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Complexity Theory and Explicit
Constructions of Ramsey Graphs
Rahul Santhanam
University of Edinburgh
Plan of the Talk
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Setting the Stage
Constructions and List Constructions
An Approach Through Logic
Testability, Combinability and Reductions
Plan of the Talk
•
•
•
•
Setting the Stage
Constructions and List Constructions
An Approach Through Logic
Testability, Combinability and Reductions
Framework
• Motivating Problem: Construct graphs on N
vertices without cliques or independent sets
of size 2 log(N) in time poly(N)
• We give elements of a structural theory of
explicit constructions emphasizing
connections to complexity theory; use the
Ramsey problem as an illustrative example
Dramatis Personae
• Protagonist: Ramsey graphs
• Secondary Characters: Hard Boolean functions
(functions of high circuit complexity), pseudorandom generators
• Minor Characters: Good error-correcting
codes, expanders, extractors, primes etc.
Brief Intro to Complexity Theory
• P = Boolean functions computable in
deterministic time poly(n)
• E = Boolean functions in time 2O(n)
• NP = Boolean functions verifiable in time poly(n)
• SIZE(s) = Boolean functions computable by
Boolean circuits of size s(n)
• SIZEA(s) = Boolean functions computed by
A-oracle circuits of size s(n)
• P is contained in SIZE(poly), NP in SIZENP(poly)
Hard Boolean Functions
• Some of the fundamental lower bound
questions in complexity theory are explicit
construction questions
– E not in SIZE(s) is equivalent to a poly(N)-time
construction of a truth table of a Boolean function
on log(N) bits which does not have circuits of size
s(log(N))
– We know that hard functions exist, but we don’t
know explicit examples
Pseudorandomness and the
Probabilistic Method
• We know that a random graph is Ramsey, a
random Boolean function is hard etc.
• Theory of pseudorandomness studies
pseudorandom objects, which are
“indistinguishable” from random ones but are
constructed using much less randomness
• We would like to argue that pseudorandom
graphs are Ramsey, pseudorandom functions
are hard etc.
Properties and their Complexity
• We model properties simply as Boolean functions
• Ramsey is in coNP
– To verify that a graph is non-Ramsey, simply guess a
subset of vertices of size 2 log(N) and check that it is a
clique or independent set
• Hard Boolean Function is in coNP
– To verify that a Boolean function (given by its truth
table) has small circuits, simply guess the circuit and
check that it is consistent with the truth table by
evaluating circuit on all inputs of size log(N)
• Primes is in P [AKS]
Pseudorandom Generators
• A quick PRG against a class C of properties is a
poly-time computable sequence of functions GN:
{0,1}O(log(N)) → {0,1}N such that for any Q in C,
Prx(Q(x) = 1) ~ Pry(Q(G(y)) = 1)
• Note that if Prx(Q(x) = 1) = 1 – o(1) (as is typical
for properties on which probabilistic method is
successful) and if G is a quick PRG against Q, then
for each N, Range(GN) is an efficiently computable
poly(N)-sized list of objects at least one of which
satisfies Q
Hardness = Pseudorandomness
• PRGs exist non-constructively for any natural
class of properties (by the probabilistic
method), but when do quick PRGs exist?
• Theorem [NW, IW, KvM] : There is a quick PRG
against SIZEA(poly) iff there are explicit
functions which are hard against A-oracle
circuits of size 2o(n)
Plan of the Talk
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•
•
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Setting the Stage
Constructions and List Constructions
An Approach Through Logic
Testability, Combinability and Reductions
Weak Explicit Constructions
• We weaken the goal of explicit constructions of
Ramsey graphs in the following ways
– Asking for an explicit list-construction, i.e., an explicit
construction of a poly(N) sized list of graphs at least
one of which is Ramsey
– Asking for a quasi-explicit construction, i.e., a
construction that works in time 2polylog(N)
• We observe that an explicit list-construction
exists under standard complexity assumptions,
and a quasi-explicit construction unconditionally
A Conditional Explicit List Construction
• Observation: If there is an explicit Boolean
function hard against NP-oracle circuits, there is
an explicit list-construction for Ramsey property
• Proof: Ramsey is in coNP, hard Boolean function
implies quick PRG, range of quick PRG is an
explicit list
• This construction is generic: same list (i.e., range
of a certain quick PRG G) works for any property
in NP or coNP
• Also, note that for a property in P, we get an
explicit construction rather than just a list
A Quasi-Explicit Construction
• Consider the proof that the Ramsey property
holds for a random graph
• This proof requires only O(log2(N))-wise
independence between choices of random edges
• There is an O(log2(N))-wise independent sample
space of size 2O(log^3(N)) of strings of length N2 – at
least one member of this sample space
represents a Ramsey graph
• Since Ramsey property is testable in time
NO(log(N)), we can go through all possibilities in
quasi-poly time and choose one which is Ramsey
Explicit = Efficiently Constructible?
