Small clique detection and approximate Nash equilibria Danny Vilenchik UCLA Joint work with Lorenz Minder

Summary Relate three problems: A.Approximating the best Nash equilibrium B.Finding a planted k -clique in a random graph G n,1/2 C.Distinguishing G n,1/2 from G n,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]

Two player game pipi qjqj (Mixed) Strategies: (independent) row player: x =( p 1, p 2,…, p n ), column player: y =( q 1, q 2,…, q n ) Payoff of row player is xAy t (column playeris xBy t ) – expectation Payoff for column player Payoff for row player Game matrix

Example: Scissors, Rock, Paper (1,-1)(-1,1)(0,0) (-1,1)(0,0)(1,-1) (0,0)(1,-1)(-1,1) This is a zero sum game In this case, total payoff is 0 No player has any incentive to deviate (payoff still 0)

Nash Equilibrium pipi qjqj 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

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

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

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 1000log n in G n,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 )log n for any ½ > 0

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 distinguishes whp between G n,1/2 and G n,1/2 with a planted clique of size > (2+ ² 1/8 )log n Corollary of our analysis: there exists a probabilistic poly-time algorithm that finds a clique of size 3log n in G n,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

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

Techniques (Hazan and Krauthgamer) A is the adjacency matrix of a random graph with a planted clique of size c 1 log n B is an n s £ n matrix, s = s ( c 1 ) The ( i, j )-entry of B is ( b i, j,- b i, j ) Goal: “neutralize” small cliques Hopefully: Small cliques are not equilibrium Large planted clique is an equilirbrium

Properties of the game Let C be the planted clique of size c 1 log n 1/| C |

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

Properties of the game 1/| C | j

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?

Properties of the game For every set of at most c 2 log n rows D (c 2 < c 1 ) 1/| D | i Row player defects

Properties of B  The average of the c 1 log n columns corresponding to the clique < 1  Or else the planted clique is not an equilibrium (row player then defects)  For every set of c 2 log n columns there is a strike of 8’s in B  Enough to exclude small cliques as equilibria

Observation Two contesting processes regarding B :  B shouldn’t have too many rows  Or else the average of c 1 log n 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 c 2 log n columns there is a strike of 8’s  Small cliques not neutralized  If you choose c 1 sufficiently large, c 2 smaller than c 1, such a B exists

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

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 ?!

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)

Our work  Optimal result means c 1 =( 2+ ½ )log n  This means that 2 < c 2 < c 1  Because the subgraph is small (c 2 log n), it has to be very dense: 1- ½  Otherwise, again, such sub graphs exist in G n,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 c 2 log n

What did we do?  Use tightest possible version of probabilistic bounds (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)

Limitation of the technique Can we hope to have a reduction from finding the maximal clique in G n,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 G n,1/2 not connected to the cliqe

Open Questions  Remove the “best” assumption  Reduction in the other direction