1. Inoculation Strategies for Victims of Viruses and the Sum-of-Squares Partition Problem James Apnes, Kevin Change, and Aleksandr Yampolskiy

2. Problem: Network of computers all connected to each other Antivirus software that prevents computer from becoming infected Virus introduced and spreads throughout Becoming infected and installing software both have costs

3. Goal: Install antivirus software on certain number of computers that minimizes cost 2 Ways to approach problem: –Individual Selection –Dictator Selection

4. Individual Selection Each user of computer makes own decision on whether or not to install software Example: cost of software, C, outweighs cost caused by virus, L Nash Equilibrium must be reached

5. Game Set-Up Graph G (V, E) where V = {0, 1, 2, …, n-1} All n nodes start vulnerable

6. Strategies For every i in G, the strategy of i is represented by a i a i is the probability of i installing software So we can also create vector ā that represents all strategies in G I a = set of nodes that are secure G a is “attack graph” G a = G - I a

7. Attack Model Virus is started at some random node Propagates through network by attacking unprotected nodes

8. Individual Costs Cost of C to install software Cost of L if node becomes infected Cost of node i is cost i (ā) = a i C + (1-a i )L*p i (ā) p i (ā) is the probability of node getting infected based on its reachability In pure strategies: p i (ā) = k i /n

9. Social Cost Equal to the sum of individual costs in network We remove m nodes at a cost of C per node and create so many connected components in the graph Each node in the i th connected component has a k i /n chance of getting infected

10. Social Cost So… Social cost = Cm + Σ(Lk i (k i /n)) = Cm + (L/n)Σ(k i 2 ) Where k i is the number of nodes in all connected components created by removing the m nodes that are in I a

11. Nash Equilibrium Computers make decision based on their local environment Reached when each node cannot change strategy to minimize cost

12. Nash Equilibrium Can find a threshold for the decisions made by the computers –t = Cn/L Example: t = 1.5 Nash Equilibrium can be found in O(n 3 ) time

13. Price of Anarchy How far the Nash equilibrium is from the optimal social cost For this system price of anarchy is Θ(n)

14. Dictator Selection A centralized solution that makes choices for the computer in the best interest of social cost Can be reduced to sum-of-squares partition problem when removing m nodes Approximation algorithm that allows us to find O(log 1.5 n)m nodes that get us in within a constant of the optimal solution

15. Sum-of-Squares We see social cost is Cm + (L/n)Σ(k i 2 ) Sum-of-Squares partition gives us approximately best answer for removing m nodes Repeat to make m 1,2, …, n

16. Sum-of-Squares Sum-of-Squares Partition Problem: –Given a graph G =(V, E), remove a set of nodes F, of at most m, in order to partition the graph into disconnected components H 1, H 2, …, H l such that Σ i (|H i | 2 ) is minimized

17. Main Goal: Theorem 12: There exists a polynomial time (O(log 1.5 n), O(1))-bicriterion approximation algorithm for the sum-of- squares partition problem Corollary 13: If OPT is the cost of the optimum solution for the inoculation problem, there exists a polynomial time approximation algorithm that finds a solution with cost at most O(log 1.5 n)OPT

18. Arora-Rao-Vazirani Algorithm Used to find the sparsest edge cut Can be extended to be used for node cuts Takes a set of nodes H and splits them into 3 sets –V 1 and V 2 –R

19. Arora-Rao-Vazirani Algorithm Approach is similar to greedy set cover algorithm Consequence of ARV algo paper: –There exists some O(log 1/2 n)-approximation for the following problem: given graph G, find a node cut that partitions the nodes of G into three sets: two set defining disconnected subgraphs with nodes sets V 1 and V 2 and a set of removed nodes R, such that the quantity (|V 1 |+(|R|/2))(|V 2 |+(|R|/2)) |R|is maximized

20. Cost-Effectiveness We have connected subgraph H with k nodes that is split into sets V 1, V 2, and R then the cut’s cost-effectiveness is: k 2 - |V 1 | 2 - |V 2 | 2 |R|

21. Cost-Effectiveness vs. Sparsity Sparsity: Cost Effectiveness:

22. Lemma 15 Let H be a graph with k nodes, if α* is the maximum cost effectiveness of all node cuts of H, the Arora-Rao-Vazirani sparse cut algorithm will find a cut with cost effectiveness at least α*/(clog 1/2 k)

23. Algorithm Input: A graph G and integer m > 0 Initialize: G 1  G, F  , l  0 1.Use sparse cut algorithm to find an approximate most cost effective cut in each connected component in G

24. Algorithm 2. Let H 1, H 2, …, H k be the components of G l in which the sparse cut algorithm found a cut that removes at most (20clog 1/2 n)m nodes If no such component exists, then halt operation and output the partition of G that results from removing all nodes in set F

25. Algorithm 3. Choose the component H j from among those considered in the previous step for which the cost-effectiveness is highest. So we have R as the set of nodes that partitioned H j into V 1 and V 2

26. Algorithm 4. Add all of the nodes in R to the set F. Now let G l +1 be the residual graph induced by removing R from G l. If |F| > (36clog 1.5 n)m, then halt and output the partition of G that results from having removed all nodes in F 5. Otherwise, l = l+1 and repeat

27. Aspnes, James, Kevin Chang, and Aleksandr Yampolskiy. "Inoculation Strategies for Victims of Viruses and the Sum-of-squares Partition Problem." Journal of Computer and System Sciences 72.6 (2006): 1077-093. Web. S. Arora, S. Rao, U. Vazirani, Expander flows, geometric embeddings and graph partitioning, in: Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222– 231. T. Leighton, S. Rao, Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms, J. Assoc. Comput. Machinery 46 (6) (1999) 787–832.