21,825 research outputs found
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
Levelable Sets and the Algebraic Structure of Parameterizations
Asking which sets are fixed-parameter tractable for a given parameterization
constitutes much of the current research in parameterized complexity theory.
This approach faces some of the core difficulties in complexity theory. By
focussing instead on the parameterizations that make a given set
fixed-parameter tractable, we circumvent these difficulties. We isolate
parameterizations as independent measures of complexity and study their
underlying algebraic structure. Thus we are able to compare parameterizations,
which establishes a hierarchy of complexity that is much stronger than that
present in typical parameterized algorithms races. Among other results, we find
that no practically fixed-parameter tractable sets have optimal
parameterizations
Resource Bounded Immunity and Simplicity
Revisiting the thirty years-old notions of resource-bounded immunity and
simplicity, we investigate the structural characteristics of various immunity
notions: strong immunity, almost immunity, and hyperimmunity as well as their
corresponding simplicity notions. We also study limited immunity and
simplicity, called k-immunity and feasible k-immunity, and their simplicity
notions. Finally, we propose the k-immune hypothesis as a working hypothesis
that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in
the Proceedings of the 3rd IFIP International Conference on Theoretical
Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France,
August 23-26, 200
Sculpting Quantum Speedups
Given a problem which is intractable for both quantum and classical
algorithms, can we find a sub-problem for which quantum algorithms provide an
exponential advantage? We refer to this problem as the "sculpting problem." In
this work, we give a full characterization of sculptable functions in the query
complexity setting. We show that a total function f can be restricted to a
promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only
if f has a large number of inputs with large certificate complexity. The proof
uses some interesting techniques: for one direction, we introduce new
relationships between randomized and quantum query complexity in various
settings, and for the other direction, we use a recent result from
communication complexity due to Klartag and Regev. We also characterize
sculpting for other query complexity measures, such as R(f) vs. R_0(f) and
R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity:
for example, a nearly quadratic relationship between Q(f) and D(f) whenever the
promise of f is small. This contrasts with the recent super-quadratic query
complexity separations, showing that the maximum gap between classical and
quantum query complexities is indeed quadratic in various settings - just not
for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if
there is any BPP-bi-immune language in BQP, then every language outside BPP can
be restricted to a promise which places it in PromiseBQP but not in PromiseBPP.
Under a weaker assumption, that some problem in BQP is hard on average for
P/poly, we show that every paddable language outside BPP is sculptable in this
way.Comment: 30 page
Simplicity of Completion Time Distributions for Common Complex Biochemical Processes
Biochemical processes typically involve huge numbers of individual reversible
steps, each with its own dynamical rate constants. For example, kinetic
proofreading processes rely upon numerous sequential reactions in order to
guarantee the precise construction of specific macromolecules. In this work, we
study the transient properties of such systems and fully characterize their
first passage (completion) time distributions. In particular, we provide
explicit expressions for the mean and the variance of the completion time for a
kinetic proofreading process and computational analyses for more complicated
biochemical systems. We find that, for a wide range of parameters, as the
system size grows, the completion time behavior simplifies: it becomes either
deterministic or exponentially distributed, with a very narrow transition
between the two regimes. In both regimes, the dynamical complexity of the full
system is trivial compared to its apparent structural complexity. Similar
simplicity is likely to arise in the dynamics of many complex multi-step
biochemical processes. In particular, these findings suggest not only that one
may not be able to understand individual elementary reactions from macroscopic
observations, but also that such understanding may be unnecessary
Exponential Time Complexity of Weighted Counting of Independent Sets
We consider weighted counting of independent sets using a rational weight x:
Given a graph with n vertices, count its independent sets such that each set of
size k contributes x^k. This is equivalent to computation of the partition
function of the lattice gas with hard-core self-repulsion and hard-core pair
interaction. We show the following conditional lower bounds: If counting the
satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time
2^{\Omega(n)} (i.e. there is a c>0 such that no algorithm can solve #3SAT in
time 2^{cn}), counting the independent sets of size n/3 of an n-vertex graph
needs time 2^{\Omega(n)} and weighted counting of independent sets needs time
2^{\Omega(n/log^3 n)} for all rational weights x\neq 0.
We have two technical ingredients: The first is a reduction from 3SAT to
independent sets that preserves the number of solutions and increases the
instance size only by a constant factor. Second, we devise a combination of
vertex cloning and path addition. This graph transformation allows us to adapt
a recent technique by Dell, Husfeldt, and Wahlen which enables interpolation by
a family of reductions, each of which increases the instance size only
polylogarithmically.Comment: Introduction revised, differences between versions of counting
independent sets stated more precisely, minor improvements. 14 page
-Generic Computability, Turing Reducibility and Asymptotic Density
Generic computability has been studied in group theory and we now study it in
the context of classical computability theory. A set A of natural numbers is
generically computable if there is a partial computable function f whose domain
has density 1 and which agrees with the characteristic function of A on its
domain. A set A is coarsely computable if there is a computable set C such that
the symmetric difference of A and C has density 0. We prove that there is a
c.e. set which is generically computable but not coarsely computable and vice
versa. We show that every nonzero Turing degree contains a set which is not
coarsely computable. We prove that there is a c.e. set of density 1 which has
no computable subset of density 1. As a corollary, there is a generically
computable set A such that no generic algorithm for A has computable domain. We
define a general notion of generic reducibility in the spirt of Turing
reducibility and show that there is a natural order-preserving embedding of the
Turing degrees into the generic degrees which is not surjective
An exponential lower bound for Individualization-Refinement algorithms for Graph Isomorphism
The individualization-refinement paradigm provides a strong toolbox for
testing isomorphism of two graphs and indeed, the currently fastest
implementations of isomorphism solvers all follow this approach. While these
solvers are fast in practice, from a theoretical point of view, no general
lower bounds concerning the worst case complexity of these tools are known. In
fact, it is an open question whether individualization-refinement algorithms
can achieve upper bounds on the running time similar to the more theoretical
techniques based on a group theoretic approach.
In this work we give a negative answer to this question and construct a
family of graphs on which algorithms based on the individualization-refinement
paradigm require exponential time. Contrary to a previous construction of
Miyazaki, that only applies to a specific implementation within the
individualization-refinement framework, our construction is immune to changing
the cell selector, or adding various heuristic invariants to the algorithm.
Furthermore, our graphs also provide exponential lower bounds in the case when
the -dimensional Weisfeiler-Leman algorithm is used to replace the standard
color refinement operator and the arguments even work when the entire
automorphism group of the inputs is initially provided to the algorithm.Comment: 21 page
The winner takes it all
We study competing first passage percolation on graphs generated by the
configuration model. At time 0, vertex 1 and vertex 2 are infected with the
type 1 and the type 2 infection, respectively, and an uninfected vertex then
becomes type 1 (2) infected at rate () times the number
of edges connecting it to a type 1 (2) infected neighbor. Our main result is
that, if the degree distribution is a power-law with exponent ,
then, as the number of vertices tends to infinity and with high probability,
one of the infection types will occupy all but a finite number of vertices.
Furthermore, which one of the infections wins is random and both infections
have a positive probability of winning regardless of the values of
and . The picture is similar with multiple starting points for the
infections
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