4,882 research outputs found
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
Let be a fixed graph on vertices. Let iff the input
graph on vertices contains as a (not necessarily induced) subgraph.
Let denote the cardinality of a maximum independent set of . In
this paper we show:
where
denotes the quantum query complexity of .
As a consequence we obtain a lower bounds for in terms of several
other parameters of such as the average degree, minimum vertex cover,
chromatic number, and the critical probability.
We also use the above bound to show that for any
, improving on the previously best known bound of . Until
very recently, it was believed that the quantum query complexity is at least
square root of the randomized one. Our bound for
matches the square root of the current best known bound for the randomized
query complexity of , which is due to Gr\"oger.
Interestingly, the randomized bound of for
still remains open.
We also study the Subgraph Homomorphism Problem, denoted by , and
show that .
Finally we extend our results to the -uniform hypergraphs. In particular,
we show an bound for quantum query complexity of the Subgraph
Isomorphism, improving on the previously known bound. For the
Subgraph Homomorphism, we obtain an bound for the same.Comment: 16 pages, 2 figure
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Approximate F_2-Sketching of Valuation Functions
We study the problem of constructing a linear sketch of minimum dimension that allows approximation of a given real-valued function f : F_2^n - > R with small expected squared error. We develop a general theory of linear sketching for such functions through which we analyze their dimension for most commonly studied types of valuation functions: additive, budget-additive, coverage, alpha-Lipschitz submodular and matroid rank functions. This gives a characterization of how many bits of information have to be stored about the input x so that one can compute f under additive updates to its coordinates.
Our results are tight in most cases and we also give extensions to the distributional version of the problem where the input x in F_2^n is generated uniformly at random. Using known connections with dynamic streaming algorithms, both upper and lower bounds on dimension obtained in our work extend to the space complexity of algorithms evaluating f(x) under long sequences of additive updates to the input x presented as a stream. Similar results hold for simultaneous communication in a distributed setting
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
Sensitivity Conjecture and Log-rank Conjecture for functions with small alternating numbers
The Sensitivity Conjecture and the Log-rank Conjecture are among the most
important and challenging problems in concrete complexity. Incidentally, the
Sensitivity Conjecture is known to hold for monotone functions, and so is the
Log-rank Conjecture for and with monotone
functions , where and are bit-wise AND and XOR,
respectively. In this paper, we extend these results to functions which
alternate values for a relatively small number of times on any monotone path
from to . These deepen our understandings of the two conjectures,
and contribute to the recent line of research on functions with small
alternating numbers
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