Deleting edges to save cows - an application of treewidth

Transcription

1 Deleting edges to save cows - an application of treewidth UCLA, 14th February 2017 Kitty Meeks Joint work with Jessica Enright (University of Stirling)

2 2/13

3 The animal contact network 3/13

4 The animal contact network 3/13

5 The animal contact network 3/13

6 The animal contact network 3/13

7 Modifying the network Vertex-deletion 4/13

8 Vertex-deletion E.g. vaccinate all animals at a particular animal holding. Modifying the network 4/13

9 Vertex-deletion E.g. vaccinate all animals at a particular animal holding. Modifying the network Edge-deletion 4/13

10 Modifying the network Vertex-deletion E.g. vaccinate all animals at a particular animal holding. Edge-deletion E.g. Double fence lines 4/13

11 Modifying the network Vertex-deletion E.g. vaccinate all animals at a particular animal holding. Edge-deletion E.g. Double fence lines Testing or quarantine for animals on a particular trade route 4/13

12 Modifying the network Vertex-deletion E.g. vaccinate all animals at a particular animal holding. Edge-deletion E.g. Double fence lines Testing or quarantine for animals on a particular trade route Cost of modifications The cost of deleting individual vertices/edges may vary; this can be captured with a weight function on vertices and/or edges. 4/13

13 How do we want to change the network? 5/13

14 Desirable properties may include: Bounded component size How do we want to change the network? 5/13

15 How do we want to change the network? Desirable properties may include: Bounded component size Bounded degree 5/13

16 How do we want to change the network? Desirable properties may include: Bounded component size Bounded degree Bounded d-neighbourhood 5/13

17 How do we want to change the network? Desirable properties may include: Bounded component size Bounded degree Bounded d-neighbourhood Low connectivity 5/13

18 How do we want to change the network? Desirable properties may include: Bounded component size Bounded degree Bounded d-neighbourhood Low connectivity We may additionally want to: consider the total number of animals in e.g. a connected component, rather than just the number of animal holdings 5/13

19 How do we want to change the network? Desirable properties may include: Bounded component size Bounded degree Bounded d-neighbourhood Low connectivity We may additionally want to: consider the total number of animals in e.g. a connected component, rather than just the number of animal holdings place more or less strict restrictions on individual animal holdings 5/13

20 Bounding the component size by deleting edges Let C h be the set of all connected graphs on h vertices. 6/13

21 Bounding the component size by deleting edges Let C h be the set of all connected graphs on h vertices. C h -FREE EDGE DELETION Input: A Graph G = (V, E) and an integer k. Question: Does there exist E E with E = k such that G \ E does not contain any H C h as an induced subgraph? 6/13

22 Bounding the component size by deleting edges Let C h be the set of all connected graphs on h vertices. C h -FREE EDGE DELETION Input: A Graph G = (V, E) and an integer k. Question: Does there exist E E with E = k such that G \ E does not contain any H C h as an induced subgraph? This problem has also been called: Min-Max Component Size Problem Minimum Worst Contamination Problem Component Order Edge Connectivity 6/13

23 Bounding the component size: what is known? This problem is NP-complete in general, even when h = 3. 7/13

24 Bounding the component size: what is known? This problem is NP-complete in general, even when h = 3. Theorem (Cai, 1996) C h -FREE EDGE DELETION can be solved in time O(h 2k n h ), where n is the number of vertices in the input graph. 7/13

25 Bounding the component size: what is known? This problem is NP-complete in general, even when h = 3. Theorem (Cai, 1996) C h -FREE EDGE DELETION can be solved in time O(h 2k n h ), where n is the number of vertices in the input graph. Theorem (Gross, Heinig, Iswara, Kazmiercaak, Luttrell, Saccoman and Suffel, 2013) C h -FREE EDGE DELETION can be solved in polynomial time when restricted to trees. 7/13

26 Treewidth is relevant (T, D) is a tree decomposition of G if T is a tree and D = {D(t) : t V(T)} is a collection of non-empty subsets of V(G) (or bags), indexed by the nodes of T, satisfying: 1 V(G) = t V(T) D(t), 2 for every e = uv E(G), there exists t V(T) such that u, v D(t), 3 for every v V(G), if T(v) is defined to be the subgraph of T induced by nodes t with v D(t), then T(v) is connected. 8/13

27 Treewidth is relevant A plot of the treewidth of the largest component in an undirected version of the cattle movement graph in Scotland in 2009 over a number of different days included: all day sets start on January 1, /13

28 New results Theorem (Enright and M., 2017) There exists an algorithm to solve C h -FREE EDGE DELETION in time O((wh) 2w n) on an input graph with n vertices whose treewidth is at most w. 9/13

29 New results Theorem (Enright and M., 2017) There exists an algorithm to solve C h -FREE EDGE DELETION in time O((wh) 2w n) on an input graph with n vertices whose treewidth is at most w. We recursively compute the minimum number of edges we delete, for each combination of: a partition of the vertices in the bag, indicating which are allowed to belong to the same component, and a function from blocks of the partition to [h], indicating the maximum number of vertices allowed so far in the component containing the block in question. 9/13

30 New results 10/13

31 New results Theorem (Enright and M., 2015+) Let Π be a monotone graph property defined by the set of forbidden subgraphs F, where max{ F : F F} exists and is equal to h. Then EDGE DELETION TO Π can be solved in time f (h, w) n on an input graph with n vertices whose treewidth is at most w, where f is an explicit computable function. 11/13

32 Algorithms in practice Graph ID Graph information Tree decomposition method CP method v(g) e(g) tw(g) Minimum Time Minimum Time Time to deletion deletion confirmation found found /13

33 Future directions Why do trade networks have small treewidth? 13/13

34 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size 13/13

35 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs 13/13

36 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs Extra structure in directed graphs 13/13

37 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs Extra structure in directed graphs Temporal connectivity 13/13

38 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs Extra structure in directed graphs Temporal connectivity New parameters to capture structure of the input 13/13

39 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs Extra structure in directed graphs Temporal connectivity New parameters to capture structure of the input More complicated or less costly goals 13/13

40 Future directions Why do trade networks have small treewidth? Budget as parameter, rather than desired component size Geographic networks - planar graphs, or powers of planar graphs Extra structure in directed graphs Temporal connectivity New parameters to capture structure of the input More complicated or less costly goals THANK YOU 13/13