1. Finding Subgraphs with Maximum Total Density and Limited Overlap
Ding Wentao

2. Definitions For graph 𝐺= 𝑉,𝐸 𝑘,𝛼 −𝐷𝑆𝐿𝑂 𝜌 𝐺 = 𝐸 𝑉
𝜌 𝐺 = 𝐸 𝑉
For 𝑆⊆𝑉, the subgraph of 𝐺 induced by 𝑆 as 𝐺 𝑆 = 𝑆,𝐸 𝑆
Where 𝐸 𝑆 = 𝑢,𝑣 ∈E 𝑢,𝑣∈𝑆
𝑘,𝛼 −𝐷𝑆𝐿𝑂
Input: graph 𝐺, integer 𝑘>0, real number 𝛼∈[0,1]
Output: a set of sets of vertices 𝑆 = { 𝑆 1 , …, 𝑆 𝑘 } with 𝑘 ≤𝑘, and 𝑆 𝑖 ⊆𝑉, such that

3. Greedy-based Approximation
Consider the disjoint case(𝛼=0):
NAIVE
𝐻←𝐺
for 𝑖=1,2,…,𝑘
𝑆 𝑖 ← the densest subgraph of 𝐻
𝐻←𝐻∖ 𝑆 𝑖
Output 𝑆

4. Finding Densest Subgraph
BasicLP
Give a solution 𝑥 , 𝑦 of value 𝑣, order it by 𝑦 𝑖 in non-increasing order, then there exists a prefix of 𝑦 corresponding to a subgraph 𝑆 with 𝜌 𝑆 ≥𝑣
A lower bound:

5. Finding Densest Subgraph
An 2-approximation algorithm within 𝑂 𝑚+𝑛 log 𝑛 time.
𝑑 𝑣 𝑖 ≤ 2 𝐸 𝐻 𝑉 𝐻 ≤2𝜌(𝐻)
𝐻←𝐺
𝑎𝑛𝑠←𝐺
for 𝑖=1,2,…,𝑘
𝑣 𝑖 ← the vertex of minimum degree
𝐻←𝐻∖ 𝑣 𝑖
if (𝜌 𝐻 >𝜌(𝑎𝑛𝑠))
𝑎𝑛𝑠←𝐻
Output 𝑎𝑛𝑠

6. Greedy-Based Approximation
Heuristic Improvement
Minimal Densest Subgraph
For any proper subgraph 𝐻 of 𝐺, 𝜌 𝐻 <𝜌 𝐺
Find a minimal densest subgraph in each iteration.
The approximation has no lower bounds:
NAIVE solution: 𝐺,∅,…,∅
Optimal solution: 𝐺 1 , 𝐺 2 ,…, 𝐺 𝑘

7. Finding Minimal Densest Subgraph
𝑇𝑟𝑦𝑅𝑒𝑚𝑜𝑣𝑒 & 𝑇𝑟𝑦𝐸𝑛ℎ𝑎𝑛𝑐𝑒
𝑇𝑟𝑦𝑅𝑒𝑚𝑜𝑣𝑒 𝑢, 𝐺 returns a densest subgraph 𝐻 excluding 𝑢 if 𝜌 𝐻 ≥𝜌(𝐺)
𝑇𝑟𝑦𝐸𝑛ℎ𝑎𝑛𝑐𝑒 𝑢,𝐺, 𝜌 𝑚𝑎𝑥 returns the densest subgraph including 𝑢 with minimum vertices
For a graph 𝐺, if 𝑇𝑟𝑦𝑅𝑒𝑚𝑜𝑣𝑒 𝑢, 𝐺 returns null, 𝑇𝑟𝑦𝐸𝑛ℎ𝑎𝑛𝑐𝑒 𝑢,𝐺, 𝜌 𝑚𝑎𝑥 returns a minimal densest subgraph

8. 𝑇𝑟𝑦𝑅𝑒𝑚𝑜𝑣𝑒 & 𝑇𝑟𝑦𝐸𝑛ℎ𝑎𝑛𝑐𝑒
𝐹𝑎𝑠𝑡𝐿𝑃: an efficient algorithm to
construct the densest subgraph
from a solution of LP.

9. 𝑇𝑟𝑦𝑅𝑒𝑚𝑜𝑣𝑒 & 𝑇𝑟𝑦𝐸𝑛ℎ𝑎𝑛𝑐𝑒

10. Finding Minimal Densest Subgraph

11. Finding Minimal Densest Subgraph
The 𝑤ℎ𝑖𝑙𝑒 loop in 𝐹𝑖𝑛𝑑𝑀𝑖𝑛𝑖𝑚𝑎𝑙 iterates 𝑂( log 𝑛 ) times
𝜖−𝑏𝑎𝑑
Vertex 𝑢 is 𝜖−𝑏𝑎𝑑 if 𝑉 𝐻 1 ≥ 1−𝜖 𝑉 𝐻 and 𝑉 𝐻 2 ≥ 1−𝜖 𝑉 𝐻
The fraction of 𝜖−𝑏𝑎𝑑 vertices in 𝑉(𝐻) is at most 2𝜖, for any 𝜖 > 0. Let 𝜖= 1 4 , we have

12. Finding Minimal Densest Subgraphs

13. Solve (𝑘,𝛼)−𝐷𝑆𝐿𝑂

14. Solve (𝑘,𝛼)−𝐷𝑆𝐿𝑂

15. Experiments Environment
on a Linux server with Intel Xeon E at 2.40GHz, while limiting the total amount of main memory available to 64 GB. Solve linear programs with the Gurobi Optimizer version All algorithms are implemented in Java.

16. Experiments
Results of group 1

17. Experiments
Results of group 1

18. Experiments
Results of group 2

19. Reference
Charikar M. Greedy approximation algorithms for finding dense components in a graph[M]//Approximation Algorithms for Combinatorial Optimization. Springer Berlin Heidelberg, 2000:
Balalau O D, Bonchi F, Chan T H, et al. Finding subgraphs with maximum total density and limited overlap[C]//Proceedings of the Eighth ACM International Conference on Web Search and Data Mining. ACM, 2015:

20. Thanks for listening
Q&A