Algorithms for Minimizing Inverse Geodesic Length - Thesis ... · Deﬁnitions Deﬁnition (Inverse Geodesic Length) Let G = (V,E) be an undirected graph, with edges of unit length.We

Algorithms for Minimizing Inverse Geodesic LengthThesis A Presentation

Joshua LauSupervisor: A/Prof. Serge Gaspers

April 9, 2018

Joshua Lau (UNSW) Algorithms for Minimizing IGL April 9, 2018 1 / 19

Outline

1 Introduction

2 TreeMinIGL

3 TreewidthIGL

4 Concluding Remarks


Definitions

Definition (Inverse Geodesic Length)Let G = (V , E ) be an undirected graph, with edges of unit length. Wedefine the Inverse Geodesic Length of G as

IGL(G) =∑

{u,v}⊆V

1d(u, v)

where d(u, v) is the length of a shortest path from u to v .

Note: If u and v are disconnected then d(u, v) =∞ so we take 1d(u,v) = 0.


Definitions

Problem (MinIGL)Input: A graph G, an integer k ≤ |V | and a target IGL TQuestion: Does there exist X ⊆ V such that |X | ≤ k and IGL(G−X ) ≤

T?

When our budget k is 0, the problem is equivalent to computing the IGL.When our target T is 0, the problem is equivalent to Vertex Cover.


ExampleConsider the following graph G when k = 2.

IGL(G) = 9× 11 + 5× 1

2 + 5× 13 + 2× 1

4 = 1323

Delete these k = 2 vertices to obtain G ′.IGL(G ′) = 3× 1

1 + 1× 12 = 31

2


ExampleConsider the following graph G when k = 2.IGL(G) = 9× 1

1 + 5× 12 + 5× 1

3 + 2× 14 = 132

3


1 + 1× 12 = 31

2


Example

Consider the following graph G when k = 2.IGL(G) = 9× 1

1 + 5× 12 + 5× 1

3 + 2× 14 = 132

3

Delete these k = 2 vertices to obtain G ′.

IGL(G ′) = 3× 11 + 1× 1

2 = 312


Example

Consider the following graph G when k = 2.IGL(G) = 9× 1

1 + 5× 12 + 5× 1

3 + 2× 14 = 132

3


1 + 1× 12 = 31

2


Background and Motivation

Higher IGL corresponds to vertices clustered closer together:IGL(Kn) =

(n2)

IGL(Pn) < n ln n, where Pn is a path with n vertices.IGL(G) = 0, when G has no edges!

Definition (Average IGL)The average IGL of a graph G = (V , E ) is given by IGL(G)/

(|V |2).

In logistics and transportation, average IGL is useful for classifyingnetworks by how efficiently resources can be moved through them.

Deletions in our example reduced average IGL from 0.65 to 0.35.




(n2)



(|V |2).






(n2)



(|V |2).






(n2)



(|V |2).





Network vulnerability: counter-terrorism and security games.Australian domestic airports.Underlying construction principle of real-world networks.Effective on both sparse and dense graphs.


Outline

1 Introduction

2 TreeMinIGL

3 TreewidthIGL



TreeMinIGL

Subexponential TreeMinIGLCan MinIGL be solved on trees with n vertices in subexponential (2o(n))time?

First useful step towards investigating potential tractability of TreeMinIGL.


Literature: Selected results

MinIGL on general graphs is NP-complete by trivial reduction fromVertex Cover when T = 0, and W [1]-hard for parameter k (Aziz,Gaspers and Najeebullah, IJCAI 2017).MinIGL is W [1]-hard for parameter treewidth via reduction fromEquitable Colouring (Aziz, Gaspers, Lee, Najeebullah, AAMAS 2018).It is unlikely that problems that are W [1]-hard are FPT.Very few problems on trees are NP-hard. Notably the Firefighterproblem is NP-complete on trees of maximum degree three (Finbowet al., Discrete Mathematics 2007)


Preliminary findings

Definition (Wiener Index)Let G = (V , E ) be a connected graph, with edges of non-negative weight.We define the Wiener index of G as∑

(G) =∑

{u,v}⊆Vd(u, v)

where d(u, v) is the length of a shortest path from u to v .

We can extend the Wiener Index to graphs with multiple componentsby taking the sum of the Wiener Index for each component.Wiener index can be easily computed in O(n) on trees.There is a simple polynomial time DP solution for minimizing thismetric on trees (allowing up to k vertex deletions).



Algorithms for MinIGL on trees:Naive O(nk+2):

(nk)

= O(nk) subsets and APSP on a tree in O(n2).

There are 2M distributions of distances from any particular vertex in atree of size M.O∗(3L), where L is the size of the largest remaining component, usingDP over a preorder traversal of the tree.

Planned approach:Hypothesize that L ≤

( nk)c for some c ≥ 1: no component is

extremely large.Proving this yields a subexponential time solution by combining thetwo solutions.




(nk)

= O(nk) subsets and APSP on a tree in O(n2).There are 2M distributions of distances from any particular vertex in atree of size M.

O∗(3L), where L is the size of the largest remaining component, usingDP over a preorder traversal of the tree.







