1. Alan Kuhnle*, Victoria G. Crawford, and My T. Thai
Scalable and Adaptive Algorithms for the Triangle Interdiction Problem on Billion-Scale Networks
Alan Kuhnle*, Victoria G. Crawford, and My T. Thai
November 20, 2017
IEEE ICDM 2017, New Orleans, LA
* Presenter

2. Network Transitivity
3 × Number of Triangles / Number of Connected Triplets
Transitivity: 0.571
Figure degrades all triangles in multiplex by removing one edge from each layer.
The smaller the objective, the less reliable the network is.

3. Transitivity and Reliability
Transitivity allows redundant paths
Bottom two nodes want to communicate

4. Transitivity and Social Influence
Complex propagation
Empirical studies (e.g. Centola1) link transitivity and influence
1 Centola, D. (2010). The Spread of Behavior in an Online Social Network Experiment.
Science, 329(5996), 1194–1197.

5. Triangle Interdiction Problem
Transitivity degradation by edge removal
Triangle Interdiction Problem (TIP)
Collection of triangles
Triangles after removal of S

6. Triangle Interdiction Problem
Transitivity: 0
Transitivity: 0.571

7. TIP Challenges TIP is an NP-hard problem
Kortsarz1 introduced a 2-approximation algorithm for TIP
Requires up to |E| LP solutions
Does not scale to large graphs
Only for static graphs
Primal-dual algorithm
List all triangles, iterate through them. If current triangle is unbroken, remove all of its edges.
Efficient 3-approximation but performs close to its worst-case guarantee
Modern networks are very large and dynamic
1 Guy Kortsarz. Approximating Maximum Subgraphs without Short Cycles. Siam Journal of Discrete Math, 24(1):255–269, 2010.

8. Our contributions TARL, a 5/2 approximation that solves one LP
Provides speed-up of up to |E| over Kortsarz.
Better worst-case guarantee than primal-dual
DART, fully dynamic 3-approximation
Usually within factor of 1.5 of optimal, better on sparse graphs
Efficiently update its solution upon edge / vertex insertion / deletion
Average update time on the order of a few microseconds
Does not need to store triangle list in memory
Runs on network with 4 billion triangles in only 37 GB memory

9. TIP IP
IP formulation of TIP:

10. TARL Description List all triangles in G
Construct and solve LP 1 optimally -> 𝑥 ∗
Break triangles containing a “low LP-weight” edge:
For each 𝑒∈𝐸
If 𝑥 ∗ 𝑒 <1/5
For each unbroken triangle containing e, add edge with largest 𝑥 ∗ value into the solution.
Break all other triangles:
Construct large (at least ½ of edges) bipartite subgraph of residual graph
Remove the complement of this graph

11. TARL TARL has performance guarantee of 5/2
It requires the solution of exactly one LP
So provides speed-up over 2-approximation of Kortsarz of up to |E|
However, the LP solution is still a large bottleneck

12. DART Overview Scalable algorithm with competitive ratio of 3
Maintains list of disjoint triangles 𝑈
Solution 𝑆⊆𝑈
Efficient dynamic data structure enables fast updates
Vertices
Edges
Triangles (in 𝑈)
Pointers from Edges to Triangles tell which triangle in U the edge belongs to.
Pointers from Triangles back to edges tell which edges form the triangle.
Pointers from Vertices to Edges tell which Edges are incident with that vertex.
Easy to update since all three of these are adjacency lists
K is number of layers in multiplex

13. DART Overview Visualization 𝑈 is maximal
Disjoint triangles, 𝑈
Solution 𝑆⊆𝑈 (red edges)
𝑊=𝑈\S (green edges)
𝑈 is maximal
𝑊 comprises all edges not forced to be in 𝑆 by adjacent triangles

14. DART Overview All DART methods rely on the following two procedures:
The AUGMENT procedure
Takes an input a single edge e
Ensures feasibility with respect to triangles containing e
Maintains disjoint triangle list U
The PRUNE procedure
DART
DART-BASE
DART-ADD
DART-REMOVE

15. DART Overview DART-BASE calls AUGMENT on each edge 𝑒
DART-ADD calls AUGMENT on newly inserted edge 𝑒
DART-REMOVE
Determines which triangle 𝑇 in 𝑈 an edge 𝑒 belongs to, if any
Removes 𝑇= {𝑒,𝑓,𝑔} from 𝑆∪𝑊
Calls AUGMENT on 𝑓,𝑔
All three DART procedures conclude with a call to PRUNE

16. DART Overview AUGMENT takes as input an edge e
Ensures 𝑆 hits all triangles containing e
Avoids adding new triangles to U if possible
Otherwise, adds new disjoint triangles to 𝑈

17. DART Overview
Have the following performance guarantee for DART

18. Results
DART returns solution close to OPT (compared on small ER network( 30 nodes)) at much faster running times (here TARL is an alg. We formulated but unscalable).

19. Visualization on small ER graph

20. Results on smaller traces

21. DART Scalability
In practice, DART is capable of running on networks with billions of triangles in under 2 hours
It can update its solution in microseconds
|S| is size of solution. Mem is max. memory used. Dynamic times are averaged over 1 million edge additions /removals. The size increase / decrease is how much solution changed after adding /removing all 1 million edges. Repeated 10 times and averaged over all.

22. Conclusions
Provided two algorithms, TARL and DART, to approximate TIP: find minimum-size set of edges whose removal breaks all triangles.
TARL
Performance guarantee of 5/2
Provides significant speed-up over Kortsarz (up to 103 times faster)
DART
Competitive guarantee of 3
Performs close to optimal on sparse networks
Fully dynamic
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