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