61 research outputs found
An EPTAS for Scheduling on Unrelated Machines of Few Different Types
In the classical problem of scheduling on unrelated parallel machines, a set
of jobs has to be assigned to a set of machines. The jobs have a processing
time depending on the machine and the goal is to minimize the makespan, that is
the maximum machine load. It is well known that this problem is NP-hard and
does not allow polynomial time approximation algorithms with approximation
guarantees smaller than unless PNP. We consider the case that there
are only a constant number of machine types. Two machines have the same
type if all jobs have the same processing time for them. This variant of the
problem is strongly NP-hard already for . We present an efficient
polynomial time approximation scheme (EPTAS) for the problem, that is, for any
an assignment with makespan of length at most
times the optimum can be found in polynomial time in the
input length and the exponent is independent of . In particular
we achieve a running time of , where
denotes the input length. Furthermore, we study three other problem
variants and present an EPTAS for each of them: The Santa Claus problem, where
the minimum machine load has to be maximized; the case of scheduling on
unrelated parallel machines with a constant number of uniform types, where
machines of the same type behave like uniformly related machines; and the
multidimensional vector scheduling variant of the problem where both the
dimension and the number of machine types are constant. For the Santa Claus
problem we achieve the same running time. The results are achieved, using mixed
integer linear programming and rounding techniques
Homomorphic encryption and some black box attacks
This paper is a compressed summary of some principal definitions and concepts
in the approach to the black box algebra being developed by the authors. We
suggest that black box algebra could be useful in cryptanalysis of homomorphic
encryption schemes, and that homomorphic encryption is an area of research
where cryptography and black box algebra may benefit from exchange of ideas
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
Parameterized Complexity of Maximum Edge Colorable Subgraph
A graph is {\em -edge colorable} if there is a coloring , such that for distinct , we have
. The {\sc Maximum Edge-Colorable Subgraph} problem
takes as input a graph and integers and , and the objective is to
find a subgraph of and a -edge-coloring of , such that . We study the above problem from the viewpoint of Parameterized
Complexity. We obtain \FPT\ algorithms when parameterized by: the vertex
cover number of , by using {\sc Integer Linear Programming}, and ,
a randomized algorithm via a reduction to \textsc{Rainbow Matching}, and a
deterministic algorithm by using color coding, and divide and color. With
respect to the parameters , where is one of the following: the
solution size, , the vertex cover number of , and l -
{\mm}(G), where {\mm}(G) is the size of a maximum matching in ; we show
that the (decision version of the) problem admits a kernel with vertices. Furthermore, we show that there is no kernel of size
, for any and computable
function , unless \NP \subseteq \CONPpoly
Centerpoints: a link between optimization and convex geometry
We introduce a concept that generalizes several different notions of a âcenterpointâ in the literature. We develop an oracle-based algorithm for convex mixed-integer optimization based on centerpoints. Further, we show that algorithms based on centerpoints are âbest possibleâ in a certain sense. Motivated by this, we establish several structural results about this concept and provide efficient algorithms for computing these points
Slide reduction, revisitedâfilling the gaps in svp approximation
We show how to generalize Gama and Nguyen's slide reduction algorithm [STOC
'08] for solving the approximate Shortest Vector Problem over lattices (SVP).
As a result, we show the fastest provably correct algorithm for
-approximate SVP for all approximation factors . This is the range of approximation factors most
relevant for cryptography
Improved Classical Cryptanalysis of SIKE in Practice
Item does not contain fulltextPKC 202
Curves over every global field violating the local-global principle
There is an algorithm that takes as input a global field k and produces a
curve over k violating the local-global principle. Also, given a global field k
and a nonnegative integer n, one can effectively construct a curve X over k
such that #X(k)=n and X has points over every completion of k.Comment: 5 page
On the shortness of vectors to be found by the Ideal-SVP quantum algorithm
The hardness of finding short vectors in ideals of cyclotomic number fields (hereafter, Ideal-SVP) can serve as a worst-case assumption for numerous efficient cryptosystems, via the average-case problems Ring-SIS and Ring-LWE. For a while, it could be assumed the Ideal-SVP problem was as hard a
Efficient Ephemeral Elliptic Curve Cryptographic Keys
We show how any pair of authenticated users can on-the-fly agree on an elliptic curve group that is unique to their communication session, unpredictable to outside observers, and secure against known attacks. Our proposal is suitable for deployment on constrained devices such as smartphones, allowing them to efficiently generate ephemeral parameters that are unique to any single cryptographic application such as symmetric key agreement. For such applications it thus offers an alternative to long term usage of standardized or otherwise pre-generated elliptic curve parameters, obtaining security against cryptographic attacks aimed at other users, and eliminating the need to trust elliptic curves generated by third parties
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