A {\em supermodel} is a satisfying assignment of a boolean formula
for which any small alteration, such as a single bit flip, can be
repaired by another small alteration, yielding a nearby
satisfying assignment. We study computational problems associated
with super models and some generalizations thereof. For general
formulas, ... more >>>

We show new tradeoffs for satisfiability and nondeterministic
linear time. Satisfiability cannot be solved on general purpose
random-access Turing machines in time $n^{1.618}$ and space
$n^{o(1)}$. This improves recent results of Lipton and Viglas and
Fortnow.

A dichotomy theorem for a class of decision problems is
a result asserting that certain problems in the class
are solvable in polynomial time, while the rest are NP-complete.
The first remarkable such dichotomy theorem was proved by
T.J. Schaefer in 1978. It concerns the ... more >>>

We study the space complexity of refuting unsatisfiable random $k$-CNFs in
the Resolution proof system. We prove that for any large enough $\Delta$,
with high probability a random $k$-CNF over $n$ variables and $\Delta n$
clauses requires resolution clause space of
$\Omega(n \cdot \Delta^{-\frac{1+\epsilon}{k-2-\epsilon}})$,
for any $0<\epsilon<1/2$. For constant $\Delta$, ... more >>>

We prove that, with high probability, the space complexity of refuting
a random unsatisfiable boolean formula in $k$-CNF on $n$
variables and $m = \Delta n$ clauses is
$O(n \cdot \Delta^{-\frac{1}{k-2}})$.

We describe a deterministic algorithm that, for constant k,
given a k-DNF or k-CNF formula f and a parameter e, runs in time
linear in the size of f and polynomial in 1/e and returns an
estimate of the fraction of satisfying assignments for f up to ... more >>>

Bayesian inference and counting satisfying assignments are important
problems with numerous applications in probabilistic reasoning. In this
paper, we show that plain old DPLL equipped with memoization can solve
both of these problems with time complexity that is at least as
good as all known algorithms. Furthermore, DPLL with memoization
more >>>

$k$-SAT is one of the best known among a wide class of random
constraint satisfaction problems believed to exhibit a threshold
phenomenon where the control parameter is the ratio, number of
constraints to number of variables. There has been a large amount of
work towards estimating ... more >>>

The satisfiability problem of Boolean Formulae in 3-CNF (3-SAT)
is a well known NP-complete problem and the development of faster
(moderately exponential time) algorithms has received much interest
in recent years. We show that the 3-SAT problem can be solved by a
probabilistic algorithm in expected time O(1,3290^n).
more >>>

We study approximation hardness and satisfiability of bounded
occurrence uniform instances of SAT. Among other things, we prove
the inapproximability for SAT instances in which every clause has
exactly 3 literals and each variable occurs exactly 4 times,
and display an explicit ... more >>>

Abstract. It is known that random k-SAT formulas with at least
(2^k*ln2)*n random clauses are unsatisfiable with high probability. This
result is simply obtained by bounding the expected number of satisfy-
ing assignments of a random k-SAT instance by an expression tending
to 0 when n, the number of variables ... more >>>

We prove results on the computational complexity of instances of 3SAT in which every variable occurs 3 or 4 times.

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions ... more >>>

We prove the first time-space tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon the known time-space tradeoffs for Sat. Let m be a positive integer, and define MODm-Sat to be the problem of determining if a given Boolean formula has ... more >>>

Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by ... more >>>

We show that tree-like OBDD proofs of unsatisfiability require an exponential increase ($s \mapsto 2^{s^{\Omega(1)}}$) in proof size to simulate unrestricted resolution, and that unrestricted OBDD proofs of unsatisfiability require an almost-exponential increase ($s \mapsto 2^{ 2^{\left( \log s \right)^{\Omega(1)}}}$) in proof size to simulate $\Res{O(\log n)}$. The ``OBDD proof ... more >>>

Consider the following two-player communication process to decide a language $L$: The first player holds the entire input $x$ but is polynomially bounded; the second player is computationally unbounded but does not know any part of $x$; their goal is to cooperatively decide whether $x$ belongs to $L$ at small ... more >>>

We give two time- and space-efficient simulations of quantum computations with
intermediate measurements, one by classical randomized computations with
unbounded error and the other by quantum computations that use an arbitrary
fixed universal set of gates. Specifically, our simulations show that every
language solvable by a bounded-error quantum algorithm running ... more >>>

Separating different propositional proof systems---that is, demonstrating that one proof system cannot efficiently simulate another proof system---is one of the main goals of proof complexity. Nevertheless, all known separation results between non-abstract proof systems are for specific families of hard tautologies: for what we know, in the average case all ... more >>>

Recently, Moser and Tardos [MT10] came up with a constructive proof of the Lovász Local Lemma. In this paper, we give another constructive proof of the lemma, based on Kolmogorov complexity. Actually, we even improve the Local Lemma slightly.

This paper characterizes alternation trading based proofs that satisfiability is not in the time and space bounded class $\DTISP(n^c, n^\epsilon)$, for various values $c<2$ and $\epsilon<1$. We characterize exactly what can be proved in the $\epsilon=0$ case with currently known methods, and prove the conjecture of Williams that $c=2\cos(\pi/7)$ is ... more >>>

We construct a PCP for NTIME(2$^n$) with constant
soundness, $2^n \poly(n)$ proof length, and $\poly(n)$
queries where the verifier's computation is simple: the
queries are a projection of the input randomness, and the
computation on the prover's answers is a 3CNF. The
previous upper bound for these two computations was
more >>>

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a simple proof of this result.

An AND-compression is ... more >>>

In this work we extend the study of solution graphs and prove that for boolean formulas in a class called CPSS, all connected components are partial cubes of small dimension, a statement which was proved only for some cases in [Schwerdtfeger 2013]. In contrast, we show that general Schaefer formulas ... more >>>

We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit $C(x_1,\ldots,x_n)$ of size $s$ and degree $d$ over a field ${\mathbb F}$, and any inputs $a_1,\ldots,a_K \in {\mathbb F}^n$,
$\bullet$ the Prover sends the Verifier the values $C(a_1), \ldots, C(a_K) \in {\mathbb F}$ and ... more >>>

Most of the known lower bounds for binary Boolean circuits with unrestricted depth are proved by the gate elimination method. The most efficient known algorithms for the #SAT problem on binary Boolean circuits use similar case analyses to the ones in gate elimination. Chen and Kabanets recently showed that the ... more >>>

We prove that if every problem in $NP$ has $n^k$-size circuits for a fixed constant $k$, then for every $NP$-verifier and every yes-instance $x$ of length $n$ for that verifier, the verifier's search space has an $n^{O(k^3)}$-size witness circuit: a witness for $x$ that can be encoded with a circuit ... more >>>