder, Motwani 1995] Graph spanners [Peleg, Schaffer 1989] Steiner point removal [Gupta 2001] Vertex sparsification [Leighton, Moitra 2010] … Distance-preserving subgraph of 𝐺 with 𝑡 branching vertices Distance-preserving minor of 𝐺 with at most 𝑡+𝑘 vertices ⇒

Minors vs Subgraphs Theorem [Krauthgamer, Zondiner 2012] Every graph on 𝑘 terminal vertices admits a distance-preserving minor with at most 𝑂 𝑘 4 vertices. They explore various classes of graphs (planar graphs, trees). ⇒ Every graph on 𝑘 terminal vertices has a distance-preserving subgraph with at most 𝑂 𝑘 4 branching vertices.

Our Results Theorem [Upper bound] Every interval graph on 𝑘 terminal vertices has a distance-preserving subgraph with at most 𝑂(𝑘 log 𝑘) branching vertices. Question Is this optimal? Answer Yes.

Our Results Theorem [Lower bound] There exists an interval graph on 𝑘 terminal vertices for which every distance-preserving subgraph has at least Ω(𝑘 log 𝑘) branching vertices.

Our Results (continued) Theorem [Solvability] Finding the optimal distance-preserving subgraph of a graph is NP-complete. Theorem [Additive approximation] Every interval graph on 𝑘 terminal vertices has a +𝟏 approximating subgraph having at most 𝑂(𝑘) branching vertices. Theorem [Vertices vs edges] There exists an interval graph on 𝑘 terminal vertices such that every optimal distance-preserving subgraph of 𝐺 has 𝑂(𝑘) branching vertices but Ω(𝑘 log 𝑘) branching edges.

Plan for this talk Proof sketch for the upper bound. Proof sketch for the lower bound.

Proof of the Upper Bound Theorem Every interval graph on 𝑘 terminal vertices admits a distance- preserving subgraph having at most 𝑂(𝑘 log 𝑘) branching vertices.

Interval Graphs Definition An interval graph is the intersection graph of a family of intervals on the real line. Intervals are ordered from left to right on the basis of their right end point. 𝑣 2 𝑣 6 𝑣 8 𝑣 1 𝑣 3 𝑣 5 𝑣 4 𝑣 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Interval Graphs 𝑣 2 𝑣 6 𝑣 1 𝑣 8 𝑣 7 𝑣 3 𝑣 5 𝑣 4 𝑣 2 𝑣 6 𝑣 8 𝑣 3 𝑣 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

The Shipping Problem Intervals represent cargo ships docking at a seaport (vertices). Certain ships need to transfer containers to and from one another (terminal vertices). 𝑣 2 𝑣 6 𝑣 8 𝑣 1 𝑣 3 𝑣 5 𝑣 4 𝑣 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

The Shipping Problem Minimize the number of transfers per container (shortest paths). Minimize the number of ships dealing with multiple transfers (vertices of degree ≥ 3). 𝑣 2 𝑣 6 𝑣 8 𝑣 1 𝑣 3 𝑣 5 𝑣 4 𝑣 7 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Shortest Paths in Interval Graphs 𝑑 Mario,Peach = ? Idea Greed is good

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs

Shortest Paths in Interval Graphs 𝑑 Mario,Peach =6 Proof Induction on path length Such a path is called a greedy shortest path.

Multiple Sources, One Destination Consider the final interval on the greedy shortest path of each source before the destination. Sources Destination

Multiple Sources, One Destination Consider the final interval on the greedy shortest path of each source before the destination. Sources Destination

Multiple Sources, One Destination Lemma All sources can use the path of the source having the leftmost final interval. Sources Destination

Multiple Sources, One Destination Lemma All sources can use the path of the source having the leftmost final interval. Sources Destination

Multiple Sources, One Destination Lemma All sources can use the path of the source having the leftmost final interval. Corollary If sources are interconnected, destination requires one vertex to connect to all the sources. Sources Destination

Proof of the Upper Bound Define 𝑇 𝑘 = maximum number of branching vertices in an optimal distance-preserving subgraph of an interval graph with 𝑘 terminal vertices. To prove 𝑇 𝑘 ≤𝑂(𝑘 log 𝑘). Terminals Non-terminals 𝐺

