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Electronic Colloquium on Computational Complexity

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All reports by Author Rahul Santhanam:

TR18-030 | 9th February 2018
Shuichi Hirahara, Igor Carboni Oliveira, Rahul Santhanam

NP-hardness of Minimum Circuit Size Problem for OR-AND-MOD Circuits

The Minimum Circuit Size Problem (MCSP) asks for the size of the smallest boolean circuit that computes a given truth table. It is a prominent problem in NP that is believed to be hard, but for which no proof of NP-hardness has been found. A significant number of works have ... more >>>

TR17-173 | 6th November 2017
Igor Carboni Oliveira, Ruiwen Chen, Rahul Santhanam

An Average-Case Lower Bound against ACC^0

In a seminal work, Williams [Wil14] showed that NEXP (non-deterministic exponential time) does not have polynomial-size ACC^0 circuits. Williams' technique inherently gives a worst-case lower bound, and until now, no average-case version of his result was known.

We show that there is a language L in NEXP (resp. EXP^NP) ... more >>>

TR16-197 | 7th December 2016
Igor Carboni Oliveira, Rahul Santhanam

Conspiracies between Learning Algorithms, Circuit Lower Bounds and Pseudorandomness

We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size at most s(n). We show:

Learning Speedups: If C[$n^{O(1)}$] admits a randomized weak learning algorithm under the uniform ... more >>>

TR16-196 | 5th December 2016
Igor Carboni Oliveira, Rahul Santhanam

Pseudodeterministic Constructions in Subexponential Time

We study {\it pseudodeterministic constructions}, i.e., randomized algorithms which output the {\it same solution} on most computation paths. We establish unconditionally that there is an infinite sequence $\{p_n\}_{n \in \mathbb{N}}$ of increasing primes and a randomized algorithm $A$ running in expected sub-exponential time such that for each $n$, on input ... more >>>

TR15-192 | 26th November 2015
Ruiwen Chen, Rahul Santhanam

Satisfiability on Mixed Instances

The study of the worst-case complexity of the Boolean Satisfiability (SAT) problem has seen considerable progress in recent years, for various types of instances including CNFs \cite{PPZ99, PPSZ05, Sch99, Sch05}, Boolean formulas \cite{San10} and constant-depth circuits \cite{IMP12}. We systematically investigate the complexity of solving {\it mixed} instances, where different parts ... more >>>

TR15-191 | 26th November 2015
Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan

Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

We show average-case lower bounds for explicit Boolean functions against bounded-depth threshold circuits with a superlinear number of wires. We show that for each integer d > 1, there is \epsilon_d > 0 such that Parity has correlation at most 1/n^{\Omega(1)} with depth-d threshold circuits which have at most
n^{1+\epsilon_d} ... more >>>

TR15-112 | 16th July 2015
Ruiwen Chen, Rahul Santhanam

Improved Algorithms for Sparse MAX-SAT and MAX-$k$-CSP

We give improved deterministic algorithms solving sparse instances of MAX-SAT and MAX-$k$-CSP. For instances with $n$ variables and $cn$ clauses (constraints), we give algorithms running in time $\poly(n)\cdot 2^{n(1-\mu)}$ for
\item $\mu = \Omega(\frac{1}{c} )$ and polynomial space solving MAX-SAT and MAX-$k$-SAT,
\item $\mu = \Omega(\frac{1}{\sqrt{c}} )$ and ... more >>>

TR14-173 | 13th December 2014
Igor Carboni Oliveira, Rahul Santhanam

Majority is incompressible by AC$^0[p]$ circuits

Revisions: 1

We consider $\cal C$-compression games, a hybrid model between computational and communication complexity. A $\cal C$-compression game for a function $f \colon \{0,1\}^n \to \{0,1\}$ is a two-party communication game, where the first party Alice knows the entire input $x$ but is restricted to use strategies computed by $\cal C$-circuits, ... more >>>

TR14-171 | 11th December 2014
Lance Fortnow, Rahul Santhanam

Hierarchies Against Sublinear Advice

We strengthen the non-deterministic time hierarchy theorem of
\cite{Cook72, Seiferas-Fischer-Meyer78, Zak83} to show that the lower bound
holds against sublinear advice. More formally, we show that for any constants
$c$ and $d$ such that $1 \leq c < d$, there is a language in $\NTIME(n^d)$
which is not in $\NTIME(n^c)/n^{1/d}$. ... more >>>

