82,799 research outputs found
More results on Shortest Linear Programs
At the FSE conference of ToSC 2018, Kranz et al. presented their results on shortest linear programs for the linear layers of
several well known block ciphers in literature. Shortest linear programs are essentially the minimum number of 2-input xor gates required to completely describe a linear system of equations. In the above paper the authors showed that the commonly used metrics like d-xor/s-xor count that are used to judge the ``lightweightedness\u27\u27 do not represent the minimum number of xor gates required to describe a given MDS matrix. In fact they used heuristic based algorithms of Boyar/Peralta and Paar to find implementations of MDS matrices with even fewer xor gates than was previously known. They proved that the AES mixcolumn matrix can be implemented with as little as 97 xor gates. In this paper we show that the values reported in the above paper
are not optimal. By suitably including random bits in the instances of the above algorithms we can achieve implementations of almost all matrices with lesser number of gates than were reported in the above paper. As a result we report an implementation of the AES mixcolumn matrix that uses only 95 xor gates.
In the second part of the paper, we observe that most standard cell libraries contain both 2 and 3-input xor gates, with the silicon area of the 3-input xor gate being smaller than the sum of the areas of two 2-input xor gates. Hence when linear circuits are synthesized by logic compilers (with specific instructions to optimize for area), most of them would return a solution circuit containing both 2 and 3-input xor gates. Thus from a practical point of view, reducing circuit size in presence of these gates is no longer equivalent to solving the shortest linear program. In this paper we show that by adopting a graph based heuristic it is possible to convert a circuit constructed with 2-input xor gates to another functionally equivalent circuit that utilizes both 2 and 3-input xor gates and occupies less hardware area. As a result we obtain more lightweight implementations of all the matrices listed in the ToSC paper
Physarum Inspired Dynamics to Solve Semi-Definite Programs
Physarum Polycephalum is a Slime mold that can solve the shortest path problem. A mathematical model based on the Physarum's behavior, known as the Physarum Directed Dynamics, can solve positive linear programs. In this paper, we will propose a Physarum based dynamic based on the previous work and introduce a new way to solve positive Semi-Definite Programming (SDP) problems, which are more general than positive linear programs. Empirical results suggest that this extension of the dynamic can solve the positive SDP showing that the nature-inspired algorithm can solve one of the hardest problems in the polynomial domain. In this work, we will formulate an accurate algorithm to solve positive and some non-negative SDPs and formally prove some key characteristics of this solver thus inspiring future work to try and refine this method
Mixed integer predictive control and shortest path reformulation
Mixed integer predictive control deals with optimizing integer and real
control variables over a receding horizon. The mixed integer nature of controls
might be a cause of intractability for instances of larger dimensions. To
tackle this little issue, we propose a decomposition method which turns the
original -dimensional problem into indipendent scalar problems of lot
sizing form. Each scalar problem is then reformulated as a shortest path one
and solved through linear programming over a receding horizon. This last
reformulation step mirrors a standard procedure in mixed integer programming.
The approximation introduced by the decomposition can be lowered if we operate
in accordance with the predictive control technique: i) optimize controls over
the horizon ii) apply the first control iii) provide measurement updates of
other states and repeat the procedure
Robust optimization with incremental recourse
In this paper, we consider an adaptive approach to address optimization
problems with uncertain cost parameters. Here, the decision maker selects an
initial decision, observes the realization of the uncertain cost parameters,
and then is permitted to modify the initial decision. We treat the uncertainty
using the framework of robust optimization in which uncertain parameters lie
within a given set. The decision maker optimizes so as to develop the best cost
guarantee in terms of the worst-case analysis. The recourse decision is
``incremental"; that is, the decision maker is permitted to change the initial
solution by a small fixed amount. We refer to the resulting problem as the
robust incremental problem. We study robust incremental variants of several
optimization problems. We show that the robust incremental counterpart of a
linear program is itself a linear program if the uncertainty set is polyhedral.
Hence, it is solvable in polynomial time. We establish the NP-hardness for
robust incremental linear programming for the case of a discrete uncertainty
set. We show that the robust incremental shortest path problem is NP-complete
when costs are chosen from a polyhedral uncertainty set, even in the case that
only one new arc may be added to the initial path. We also address the
complexity of several special cases of the robust incremental shortest path
problem and the robust incremental minimum spanning tree problem
A Reference Interpreter for the Graph Programming Language GP 2
GP 2 is an experimental programming language for computing by graph
transformation. An initial interpreter for GP 2, written in the functional
language Haskell, provides a concise and simply structured reference
implementation. Despite its simplicity, the performance of the interpreter is
sufficient for the comparative investigation of a range of test programs. It
also provides a platform for the development of more sophisticated
implementations.Comment: In Proceedings GaM 2015, arXiv:1504.0244
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial
Stratified Negation in Limit Datalog Programs
There has recently been an increasing interest in declarative data analysis,
where analytic tasks are specified using a logical language, and their
implementation and optimisation are delegated to a general-purpose query
engine. Existing declarative languages for data analysis can be formalised as
variants of logic programming equipped with arithmetic function symbols and/or
aggregation, and are typically undecidable. In prior work, the language of
was proposed, which is sufficiently powerful to
capture many analysis tasks and has decidable entailment problem. Rules in this
language, however, do not allow for negation. In this paper, we study an
extension of limit programs with stratified negation-as-failure. We show that
the additional expressive power makes reasoning computationally more demanding,
and provide tight data complexity bounds. We also identify a fragment with
tractable data complexity and sufficient expressivity to capture many relevant
tasks.Comment: 14 pages; full version of a paper accepted at IJCAI-1
Computational Approaches for Stochastic Shortest Path on Succinct MDPs
We consider the stochastic shortest path (SSP) problem for succinct Markov
decision processes (MDPs), where the MDP consists of a set of variables, and a
set of nondeterministic rules that update the variables. First, we show that
several examples from the AI literature can be modeled as succinct MDPs. Then
we present computational approaches for upper and lower bounds for the SSP
problem: (a)~for computing upper bounds, our method is polynomial-time in the
implicit description of the MDP; (b)~for lower bounds, we present a
polynomial-time (in the size of the implicit description) reduction to
quadratic programming. Our approach is applicable even to infinite-state MDPs.
Finally, we present experimental results to demonstrate the effectiveness of
our approach on several classical examples from the AI literature
Two Results on Slime Mold Computations
We present two results on slime mold computations. In wet-lab experiments
(Nature'00) by Nakagaki et al. the slime mold Physarum polycephalum
demonstrated its ability to solve shortest path problems. Biologists proposed a
mathematical model, a system of differential equations, for the slime's
adaption process (J. Theoretical Biology'07). It was shown that the process
convergences to the shortest path (J. Theoretical Biology'12) for all graphs.
We show that the dynamics actually converges for a much wider class of
problems, namely undirected linear programs with a non-negative cost vector.
Combinatorial optimization researchers took the dynamics describing slime
behavior as an inspiration for an optimization method and showed that its
discretization can -approximately solve linear programs with
positive cost vector (ITCS'16). Their analysis requires a feasible starting
point, a step size depending linearly on , and a number of steps
with quartic dependence on , where is
the difference between the smallest cost of a non-optimal basic feasible
solution and the optimal cost ().
We give a refined analysis showing that the dynamics initialized with any
strongly dominating point converges to the set of optimal solutions. Moreover,
we strengthen the convergence rate bounds and prove that the step size is
independent of , and the number of steps depends logarithmically
on and quadratically on
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