Proof of the Upper Bound Define 𝑇 𝑘 = maximum number of branching vertices in an optimal distance-preserving subgraph of an interval graph with 𝑘 terminal vertices. To prove 𝑇 𝑘 ≤𝑂(𝑘 log 𝑘). 𝐺 1 𝐺 2

Proof of the Upper Bound To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. Sources Destination 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. Sources Destination 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. Sources Destination 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. Sources Destination 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations To prove 𝑇 𝑘 ≤𝑂 𝑘 log 𝑘 . Proof By induction. Sources Destination 𝐺 1 𝐺 2

Multiple Sources, Multiple Destinations Shortest paths from terminals in 𝐺 1 can be connected to terminals in 𝐺 2 using 𝑂(𝑘) additional branching vertices. 𝐺 1 𝐺 2

Completing the Proof 𝑇 𝑘 ≤𝑇 𝑘 2 +𝑇 𝑘 2 +𝑂(𝑘) 𝐺 1 𝐺 2 Both terminal vertices lie in 𝐺 2 Both terminal vertices lie in 𝐺 1 One terminal lies in 𝐺 1 , other in 𝐺 2 𝑇 𝑘 ≤𝑇 𝑘 2 +𝑇 𝑘 2 +𝑂(𝑘) 𝑇 𝑘 2 𝑇 𝑘 2 𝐺 1 𝐺 2

Proof of the Lower Bound Theorem For every 𝑘, there exists an interval graph on 𝑘 terminal vertices for which every distance-preserving subgraph has at least Ω(𝑘 log 𝑘) branching vertices.

The Interval Graph Interval starting points: 1,2,…, 𝑘 2 𝑘=8 1 2 3 4 5 6 7 8 9 . . . . . . 32 Interval starting points: 1,2,…, 𝑘 2 Interval length: 𝑘 𝑘=8

The Interval Graph Interval starting points: 1,2,…, 𝑘 2 𝑘=8 1 2 3 4 5 6 7 8 9 . . . . . . 32 Interval starting points: 1,2,…, 𝑘 2 Interval length: 𝑘 𝑘=8

A New Representation Represent each interval by a tile at its starting point. Each tile is a vertex. 𝑘=8

A New Representation Represent each interval by a tile at its starting point. Each tile is a vertex. 𝑘=8

A New Representation The terminals are strategically placed at 𝑘 of the 𝑘×𝑘 locations. 𝑘 𝑘 𝑘 𝑘

A New Representation Terminals Non-terminals The terminals are strategically placed at 𝑘 of the 𝑘×𝑘 locations. 𝑘 𝑘 𝑘 𝑘

Rearranging the Vertices in a Grid 𝑘 𝑘 𝑘 𝑘 View as a 𝑘×𝑘 matrix with 0-1 entries. Non-terminals are 0, terminals are 1. Permutation matrix.

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 000 001 010 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 000 001 𝑟𝑒 𝑣 3 000 =000 010 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 000 001 𝑟𝑒 𝑣 3 001 =100 010 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 000 001 𝑟𝑒 𝑣 3 010 =010 010 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 000 001 𝑟𝑒 𝑣 3 011 =110 010 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 00 1 1 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 00 1 1 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 00 1 1 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 01 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 01 011 100 101 110 111

The Bit-Reversal Permutation Matrix 000 001 010 011 100 101 110 111 Friends Two terminals are called friends if their column indices differ in exactly one bit. 000 001 010 01 011 100 101 Lemma Preserving distance between a pair of friends requires a unique branching vertex. 110 111

Completing the Proof Lemma Preserving distance between a pair of friends requires a unique branching vertex. + The number of pairs of friends is 𝑘 log 𝑘 2 ⇒ Theorem Preserving distances between all pairs of friends requires Ω 𝑘 log 𝑘 branching vertices.

Conclusion Subsequent Work Extended the result to bi-interval graphs. A polynomial-time algorithm for optimal single-source distance- preserving subgraphs of interval graphs (NP-complete in general). Open Problems Is this problem FPT (with parameter 𝑘)? Generalizations of interval graphs (perfect graphs, chordal graphs).

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