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Faster algorithm for counting of the integer points number in Δ-modular polyhedra [article]

D. V. Gribanov, D. S. Malyshev
2022 arXiv   pre-print
Given bounds can be used to obtain faster algorithms for the ILP feasibility problem, and for the problem to count integer points in a simplex or in an unbounded Subset-Sum polytope.  ...  We show that |𝒫∩ℤ^n| can be computed with an algorithm, having the arithmetic complexity bound O( ν(d,m,Δ) · d^3 ·Δ^4 ·log(Δ) ), where d = (𝒫) and ν(d,m,Δ) is the maximal possible number of vertices  ...  Corollary 1 The arithmetic complexity of an algorithm by Theorem 2 can be bounded with the following relations: The first bound can be used to count the number of integer points in a simplex or the number  ... 
arXiv:2110.01732v4 fatcat:mw55oxqemjhjvngjrzcdxfzcsy

Dynamic enforcement of knowledge-based security policies using probabilistic abstract interpretation

Piotr Mardziel, Stephen Magill, Michael Hicks, Mudhakar Srivatsa
2013 Journal of Computer Security  
We focus on developing probabilistic polyhedra in particular, to support numeric programs.  ...  We also show that, for our benchmarks, restricting constraints to octagons or intervals, rather than full polyhedra, can dramatically improve performance while incurring little to no loss in precision.  ...  The views and conclusions contained in this document are those of the author(s) and should not be interpreted as representing the official policies, either expressed or implied, of the U.S.  ... 
doi:10.3233/jcs-130469 fatcat:xujrukivkfhypbagdrcpyyltsi

Inference in Higher Order MRF-MAP Problems with Small and Large Cliques

Ishant Shanu, Chetan Arora, S.N. Maheshwari
2018 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition  
This allows us to combine the two styles in a joint framework exploiting the strength of both of them.  ...  Inference in a general MRF-MAP problem is NP Hard, but can be performed in polynomial time for the special case when potential functions are submodular.  ...  Acknowledgement: Chetan Arora has been supported by Infosys Center for AI, Visversaraya Young Faculty Research Fellowship, and EMR Funding by DST-SERB, Government of India.  ... 
doi:10.1109/cvpr.2018.00822 dblp:conf/cvpr/Shanu0M18 fatcat:sytjbkof4vgehkapyxqvfsiasm

SubPolyhedra: A (More) Scalable Approach to Infer Linear Inequalities [chapter]

Vincent Laviron, Francesco Logozzo
2008 Lecture Notes in Computer Science  
Hints can be automatically generated or provided by the user in the form of annotations. We implemented SubPoly on the top of Clousot, a generic abstract interpreter for .Net.  ...  SubPoly is as expressive as Polyhedra, but it drops some of the deductive power to achieve scalability.  ...  At the join point we apply Algorithm 1.  ... 
doi:10.1007/978-3-540-93900-9_20 fatcat:smwszmoe25eebgltl2xzvsxtq4

A complete, exact and efficient implementation for computing the edge-adjacency graph of an arrangement of quadrics

Michael Hemmer, Laurent Dupont, Sylvain Petitjean, Elmar Schömer
2011 Journal of symbolic computation  
We enhanced an implementation for an exact parameterization of the intersection curves of two quadrics, such that we can compute the exact parameter value for intersection points and from that the edgeadjacency  ...  Our implementation is complete in the sense that it can handle all kinds of inputs including all degenerate ones, i.e. singularities or tangential intersection points.  ...  For the chosen instances the number of vertices in the arrangement on Q 1 is quadratic in the number of quadrics. Therefore, both algorithms are linear in the number of vertices.  ... 
doi:10.1016/j.jsc.2010.11.002 fatcat:k5xkc3i3fzfabjhcrlffqjsmdm

Strongly polynomial and fully combinatorial algorithms for bisubmodular function minimization

S. Thomas McCormick, Satoru Fujishige
2008 Mathematical programming  
This new method gives a somewhat simpler strongly polynomial SFM algorithm, as well as the first combinatorial strongly polynomial algorithm for BSFM.  ...  Recently Fujishige and Iwata showed how to extend the Iwata, Fleischer, and Fujishige (IFF) algorithm for submodular function minimization (SFM) to bisubmodular function minimization (BSFM).  ...  Acknowledgment We thank Satoru Iwata for his substantial help with this material.  ... 
doi:10.1007/s10107-008-0242-9 fatcat:utoiruppevbu7gqc2xi4xmmm6i

Synthesising Interprocedural Bit-Precise Termination Proofs (extended version) [article]

Hong-Yi Chen, Cristina David, Daniel Kroening, Peter Schrammel, Björn Wachter
2015 arXiv   pre-print
Our experimental results show that our tool 2LS outperforms state-of-the-art alternatives, and demonstrate the clear advantage of interprocedural reasoning over monolithic analysis in terms of efficiency  ...  We present a modular termination analysis for C programs using template-based interprocedural summarisation.  ...  Since the domains of x, x ′ in Algorithm 4 and of x in in Algorithm 3 might be large, we limit also the number of iterations (default 20) of the while loops in these algorithms.  ... 
arXiv:1505.04581v1 fatcat:4e6whaznabc5dimy5bhf4e2dre

