2,943 research outputs found
The zero-error randomized query complexity of the pointer function
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to
prove separation results among various measures of complexity such as
deterministic, randomized and quantum query complexities, exact and approximate
polynomial degrees, etc. In particular, the widest possible (quadratic)
separations between deterministic and zero-error randomized query complexity,
as well as between bounded-error and zero-error randomized query complexity,
have been obtained by considering {\em
variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function.
However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the
precise zero-error complexity of the original pointer function was not known.
We show a lower bound of on the zero-error
randomized query complexity of the pointer function on bits;
since an upper bound is already known
\cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a
factor of \polylog\, n
Greedy Algorithm for Inference of Decision Trees from Decision Rule Systems
Decision trees and decision rule systems play important roles as classifiers,
knowledge representation tools, and algorithms. They are easily interpretable
models for data analysis, making them widely used and studied in computer
science. Understanding the relationships between these two models is an
important task in this field. There are well-known methods for converting
decision trees into systems of decision rules. In this paper, we consider the
inverse transformation problem, which is not so simple. Instead of constructing
an entire decision tree, our study focuses on a greedy polynomial time
algorithm that simulates the operation of a decision tree on a given tuple of
attribute values.Comment: arXiv admin note: substantial text overlap with arXiv:2305.01721,
arXiv:2302.0706
Hard Problems on Random Graphs
Many graph properties are expressible in first order logic. Whether a graph contains a clique or a dominating set of size k are two examples. For the solution size as its parameter the first one is W[1]-complete and the second one W[2]-complete meaning that both of them are hard problems in the worst-case. If we look at both problem from the aspect of average-case complexity, the picture changes. Clique can be solved in expected FPT time on uniformly distributed graphs of size n, while this is not clear for Dominating Set. We show that it is indeed unlikely that Dominating Set can be solved efficiently on random graphs: If yes, then every first-order expressible graph property can be solved in expected FPT time, too. Furthermore, this remains true when we consider random graphs with an arbitrary constant edge probability. We identify a very simple problem on random matrices that is equally hard to solve on average: Given a square boolean matrix, are there k rows whose logical AND is the zero vector? The related Even Set problem on the other hand turns out to be efficiently solvable on random instances, while it is known to be hard in the worst-case
Towards Better Separation between Deterministic and Randomized Query Complexity
We show that there exists a Boolean function which observes the following
separations among deterministic query complexity , randomized zero
error query complexity and randomized one-sided error query
complexity : and
. This refutes the conjecture made by Saks
and Wigderson that for any Boolean function ,
. This also shows widest separation between
and for any Boolean function. The function was defined by
G{\"{o}}{\"{o}}s, Pitassi and Watson who studied it for showing a separation
between deterministic decision tree complexity and unambiguous
non-deterministic decision tree complexity. Independently of us, Ambainis et al
proved that different variants of the function certify optimal (quadratic)
separation between and , and polynomial separation between
and . Viewed as separation results, our results are subsumed
by those of Ambainis et al. However, while the functions considerd in the work
of Ambainis et al are different variants of , we work with the original
function itself.Comment: Reference adde
Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs
We show that given an embedding of an O(log n) genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists.
As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for O(log n) genus bipartite graphs
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