Small clique detection and approximateSmall clique detection and approximateNash equilibriaNash equilibria
Danny VilenchikDanny Vilenchik
UCLAUCLA
Joint work with Lorenz MinderJoint work with Lorenz Minder
Summary
Relate three problems:
A. Approximating the best Nash equilibrium
B. Finding a planted k-clique in a random graph Gn,1/2
C. Distinguishing Gn,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]
Two player game
),( bapi
qj
(Mixed) Strategies: (independent)
row player: x=(p1,p2,…,pn),
column player: y=(q1,q2,…,qn)
Payoff of row player is xAyt (column playeris xByt) – expectation
Payoff for column player
Payoff for row player
Game matrix
Example: Scissors, Rock, Paper
(0,0)-(1,1)(1-,1)
(1-,1)(0,0)-(1,1)
-(1,1)(1-,1)(0,0)
3/1
3/1
3/1
3/1 3/1 3/1
• 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
),( bapi
qj
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
xAyxAyxAyAyxyx ''','
'''',' xAyxAyxAyAyxyx
yBAx 2
1
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)
Gn,1/2
Largest clique is whp of size(2-o(1))logn
Plant a clique of size kGenerate Gn,1/2 independently
What is known for these problems?
Can find planted k-clique in O(nk)
Can find planted k-clique in poly time if k=(n1/2) [AKS’98]
Hard to distinguish between Gn,1/2 from Gn,1/2,k for k=(2-²)logn [JP’98]
Can efficiently compute a 0.34-equilinrium [TP’07]
Can compute (best) ²-equilibrium
in time [LMM’03]2/log nn
Currently no polynomial algorithm for planted O(logn)-clique
No polynomial algorithm to find a clique of size > logn in Gn,1/2
NP-Hard to compute best-Nash
Is there a PTAS for best-Nash?
Can find planted O(logn)-clique in O(nlogn)
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
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
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
Techniques
Goal: Given a graph G, incorporate it into a game so that the ²-best Nash relates to its maximum clique
First try:
0,01,10,01,1
1,11,11,10,0
0,01,11,10,0
0,01,11,11,1Game matrix is just the adjacency matrix
The value of the best Nash is 1
A)G(
1/2
1/2
1/2 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 c1logn
B is an ns £ n matrix, s=s(c1)
The (i,j)-entry of B is (bi,j,-bi,j)
0B
BA T
Goal: “neutralize” small cliques
9/18
9/80, jib
Hopefully:
• Small cliques are not equilibrium
• Large planted clique is an equilirbrium
Properties of the game
Let C be the planted clique of size c1log n
0B
BT
1,11,10,00,00,0
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,11/|C|
1/|C|
1/|C|
1/|C|
1/|C| 1/|C| 1/|C| 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
0B
BT
1,11,10,00,00,0
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,1
1/|C|
1/|C|
1/|C|
1/|C|
1/|C| 1/|C| 1/|C| 1/|C|
j
1||
1
CiijbC
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
0B
BT
1,11,10,00,00,0
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,1
0,01,11,11,11,1
For every set of at most c2log n rows D (c2 < c1)
1/|D| 1/|D|
i
8: ijbj
Row player defects
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
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
Main Point of Analysis
Plant a clique of size c1log n Recover a graph of size f size c2log n
and density 0.55Such density and size do not exist in Gn,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 ?!
0B
BA T
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 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
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 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
Open Questions
Remove the “best” assumption
Reduction in the other direction