TR13-108 | 9th August 2013
Rahul Santhanam, Ryan Williams

New Algorithms for QBF Satisfiability and Implications for Circuit Complexity

Revisions: 1

We revisit the complexity of the satisfiability problem for quantified Boolean formulas. We show that satisfiability
of quantified CNFs of size $\poly(n)$ on $n$ variables with $O(1)$
quantifier blocks can be solved in time $2^{n-n^{\Omega(1)}}$ by zero-error
randomized algorithms. This is the first known improvement over brute force search in ... more >>>

TR12-108 | 4th September 2012
Arkadev Chattopadhyay, Rahul Santhanam

Lower Bounds on Interactive Compressibility by Constant-Depth Circuits

We formulate a new connection between instance compressibility \cite{Harnik-Naor10}), where the compressor uses circuits from a class $\C$, and correlation with
circuits in $\C$. We use this connection to prove the first lower bounds
on general probabilistic multi-round instance compression. We show that there
is no
probabilistic multi-round ... more >>>

TR12-084 | 3rd July 2012
Rahul Santhanam

Ironic Complicity: Satisfiability Algorithms and Circuit Lower Bounds

I discuss recent progress in developing and exploiting connections between
SAT algorithms and circuit lower bounds. The centrepiece of the article is
Williams' proof that $NEXP \not \subseteq ACC^0$, which proceeds via a new
algorithm for $ACC^0$-SAT beating brute-force search. His result exploits
a formal connection from non-trivial SAT algorithms ... more >>>

TR12-077 | 10th June 2012
Chiranjit Chakraborty, Rahul Santhanam

Instance Compression for the Polynomial Hierarchy and Beyond

Comments: 2

We define instance compressibility for parametric problems in PH and PSPACE. We observe that

the problem \Sigma_{i}CircuitSAT of deciding satisfiability of a quantified Boolean circuit with i-1 alternations of quantifiers starting with an existential uantifier is complete for parametric problems in \Sigma_{i}^{p} with respect to W-reductions, and that analogously ... more >>>

TR12-059 | 14th May 2012
Rahul Santhanam, Ryan Williams

Uniform Circuits, Lower Bounds, and QBF Algorithms

We explore the relationships between circuit complexity, the complexity of generating circuits, and circuit-analysis algorithms. Our results can be roughly divided into three parts:

1. Lower Bounds Against Medium-Uniform Circuits. Informally, a circuit class is ``medium uniform'' if it can be generated by an algorithmic process that is somewhat complex ... more >>>

TR11-135 | 9th October 2011
Maurice Jansen, Rahul Santhanam

Stronger Lower Bounds and Randomness-Hardness Tradeoffs using Associated Algebraic Complexity Classes

We associate to each Boolean language complexity class $\mathcal{C}$ the algebraic class $a\cdot\mathcal{C}$ consisting of families of polynomials $\{f_n\}$ for which the evaluation problem over the integers is in $\mathcal{C}$. We prove the following lower bound and randomness-to-hardness results:

1. If polynomial identity testing (PIT) is in $NSUBEXP$ then $a\cdot ... more >>>

TR11-133 | 4th October 2011
Maurice Jansen, Rahul Santhanam

Marginal Hitting Sets Imply Super-Polynomial Lower Bounds for Permanent

Suppose $f$ is a univariate polynomial of degree $r=r(n)$ that is computed by a size $n$ arithmetic circuit.
It is a basic fact of algebra that a nonzero univariate polynomial of degree $r$ can vanish on at most $r$ points. This implies that for checking whether $f$ is identically zero, ... more >>>

TR11-131 | 29th September 2011
Rahul Santhanam, Srikanth Srinivasan

On the Limits of Sparsification

Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for $k$-CNFs:
every k-CNF is a sub-exponential size disjunction of $k$-CNFs with a linear
number of clauses. This lemma has subsequently played a key role in the study
of the exact complexity of the satisfiability problem. A natural question is
more >>>

