Small clique detection and approximate Nash equilibria

1. Small clique detection and approximateNash equilibria Danny Vilenchik UCLA Joint work with Lorenz Minder

2. Summary Relate three problems: • Approximating the best Nash equilibrium • Finding a planted k-clique in a random graph Gn,1/2 • DistinguishingGn,1/2 from Gn,1/2 with slightly larger planted clique • Executive summary: • A is at least as hard as B (for sufficiently large constant k) [Hazan & Krauthgamer 2009] • A is at least as hard as C [joint work with L. Minder, 2009]

3. Two player game Payoff for column player qj Game matrix pi (Mixed) Strategies: (independent) row player: x=(p1,p2,…,pn), column player: y=(q1,q2,…,qn) Payoff of row player isxAyt (column playerisxByt) – expectation Payoff for row player

4. Example: Scissors, Rock, Paper • This is a zero sum game • In this case, total payoff is 0 • No player has any incentive to deviate (payoff still 0)

5. Nash Equilibrium qj pi A strategy (x,y) is a Nash-equilibrium if A strategy (x,y)is an ²-Nash-equilibrium if The value of a strategy (x,y) is The best equilibrium is the one with maximal value (say m) An ²-best ²-equilibrium is: 1. An ²-equlibrium 2. Has value at least m-²

6. Planted k-clique (Jerrum, Kucera) Gn,1/2 Plant a clique of size k Generate Gn,1/2 independently Largest clique is whp of size (2-o(1))logn

7. What is known for these problems? Can efficiently compute a 0.34-equilinrium [TP’07] Can find planted k-clique in O(nk) Can find planted k-clique in poly time if k=(n1/2) [AKS’98] Can compute (best) ²-equilibrium in time [LMM’03] Can find planted O(logn)-clique in O(nlogn) Hard to distinguish between Gn,1/2 from Gn,1/2,k for k=(2-²)logn [JP’98] NP-Hard to compute best-Nash Currently no polynomial algorithm for planted O(logn)-clique Is there a PTAS for best-Nash? No polynomial algorithm to find a clique of size > logn in Gn,1/2

8. Hardness Result for ²-best Nash Hazan and Krauthgamer show (SODA 2009): If there exists poly-time algorithm that finds the ²-best Nash then there exists a probabilistic poly-time algorithm that finds a clique of size 1000logn in Gn,1/2,1000log n • This result relates seemingly unrelated problems • How far can this technique be stretched? • Optimal would be a planted clique of size (2+½)logn for any ½ > 0

9. Hardness Result for ²-best Nash Our result (with Lorenz Minder) If there exists poly-time algorithm that finds the ²-best Nash then There exists a poly-time algorithm that distinguisheswhp between Gn,1/2 and Gn,1/2 with a planted clique of size > (2+²1/8)log n Corollary of our analysis: there exists a probabilistic poly-time algorithm that finds a clique of size 3logn in Gn,1/2,3log n In some sense this is the best one can expect. If k < 2logn, the two distributions may be info. theoret. indist. ) bound too tight

10. Techniques Goal: Given a graphG, incorporate it into a game so that the ²-best Nash relates to its maximum clique First try: 1/2 1/2 1 1 Game matrix is just the adjacency matrix The value of the best Nash is 1 1/2 1/2 A)G( Conclusion: need to “neutralize” small cliques

11. Techniques (Hazan and Krauthgamer) Goal: “neutralize” small cliques Ais the adjacency matrix of a random graph with a planted clique of sizec1logn B is an ns£n matrix, s=s(c1) The (i,j)-entry of B is (bi,j,-bi,j) • Hopefully: • Small cliques are not equilibrium • Large planted clique is an equilirbrium

12. Properties of the game Let C be the planted clique of size c1log n 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C|

13. Techniques (Hazan and Krauthgamer) • The value of the strategy is 1 • Why is it a Nash-equilibrium? • The matrix B may interfere now

14. Properties of the game 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| 1/|C| j

15. Techniques (Hazan and Krauthgamer) • The value of the strategy is 1 • Why is it a Nash-equilibrium? • The matrix B may interfere now • The best Nash is of value at least 1 • How about “neutralizing” small cliques?

16. Properties of the game For every set of at most c2log n rows D (c2 < c1) 1/|D| 1/|D| Row player defects i

17. Properties of B • The average of the c1logn columns corresponding to the clique < 1 • Or else the planted clique is not an equilibrium (row player then defects) • For every set of c2logn columns there is a strike of 8’s in B • Enough to exclude small cliques as equilibria

18. Observation Two contesting processes regarding B: • B shouldn’t have too many rows • Or else the average of c1logn columns > 1 (at some row) • Planted clique is not an equilibrium (row player then defects) • B shouldn’t have too few rows • Otherwise not for every set of c2logn columns there is a strike of 8’s • Small cliques not neutralized • If you choose c1 sufficiently large, c2 smaller than c1, such a B exists

19. Main Point of Analysis Recover a graph of size f size c2log n and density 0.55 Plant a clique of size c1log n Such density and size do not exist in Gn,1/2whp) must intersect planted clique on many vertices )use greedy to complete to the planted clique

20. Main points in the analysis If the strategy (x,y) is an ε-best Nash equilibrium then: Fact 1: both players put most of their probability mass on A Why? The game outside A is 0-sum. So if one player has 2δ-probability outside A, the value of the game cannot exceed (2-2δ)/2=1- δ (maximal value on A is 1) But, we know that the best Nash has value 1, so δ< ε Here we use the fact that we are given a best Nash equilibrium. OPEN PROBLEM: can you let go of the “best” assumption ?!

21. Main points in the analysis If the strategy (x,y) is an ε’-best Nash equilibrium played on A then: Fact 2: Small sets of indices cannot be assigned with probability > 1/8 Why? By the second property of B, a strike of 8’s will cause a player to defect Fact 3: Sets of large probability correspond to high payoff, and in turn to dense subgraphs. Again, here we use the fact that the equilibrium has value 1 (since it is the best one)

22. Our work • Optimal result means c1=(2+½)log n • This means that 2 < c2<c1 • Because the subgraph is small (c2logn), it has to be very dense: 1-½ • Otherwise, again, such sub graphs exist in Gn,1/2 • Need to preserve the separation properties of the game • The planted clique is a Nash equilibrium of value 1 • Probability is placed on sets of size at least c2log n

23. What did we do? • Use tightest possible version of probabilisticbounds (Chernoff in our case) • Optimize over values of Bernoulli variables (in the matrix B) • Two contesting processes in B • Tighter analysis of other game properties • However, we only get detection of small cliques • To find a planted clique we need to plant a clique of size 3logn • (we don’t know an algorithm that finds a planted clique when given a piece of it of size < logn)

24. Limitation of the technique Can we hope to have a reduction from finding the maximal clique in Gn,1/2? • Probably not • The main reason: the technique relates value of equilibrium to density) value cannot exceed 1-², and there are plenty of such dense subgaphs in Gn,1/2 not connected to the cliqe

25. Open Questions • Remove the “best” assumption • Reduction in the other direction