3 research outputs found
Sensitivity, Affine Transforms and Quantum Communication Complexity
We study the Boolean function parameters
sensitivity (), block sensitivity (), and alternation () under
specially designed affine transforms. For a function f:\F_2^n\to \{0,1\}, and
for M \in \F_2^{n\times n} and b\in \F_2^n, the result of the
transformation is defined as \forall x\in\F_2^n, g(x)=f(Mx+b).
We study alternation under linear shifts ( is the identity matrix) called
the shift invariant alternation (denoted by ). We exhibit an explicit
family of functions for which is . We show an
affine transform , such that the corresponding function satisfies
, using which we proving that for , the
bounded error quantum communication complexity of with prior entanglement,
. Our proof builds on ideas from
Sherstov (2010) where we use specific properties of the above affine
transformation. We show,
* For a prime and , any with
must satisfy . Here, denotes the degree of
the multilinear polynomial of over \F_p.
* For any such that there exists primes and with for , the deterministic communication
complexity - and are polynomially related. In particular,
this holds when . Thus, for this class of functions, this
answers an open question (see Buhrman and deWolf (2001)) about the relation
between the two measures.
We construct linear transformation , such that satisfies, . Using this, we exhibit a family of Boolean functions that rule out a
potential approach to settle the XOR Log-Rank conjecture via a proof of
Sensitivity conjecture [Hao Huang (2019)].Comment: 19 pages, 1 figure. Added a new lower bound for shifted alternation
(in Section 3) and an application to the existence of family of functions
under linear transforms (in Section 5
Parity Decision Tree Complexity is Greater Than Granularity
We prove a new lower bound on the parity decision tree complexity
of a Boolean function . Namely, granularity of the
Boolean function is the smallest such that all Fourier coefficients of
are integer multiples of . We show that .
This lower bound is an improvement of lower bounds through the sparsity of
and through the degree of over . Using our lower bound we
determine the exact parity decision tree complexity of several important
Boolean functions including majority and recursive majority. For majority the
complexity is , where is the number of
ones in the binary representation of . For recursive majority the complexity
is . Finally, we provide an example of a function for which our
lower bound is not tight.
Our results imply new lower bound of on the
multiplicative complexity of majority.Comment: Compared to the previous version we added a comparison of the
complexity measures discussed to the degree of Boolean functions over
. We removed the section on as a non-instructive
example. We added the connection to multiplicative complexit
Exact Algorithms With Worst-case Guarantee For Scheduling: From Theory to Practice
This PhD thesis summarizes research works on the design of exact algorithms
that provide a worst-case (time or space) guarantee for NP-hard scheduling
problems. Both theoretical and practical aspects are considered with three main
results reported. The first one is about a Dynamic Programming algorithm which
solves the F3Cmax problem in O*(3^n) time and space. The algorithm is easily
generalized to other flowshop problems and single machine scheduling problems.
The second contribution is about a search tree method called Branch & Merge
which solves the 1||SumTi problem with the time complexity converging to
O*(2^n) and in polynomial space. Our third contribution aims to improve the
practical efficiency of exact search tree algorithms solving scheduling
problems. First we realized that a better way to implement the idea of Branch &
Merge is to use a technique called Memorization. By the finding of a new
algorithmic paradox and the implementation of a memory cleaning strategy, the
method succeeded to solve instances with 300 more jobs with respect to the
state-of-the-art algorithm for the 1||SumTi problem. Then the treatment is
extended to another three problems 1|ri|SumCi, 1|dtilde|SumwiCi and F2||SumCi.
The results of the four problems all together show the power of the
Memorization paradigm when applied on sequencing problems. We name it Branch &
Memorize to promote a systematic consideration of Memorization as an essential
building block in branching algorithms like Branch and Bound. The method can
surely also be used to solve other problems, which are not necessarily
scheduling problems.Comment: 156 pages, PhD thesi