(nk)

= O(nk) subsets and APSP on a tree in O(n2).There are 2M distributions of distances from any particular vertex in atree of size M.O∗(3L), where L is the size of the largest remaining component, usingDP over a preorder traversal of the tree.







(nk)

= O(nk) subsets and APSP on a tree in O(n2).There are 2M distributions of distances from any particular vertex in atree of size M.O∗(3L), where L is the size of the largest remaining component, usingDP over a preorder traversal of the tree.






Can we do better than APSP (O(n2)) when evaluating the IGL for everysubset?

Yes!

TreeIGLThe IGL of a tree with n vertices and unit length edges can be computedin O(n log2 n)

1 Divide-and-conquer (centroid decomposition);2 Express paths from the root as polynomials (degree at most n);3 Fast polynomial multiplication using Fast Fourier Transform (FFT) to

combine paths.

Hence, we can also compute the IGL of unweighted forests in O(n log2 n).



Can we do better than APSP (O(n2)) when evaluating the IGL for everysubset?Yes!



combine paths.




Can we do better than APSP (O(n2)) when evaluating the IGL for everysubset?Yes!



combine paths.



Outline

1 Introduction

2 TreeMinIGL

3 TreewidthIGL



TreewidthIGL

Naturally, we wish to extend this result to graphs with bounded treewidth.

FP Subquadratic TreewidthIGLCan the IGL of a graph with treewidth w be computed in fixed parametersubquadratic time (with respect to parameter w)?

Also useful for the analysis and classification of real-world graphs.


TreewidthIGL

Naturally, we wish to extend this result to graphs with bounded treewidth.

FP Subquadratic TreewidthIGLCan the IGL of a graph with treewidth w be computed in fixed parametersubquadratic time (with respect to parameter w)?

Also useful for the analysis and classification of real-world graphs.



Distances on graphs with bounded treewidth (treewidth at most w)For any fixed w ′, can detect if tw(G) ≤ w ′ and produce acorresponding tree decomposition, in O(n) for FP w ′ (Bodlaender,SIAM J. Computing, 1996). Hence, we can also find tw(G) and thecorresponding decomposition in O(n).

Cabello and Knauer, Computational Geometry 2009:

Techniques for divide and conquer on graphs with bounded treewidth(analogue to centroid decomposition)Compute Wiener index on connected, weighted graphs inO(n logw−1 n) = O(n1+o(1)) by reduction to orthogonal rangesearching.

Adapted to radius/diameter on weighted graphs (Abboud, VassilevskaW., Wang, SODA 2016).APSP can be computed on weighted graphs in O(n2w) (de Weerdt,van der Krogt, JAIR 2012).



Distances on graphs with bounded treewidth (treewidth at most w)For any fixed w ′, can detect if tw(G) ≤ w ′ and produce acorresponding tree decomposition, in O(n) for FP w ′ (Bodlaender,SIAM J. Computing, 1996). Hence, we can also find tw(G) and thecorresponding decomposition in O(n).Cabello and Knauer, Computational Geometry 2009:






Techniques for divide and conquer on graphs with bounded treewidth(analogue to centroid decomposition)

Compute Wiener index on connected, weighted graphs inO(n logw−1 n) = O(n1+o(1)) by reduction to orthogonal rangesearching.











Adapted to radius/diameter on weighted graphs (Abboud, VassilevskaW., Wang, SODA 2016).

APSP can be computed on weighted graphs in O(n2w) (de Weerdt,van der Krogt, JAIR 2012).








It is likely that computing the IGL of a graph is as hard as computing itsdistance distribution.

Planned approach:More information than diameter, less than APSP, so one wouldexpect the existence a solution with complexity in between.Unit length edges mean this problem is more approachableAdapt algorithm to graphs with bounded treewidth: use Cabello andKnauer’s decomposition.


Outline

1 Introduction

2 TreeMinIGL

3 TreewidthIGL



Next steps

Immediate steps:Investigate hypothesized bound on size L of largest remainingcomponent.Examine the polynomial multiplication method under orthogonalrange constraints, similar to Cabello and Knauer.

Future work (beyond scope of this Thesis):Attempt to prove/disprove NP-hardness of TreeMinIGLExtend subexponential algorithm for TreeMinIGL to graphs withbounded treewidth (albeit it is unlikely that it is FPT, since it isW [1]-hard).Compare hardness of computing distance distributions againstcomputing IGL.


Next steps

Immediate steps:Investigate hypothesized bound on size L of largest remainingcomponent.Examine the polynomial multiplication method under orthogonalrange constraints, similar to Cabello and Knauer.

Future work (beyond scope of this Thesis):Attempt to prove/disprove NP-hardness of TreeMinIGLExtend subexponential algorithm for TreeMinIGL to graphs withbounded treewidth (albeit it is unlikely that it is FPT, since it isW [1]-hard).Compare hardness of computing distance distributions againstcomputing IGL.


Documents

Algorithms for Minimizing Inverse Geodesic Length - Thesis ... · Deﬁnitions Deﬁnition (Inverse Geodesic Length) Let G = (V,E) be an undirected graph, with edges of unit length.We