466 research outputs found
The structure of a logarithmic advice class
The complexity class Full-P / log, corresponding to a form of logarithmic
advice for polynomial time, is studied.
In order to understand the inner structure of this class, we characterize
Full-P /log in terms of Turing reducibility to a special family of
sparse sets. Other characterizations of Full-P / log, relating it to sets
with small information content, were already known. These used tally
sets whose words follow a given regular pattern and tally sets that are
regular in a resource-bounded Kolmogorov complexity sense.
We obtain here relationships between the equivalence classes of
the mentioned tally and sparse sets under various reducibiities, which
provide new knowledge about the logarithmic advice class.
Another way to measure the amount of information encoded in a
language in a nonuniform class, is to study the relative complexity of
computing advice functions for this language. We prove bounds on the
complexity of ad vice functions for Full-P / log and for other subclasses
of it. As a consequence, Full-P / log is located in the Extended Low
Hierarchy
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
The Structure of logarithmic advice complexity classes
A nonuniform class called here Full-P/log, due to Ko, is studied.
It corresponds to polynomial time with logarithmically long
advice. Its importance lies in the structural properties it enjoys,
more interesting than those of the alternative class P/log;
specifically, its introduction was motivated by the need of
a logarithmic advice class closed under polynomial-time deterministic
reductions. Several characterizations of Full-P/log are shown,
formulated in terms of various sorts of tally sets with very
small information content. A study of its inner structure is
presented, by considering the most usual reducibilities and
looking for the relationships among the corresponding reduction and
equivalence classes defined from these special tally sets.Postprint (published version
Finding weakly reversible realizations of chemical reaction networks using optimization
An algorithm is given in this paper for the computation of dynamically
equivalent weakly reversible realizations with the maximal number of reactions,
for chemical reaction networks (CRNs) with mass action kinetics. The original
problem statement can be traced back at least 30 years ago. The algorithm uses
standard linear and mixed integer linear programming, and it is based on
elementary graph theory and important former results on the dense realizations
of CRNs. The proposed method is also capable of determining if no dynamically
equivalent weakly reversible structure exists for a given reaction network with
a previously fixed complex set.Comment: 18 pages, 9 figure
Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting
In this paper we establish links between, and new results for, three problems
that are not usually considered together. The first is a matrix decomposition
problem that arises in areas such as statistical modeling and signal
processing: given a matrix formed as the sum of an unknown diagonal matrix
and an unknown low rank positive semidefinite matrix, decompose into these
constituents. The second problem we consider is to determine the facial
structure of the set of correlation matrices, a convex set also known as the
elliptope. This convex body, and particularly its facial structure, plays a
role in applications from combinatorial optimization to mathematical finance.
The third problem is a basic geometric question: given points
(where ) determine whether there is a centered
ellipsoid passing \emph{exactly} through all of the points.
We show that in a precise sense these three problems are equivalent.
Furthermore we establish a simple sufficient condition on a subspace that
ensures any positive semidefinite matrix with column space can be
recovered from for any diagonal matrix using a convex
optimization-based heuristic known as minimum trace factor analysis. This
result leads to a new understanding of the structure of rank-deficient
correlation matrices and a simple condition on a set of points that ensures
there is a centered ellipsoid passing through them.Comment: 20 page
Recommended from our members
New Applications of the Nearest-Neighbor Chain Algorithm
The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together
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 ≤ s(n). We show:
Learning Speedups. If C[poly(n)] admits a randomized weak learning algorithm under the
uniform distribution with membership queries that runs in time 2n/nω(1), then for every k ≥ 1
and ε > 0 the class C[n
k
] can be learned to high accuracy in time O(2n
ε
). There is ε > 0 such that
C[2n
ε
] can be learned in time 2n/nω(1) if and only if C[poly(n)] can be learned in time 2(log n)
O(1)
.
Equivalences between Learning Models. We use learning speedups to obtain equivalences
between various randomized learning and compression models, including sub-exponential
time learning with membership queries, sub-exponential time learning with membership and
equivalence queries, probabilistic function compression and probabilistic average-case function
compression.
A Dichotomy between Learnability and Pseudorandomness. In the non-uniform setting,
there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure
pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-C[n
k
] admits a
randomized weak learning algorithm with membership queries under the uniform distribution
that runs in time 2n/nω(1), then for each k ≥ 1, BPE * (depth-d)-C[n
k
]. If for some ε > 0 there
are P-natural proofs useful against C[2n
ε
], then ZPEXP * C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆
i.o.Circuit[n
k
], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2n/3
], then ZPEXP ⊆
i.o.ESUBEXP.
Hardness Results for MCSP. All functions in non-uniform NC1
reduce to the Minimum
Circuit Size Problem via truth-table reductions computable by TC0
circuits. In particular, if
MCSP ∈ TC0
then NC1 = TC0
- …