eric allender rutgers university graph automorphism & circuit size joint work with joshua a....

Eric AllenderRutgers University

Graph Automorphism & Circuit Size

Graph Automorphism & Circuit Size

Joint work with

Joshua A. Grochow and Cristopher Moore (SFI)

Simons Workshop, September 29, 2015

Eric Allender: Graph Automorphism & Circuit Size < 2 >

The ContextThe Context

The Minimum Circuit Size Problem (MCSP) = {(f,i) : f is the truth-table of a function that has a circuit of size ≤ i}.

In NP, but not known (or widely believed) to be NP-complete. [Kabanets, Cai], [Murray, Williams], [A, Holden, Kabanets]

Lots of reasons to believe it’s not in P.


More ContextMore Context

Factoring is in ZPPMCSP. Graph Isomorphism is in RPMCSP. Every promise problem in SZK is in

(Promise) BPPMCSP.

A motivating question: Is Graph Isomorphism in ZPPMCSP?


More ContextMore Context

Factoring is in ZPPMCSP. Graph Isomorphism is in RPMCSP. Every promise problem in SZK is in

(Promise) BPPMCSP.

An obstacle: EACH of these reductions follows the same route….


Yet More ContextYet More Context

The well-trodden path: MCSP is a wonderful test to distinguish

random from pseudorandom distributions. Thus, via [HILL], MCSP is an oracle that

allows a probabilistic algorithm to invert poly-time functions with high probability.

Note that this approach can’t show a result like “A is in ZPPMCSP” unless we already know that A is in NP∩coNP.


Context, Context, ContextContext, Context, Context

MCSP is more like a family of problems, than a single problem.

For instance “size” could mean “# of wires” or “# of gates”, or “# of bits to describe the circuit”, etc.

None of these is known to be reducible to any other – but all can stand in for “MCSP”.

One more such variant: MKTP = {(x,i) : KT(x) ≤ i}


What Can We Show?What Can We Show?

Graph Automorphism is in ZPPMKTP. As observed on an earlier slide, this involves a

different type of reduction than all earlier reductions to MKTP or MCSP (since Graph Automorphism is not known to be in NP∩coNP).

We are unable to extend this, to show Graph Automorphism is in ZPPMCSP.

– This is a new phenomenon; all other reductions to MKTP carried over to MCSP.


What Can We Show?What Can We Show?

Graph Automorphism is in ZPPMKTP. We are also unable to extend this, to ZPP-

reduce Graph Isomorphism to MKTP. …although we can adapt our framework to

show that Graph Isomorphism is in BPPMKTP. …which is weaker than the previously-known

result: Graph Isomorphism is in RPMKTP– but it may be useful to build up some tools for providing reductions to MKTP and MCSP.


The Proof (1)The Proof (1)

Theorem: Graph Automorphism is in ZPPMKTP. It suffices to solve the Graph Isomorphism

problem, restricted to rigid graphs. [KST] That is: Consider the promise problem with

– YES Instances: {(G0,G1) : G0 ≡ G1}.

– NO Instances: {(G0,G1) : G0 and G1 are rigid and are not isomorphic}

Might seem odd to impose rigidity only on the NO instances – but this stronger result holds with the same proof.



Theorem: Graph Automorphism is in ZPPMKTP. It suffices to solve the Graph Isomorphism

problem, restricted to rigid graphs. [KST] That is: Consider the promise problem with

– YES Instances: {(G0,G1) : G0 ≡ G1}.

– NO Instances: {(G0,G1) : G0 and G1 are rigid and are not isomorphic}

We already have GI in RPMKTP. Thus we need to solve this promise problem in coRPMKTP.



On input (G0,G1)

– Randomly pick a bit string w=w1w2…wt.

– Pick random permutations π1…πt.

– Let z= π1(Gw1)π2(Gw2

)…πt(Gwt)

If G0 and G1 are not isomorphic, then z allows us to reconstruct w and π1…πt, so that z has (non-time-bounded) K-complexity around t+ts (where s = log n!), whp. Hence KT(z) > t+ts.

Otherwise, KT(z) is around n2+ts.



On input (G0,G1)

– Randomly pick a bit string w=w1w2…wt.

– Pick random permutations π1…πt.

– Let z= π1(Gw1)π2(Gw2

)…πt(Gwt)

If G0 and G1 are not isomorphic, then z allows us to reconstruct w and π1…πt, so that z has (non-time-bounded) K-complexity around t+ts (where s = log n!), whp. Hence KT(z) > t+ts.

Otherwise, KT(z) is around n2+ts. QED


Why does this break for MCSP?Why does this break for MCSP?

The easiest way to answer this, is to present a version of MCSP where the proof does go through (and also to explain a bit about how KT complexity is defined).

KT(x) = min{|d|+t : U can construct x from description d in time t}




KT(x) = min{|d|+t : Ud(i) = the ith bit of x, and runs in time t}

A multiplexer circuit has AND, OR and NOT gates, and also INDEX gates (of fan-in log m) that access a (fixed) array of size m.

They can simulate U efficiently. [Store d in the array.]




KT(x) = min{|d|+t : Ud(i) = the ith bit of x, and runs in time t}

In contrast, implementing a multiplexer using AND and OR requires size O(m) = O(|d|) for each query that U makes to d.

…which kills the argument!


Gap amplificationGap amplification

The way to modify this approach, to make it work for MCSP, would be to “amplify” the gap between KT ≥ t(s+1) and KT ≤ ts, to something like KT ≥ b and KT ≤ bε for some ε > 0.

This would pay other dividends. It would give a way to reduce some of the different variants of MCSP to each other.

This would provide a notion of “robustness” for MCSP – which is currently lacking. (Different versions of the problem might have different complexity.)


Different versions of MCSPDifferent versions of MCSP

For instance, is it possible that one version of MCSP is NP-complete, while another version is in P?


Open QuestionsOpen Questions

Is Graph Automorphism in ZPPMCSP? How about Graph Isomorphism? …or all of SZK? Is there some way to relate the complexities of

MKTP and (the many versions of) MCSP? …through Gap Amplification, or via some

other means? And there are tons of other important

questions about MCSP.


Thank you!Thank you!

eric allender rutgers university graph automorphism & circuit size joint work with joshua a....

Documents