• The previous construction cheats in some
sense – we don’t really “get our hands” on a
graph which is Ramsey
• However, the PRG-based approach does give
candidate explicit constructions for which
there is a natural combinatorial interpretation
Evidence for Explicit Constructions?
• Is there a plausible complexity-theoretic
hypothesis under which we can prove there is an
explicit construction of Ramsey graphs?
• Hypothesis: There is poly-time computable
sequence {xN} of strings, |xN| = N, such that for all
“super-efficient” f and almost all N, |y| < N –
ω(1) implies f(y) ≠ xN
• When interpreted as a graph, xN would be
Ramsey (since non-Ramsey graphs can be
compressed)
• How reasonable is hypothesis?
Plan of the Talk
•
•
•
•
Setting the Stage
Constructions and List Constructions
An Approach Through Logic
Testability, Combinability and Reductions
Narrowing the Focus
• Question: Is there an explicit list-construction
for Ramsey?
• Note that to get this, we just need a PRG
against the Ramsey property, not against all of
coNP or SIZENP(poly)
• Approach: Consider smaller natural families of
properties containing Ramsey and try to show
PRGs against them, eg., families based on
logical definability
EMSO and the Ramsey property
• Consider Existential Monadic Second Order
logic on graphs with arithmetic (+, *)
• The negation of the Ramsey property is
expressible in this logic
– “There exists a set S of size at least 2 log(N) such
that S is either a clique or an independent set”
• Is there a PRG against the class of properties
definable in this logic (or even for EMSO
without arithmetic)?
PRGs for Logics
• One could hope to unconditionally construct a
quick PRG for any logic for which
inexpressibility results are known
• There has been a lot of work on 0-1 laws and
limit laws. One could hope to get PRGs even
for logics for which such laws do not hold
Plan of the Talk
•
•
•
•
Setting the Stage
Constructions and List Constructions
An Approach Through Logic
Testability, Combinability and Reductions
Testability
• Question: Is the Ramsey property in P?
• We care about this because if so
– We can make the standard probabilistic
construction zero-error
– We can prove that explicit constructions exist
under a standard complexity-theoretic hypothesis
• [RR] showed that if Factoring is hard, then
testing whether a Boolean function has high
circuit complexity can’t be done in poly time
The Planted Clique Problem
• Consider the following distributions
– D: Erdos-Renyi graph with edge prob ½
– D’: Erdos-Renyi graph with edge prob ½ + random
planted clique of size 3 log(n)
• Planted Clique Problem: Is there a polynomialtime algorithm to distinguish D and D’?
• Widely studied problem [K] [J] [AKS] [FK]
[AAKMRX] with applications to crypto [JP]
Hardness of Ramsey Property
• Observation: If Planted Clique problem is
hard, then Ramsey property is not in P
• Proof: A graph drawn from D is Ramsey with
high probability, a graph drawn from D’ is not
• Question: Is Ramsey property hard under
more standard crypto assumptions?
Combinability
• Given a poly-sized list of objects at least one
of which satisfies property Q, can we produce
in poly time a single object satisfying property
Q?
• Clearly yes if Q is poly-time testable. But how
about properties such as Ramsey property?
• [N] uses graph products to give explicit
Ramsey construction with weak parameters
Combinability (contd)
• Combinability holds for Hard Boolean
Function property – just concatenate truth
tables of Boolean functions in the list
• Is there a nice characterization of properties
for which combinability holds?
Reductions Among Explicit
Constructions
• Say that property Q reduces to property R (Q <= R) if
there is a polynomial-time oracle machine which, when
given oracle access to explicit constructions of R,
produces an explicit construction for Q
• Hardness=Pseudorandomness theorem shows that
PRG <= Hard Boolean Function and Hard Boolean
Function <= PRG; also that any property in P reduces to
Hard Boolean Function (against NP-oracle circuits)
• [V] surveys reductions between expanders, errorcorrecting codes, extractors...
Open Questions
• Is there an explicit list-construction for
Ramsey?
• Are there PRGs against MSO or other logics
where inexpressibility results are known?
• More evidence that Ramsey is not efficiently
testable?
• Is the Ramsey property combinable?
Thank You