60 research outputs found

### Quantum Computation by Adiabatic Evolution

We give a quantum algorithm for solving instances of the satisfiability
problem, based on adiabatic evolution. The evolution of the quantum state is
governed by a time-dependent Hamiltonian that interpolates between an initial
Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian,
whose ground state encodes the satisfying assignment. To ensure that the system
evolves to the desired final ground state, the evolution time must be big
enough. The time required depends on the minimum energy difference between the
two lowest states of the interpolating Hamiltonian. We are unable to estimate
this gap in general. We give some special symmetric cases of the satisfiability
problem where the symmetry allows us to estimate the gap and we show that, in
these cases, our algorithm runs in polynomial time.Comment: 24 pages, 12 figures, LaTeX, amssymb,amsmath, BoxedEPS packages;
email to [email protected]

### How many functions can be distinguished with k quantum queries?

Suppose an oracle is known to hold one of a given set of D two-valued
functions. To successfully identify which function the oracle holds with k
classical queries, it must be the case that D is at most 2^k. In this paper we
derive a bound for how many functions can be distinguished with k quantum
queries.Comment: 5 pages. Lower bound on sorting n items improved to (1-epsilon)n
quantum queries. Minor changes to text and corrections to reference

### Intermediate problems in modular circuits satisfiability

In arXiv:1710.08163 a generalization of Boolean circuits to arbitrary finite
algebras had been introduced and applied to sketch P versus NP-complete
borderline for circuits satisfiability over algebras from congruence modular
varieties. However the problem for nilpotent (which had not been shown to be
NP-hard) but not supernilpotent algebras (which had been shown to be polynomial
time) remained open.
In this paper we provide a broad class of examples, lying in this grey area,
and show that, under the Exponential Time Hypothesis and Strong Exponential
Size Hypothesis (saying that Boolean circuits need exponentially many modular
counting gates to produce boolean conjunctions of any arity), satisfiability
over these algebras have intermediate complexity between $\Omega(2^{c\log^{h-1}
n})$ and $O(2^{c\log^h n})$, where $h$ measures how much a nilpotent algebra
fails to be supernilpotent. We also sketch how these examples could be used as
paradigms to fill the nilpotent versus supernilpotent gap in general.
Our examples are striking in view of the natural strong connections between
circuits satisfiability and Constraint Satisfaction Problem for which the
dichotomy had been shown by Bulatov and Zhuk

### Fine-Grained Reductions from Approximate Counting to Decision

In this paper, we introduce a general framework for fine-grained reductions
of approximate counting problems to their decision versions. (Thus we use an
oracle that decides whether any witness exists to multiplicatively approximate
the number of witnesses with minimal overhead.) This mirrors a foundational
result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the
polynomial-time setting, and a similar result of M\"uller (IWPEC 2006) in the
FPT setting. Using our framework, we obtain such reductions for some of the
most important problems in fine-grained complexity: the Orthogonal Vectors
problem, 3SUM, and the Negative-Weight Triangle problem (which is closely
related to All-Pairs Shortest Path).
We also provide a fine-grained reduction from approximate #SAT to SAT.
Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for
some $1<c<2$ and all $k$ there is an $O(c^n)$-time algorithm for k-SAT. Then we
prove that for all $k$, there is an $O((c+o(1))^n)$-time algorithm for
approximate #$k$-SAT. In particular, our result implies that the Exponential
Time Hypothesis (ETH) is equivalent to the seemingly-weaker statement that
there is no algorithm to approximate #3-SAT to within a factor of $1+\epsilon$
in time $2^{o(n)}/\epsilon^2$ (taking $\epsilon > 0$ as part of the input).Comment: An extended abstract was presented at STOC 201

### Credimus

We believe that economic design and computational complexity---while already
important to each other---should become even more important to each other with
each passing year. But for that to happen, experts in on the one hand such
areas as social choice, economics, and political science and on the other hand
computational complexity will have to better understand each other's
worldviews.
This article, written by two complexity theorists who also work in
computational social choice theory, focuses on one direction of that process by
presenting a brief overview of how most computational complexity theorists view
the world. Although our immediate motivation is to make the lens through which
complexity theorists see the world be better understood by those in the social
sciences, we also feel that even within computer science it is very important
for nontheoreticians to understand how theoreticians think, just as it is
equally important within computer science for theoreticians to understand how
nontheoreticians think

### Theory of Computation

A more extensive and theoretical treatment of the material in 6.045J/18.400J, emphasizing computability and computational complexity theory. Regular and context-free languages. Decidable and undecidable problems, reducibility, recursive function theory. Time and space measures on computation, completeness, hierarchy theorems, inherently complex problems, oracles, probabilistic computation, and interactive proof systems

### Theory of Computation

This graduate level course is more extensive and theoretical treatment of the material in Computability, and Complexity (6.045J / 18.400J). Topics include Automata and Language Theory, Computability Theory, and Complexity Theory

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