Polyhedral Omega: a New Algorithm for Solving Linear Diophantine Systems

Felix Breuer, Zafeirakis Zafeirakopoulos
2017 Annals of Combinatorics  
For example, they can be used to prove theorems in number theory and com- binatorics [4, 5, 50], compute volumes [16], count integer points in polyhedra [17], to maximize non-linear functions over lattice  ...  .: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19(4), 769–779 (1994) 18.  ... 
doi:10.1007/s00026-017-0349-x fatcat:72ojxpeo3fgyxdkydbnqaoyke4

Integer Programming and Algorithmic Geometry of Numbers [chapter]

Friedrich Eisenbrand
2009 50 Years of Integer Programming 1958-2008  
I also want to thank Johannes Blömer for several discussions on shortest and closest vectors.  ...  Acknowledgments I am grateful to Damien Stehlé for numerous comments and suggestions which helped me a lot to improve this manuscript.  ...  Barvinok and Woods [7] extended the counting algorithm of Barvinok [8] such that it computes a short rational generating function for an integer projection of the set of integer points in a polytope  ... 
doi:10.1007/978-3-540-68279-0_14 fatcat:c7iusb6esbgpnbjnmohiokwegy

Exploring the Impact of Affine Loop Transformations in Qubit Allocation [article]

Martin Kong
2020 arXiv   pre-print
In this paper we explore the synergies and impact of affine loop transformations in the context of qubit allocation and mapping.  ...  Our results demonstrate that affine transformations using global optimization criteria can cooperate effectively in several scenarios with quantum qubit mapping algorithms to reduce the circuit depth,  ...  Figure 14 . adder-maj-uma and cuccaro-adder-6bit circuits • rd84_142: Counts the number of ones in the input [79] .  ... 
arXiv:2010.11999v1 fatcat:resho5oxnrf2blaymhw2lfnw44

From Array Domains to Abstract Interpretation Under Store-Buffer-Based Memory Models [chapter]

Thibault Suzanne, Antoine Miné
2016 Lecture Notes in Computer Science  
Whereas the usual method for this kind of programs implements a program transformation to come back to an analysis under a sequentially consistent model, the novelty of our work consists in applying abstract  ...  We discuss an application to fence removal and show that our implementation is usually able to remove as many or more fences, with respect to the state of the art, on concurrent algorithms designed for  ...  A concrete domain for PSO programs V is the numerical set in which the variables are valued, for instance Z or Q (respectively the set of integers and rational numbers).  ... 
doi:10.1007/978-3-662-53413-7_23 fatcat:jfreslno2nfehcgqff35azfwqi

MCPA: Program Analysis as Machine Learning [article]

Marcel Böhme
2019 arXiv   pre-print
However, the analytical approach is ill-equiped to handle implementations of complex, large-scale, heterogeneous software systems we see in the real world.  ...  We demonstrate the pertinent concepts of MCPA using three applications: (ϵ,δ)-approximate quantitative analysis, (ϵ,δ)-approximate software verification, and (ϵ,δ)-approximate patch verification.  ...  David Rosenblum for his inspiring ASE'16 keynote speech on probabilistic thinking [36] .  ... 
arXiv:1911.04687v1 fatcat:67lob2qahfdxbhmrq5m7prydra

SubPolyhedra: a family of numerical abstract domains for the (more) scalable inference of linear inequalities

Vincent Laviron, Francesco Logozzo
2011 International Journal on Software Tools for Technology Transfer (STTT)  
Abstract domains in SubPoly are as expressive as Polyhedra, but they drop some of the deductive power to achieve scalability.  ...  The specification usually includes the absence of runtime exceptions (division by zero, integer overflow, array index out of bounds . . . ) and programmer annotations in the form of preconditions, postconditions  ...  Acknowledgments Thanks to Lev Nachmanson for providing us the Simplex implementation. Thanks to Manuel Fähndrich, Jérôme Feret, Corneliu Popeea for the useful discussions.  ... 
doi:10.1007/s10009-011-0199-5 fatcat:nq3sozwbyzeutm6eve5j3pbgoe

Learning with Submodular Functions: A Convex Optimization Perspective [article]

Francis Bach
2013 arXiv   pre-print
This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance.  ...  In this monograph, we present the theory of submodular functions from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization  ...  The suggestions of the reviewers were greatly appreciated and have significantly helped improve the manuscript.  ... 
arXiv:1111.6453v2 fatcat:qsbgrxoot5f7jhss4otffr3izy

Learning with Submodular Functions: A Convex Optimization Perspective

Francis Bach
2013 Foundations and Trends® in Machine Learning  
The suggestions of the reviewers were greatly appreciated and have significantly helped improve the manuscript.  ...  The author would like to thank Thibaut Horel, Stefanie Jegelka, Rodolphe Jenatton, Armand Joulin, Simon Lacoste-Julien, Julien Mairal and Guillaume Obozinski for discussions related to submodular functions  ...  -Counting elements in a partitions: Given a partition of V into m sets G 1 , . . . , G m , then the function F that counts for a set A the number of elements in the partition which intersects A is submodular  ... 
doi:10.1561/2200000039 fatcat:kk7w6zsnsnbp3eoa6b5ol3bxbq
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