TR09-064 | 3rd August 2009
Harry Buhrman, Lance Fortnow, Rahul Santhanam

Unconditional Lower Bounds against Advice

We show several unconditional lower bounds for exponential time classes
against polynomial time classes with advice, including:
\item For any constant $c$, $\NEXP \not \subseteq \P^{\NP[n^c]}/n^c$
\item For any constant $c$, $\MAEXP \not \subseteq \MA/n^c$
\item $\BPEXP \not \subseteq \BPP/n^{o(1)}$

It was previously unknown even whether $\NEXP \subseteq ... more >>>

TR07-096 | 8th October 2007
Lance Fortnow, Rahul Santhanam

Infeasibility of Instance Compression and Succinct PCPs for NP

We study the notion of "instance compressibility" of NP problems [Harnik-Naor06], closely related to the notion of kernelization in parameterized complexity theory [Downey-Fellows99, Flum-Grohe06, Niedermeier06]. A language $L$ in NP is instance compressible if there
is a polynomial-time computable function $f$ and a set $A$ such that
for each instance ... more >>>

TR07-005 | 17th January 2007
Rahul Santhanam

Circuit Lower Bounds for Merlin-Arthur Classes

We show that for each k > 0, MA/1 (MA with 1 bit of advice) does not have circuits of size n^k. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP_{||}^{NP}.

We extend our main result in several ways. For ... more >>>

TR07-004 | 12th January 2007
Lance Fortnow, Rahul Santhanam

Time Hierarchies: A Survey

We survey time hierarchies, with an emphasis on recent attempts to prove hierarchies for semantic classes.

more >>>

TR06-157 | 14th December 2006
Lance Fortnow, Rahul Santhanam

Fixed-Polynomial Size Circuit Bounds

We explore whether various complexity classes can have linear or
more generally $n^k$-sized circuit families for some fixed $k$. We

1) The following are equivalent,
- NP is in SIZE(n^k) (has O(n^k)-size circuit families) for some k
- P^NP|| is in SIZE(n^k) for some k
- ONP/1 is in ... more >>>

TR06-154 | 13th December 2006
Joshua Buresh-Oppenheim, Valentine Kabanets, Rahul Santhanam

Uniform Hardness Amplification in NP via Monotone Codes

We consider the problem of amplifying uniform average-case hardness
of languages in $\NP$, where hardness is with respect to $\BPP$
algorithms. We introduce the notion of \emph{monotone}
error-correcting codes, and show that hardness amplification for
$\NP$ is essentially equivalent to constructing efficiently
\emph{locally} encodable and \emph{locally} list-decodable monotone
codes. The ... more >>>

TR06-003 | 8th January 2006
Joshua Buresh-Oppenheim, Rahul Santhanam

Making Hard Problems Harder

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as ... more >>>

TR04-098 | 5th November 2004
Lance Fortnow, Rahul Santhanam, Luca Trevisan

Promise Hierarchies

We show that for any constant a, ZPP/b(n) strictly contains
ZPTIME(n^a)/b(n) for some b(n) = O(log n log log n). Our techniques
are very general and give the same hierarchy for all the common
promise time classes including RTIME, NTIME \cap coNTIME, UTIME,

We show a ... more >>>

TR02-038 | 5th June 2002
Rahul Santhanam

Resource Tradeoffs and Derandomization

Revisions: 1

We consider uniform assumptions for derandomization. We provide
intuitive evidence that BPP can be simulated non-trivially in
deterministic time by showing that (1) P \not \subseteq i.o.i.PLOYLOGSPACE
implies BPP \subseteq SUBEXP (2) P \not \subseteq SUBPSPACE implies BPP
= P. These results extend and complement earlier work of ... more >>>

TR01-022 | 15th February 2001
Rahul Santhanam

On segregators, separators and time versus space

We give the first extension of the result due to Paul, Pippenger,
Szemeredi and Trotter that deterministic linear time is distinct from
nondeterministic linear time. We show that DTIME(n \sqrt(log^{*}(n)))
\neq NTIME(n \sqrt(log^{*}(n))). We show that atleast one of the
following statements holds: (1) P \neq L ... more >>>

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