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On a Simple Connection Between Δ-modular ILP and LP, and a New Bound on the Number of Integer Vertices
[article]

2022
*
arXiv
*
pre-print

Using this fact, we prove

arXiv:2203.03907v1
fatcat:6vh7vbinzff6ti525r6keemd6m
*the*inequality |vert(P_I)| ≤ 2 ·mn·*Δ*^n-1, for*the**number**of**vertices**of*P_I, which is better, than*the*state*of**the*art*bound*for*Δ*= O(n^2). ... We show that*a*similar fact holds for*the**integer*polyhedron P_I = conv(P ∩ Z^n) if we additionally suppose that P is*Δ*-*modular*, for some*Δ*∈{1,2,...}. ...*A**simple**connection**between*∆-*modular**ILP**and**LP*Let*A*∈ Z m×n , rank(*A*) = n, b ∈ Z m ,*and*P be*the*n-dimensional polyhedron, defined by*the*system Ax ≤ b. Let F be*a*k-dimensional face*of*P. ...##
###
An FPTAS for the Δ-modular multidimensional knapsack problem
[article]

2021
*
arXiv
*
pre-print

In this note, we show that

arXiv:2103.07257v4
fatcat:ls3utbfo7jbetg2kpmuv3q4zha
*the*m-dimensional knapsack problem with*a**Δ*-*modular*constraints matrix admits an FPTAS, whose complexity*bound*depends*on**Δ*linearly. ... Strictly speaking,*the*considered problem can be solved by an exact polynomial-time algorithm, when m is fixed*and**Δ*grows as*a*polynomial*on*n. ... Any optimal vertex solution*of**the**LP*problem has at most m non-zero coordinates, so we have*the*following*bound**on**the*l 1 -norm*of*an optimal*ILP*solution z * − ⌊x * ⌋*of**the**new*problem: z * − ⌊x * ...##
###
On Δ-Modular Integer Linear Problems In The Canonical Form And Equivalent Problems
[article]

2022
*
arXiv
*
pre-print

*the*

*number*

*of*rows

*of*

*A*

*and*A_max. ... We assume that

*the*

*integer*matrix

*A*has

*the*rank n, (n + m) rows, n columns,

*and*parameterize

*the*problem by m

*and*

*Δ*(

*A*), where

*Δ*(

*A*) is

*the*maximum

*of*n × n sub-determinants

*of*

*A*, taken in

*the*absolute ... Paat for useful discussions during preparation

*of*this article. Additionally,

*the*authors would like to thank anonymous reviewers for their useful comments

*and*remarks. ...

##
###
Finding Community Structure with Performance Guarantees in Complex Networks
[article]

2011
*
arXiv
*
pre-print

Our experiments show that

arXiv:1108.4034v1
fatcat:lropckotqrhlxizqk57meapu2i
*the*rounding algorithm returns*the*optimal solutions in most cases*and*are very scalable, that is, it can run*on**a*network*of**a*few thousand nodes whereas*the**LP*solution in*the*...*The*first algorithm comes with*a*priori guarantee that*the**modularity**of*found community structure is within*a*constant factor*of**the*optimal*modularity*when*the*network has*the*power-law degree distribution ... We can rewrite*the**modularity*as Q opt = 1 2m*δ*opt ij =1 B ij Construct*a*k-division*of**the*network by randomly assigning communities in*δ*opt into*one**of*k*new*"super" communities. ...##
###
Toward Optimal Community Detection: From Trees to General Weighted Networks

2014
*
Internet Mathematics
*

By exploiting

doi:10.1080/15427951.2014.950875
fatcat:snz3qfohnnbdvpcw7hrupwvjly
*the*graph*connectivity*structure, sparse metrics helps to reduce substantially*the**number**of*constraints, thus, vastly improve*the*running time for solving linear programming*and**integer*... In this paper, we study exact*and*theoretically near-optimal algorithms for maximizing*modularity*. In*the*first part, we investigate*the*complexity*and*approximability*of**the*problem*on*tree graphs. ...*The*set*of*edges in*the**new*graph are identical to*the*edges in F . Since T is*connected**and*cycle-free, T*A*is also*connected**and*cycle-free. Thus T*A*is*a*tree with k*vertices*. ...##
###
Graph Clustering in All Parameter Regimes

2020
*
International Symposium on Mathematical Foundations of Computer Science
*

This

doi:10.4230/lipics.mfcs.2020.39
dblp:conf/mfcs/GanGVWZ20
fatcat:ssmnwfipo5gpvcdpam7xoitomu
*bound*is tight*on*star graphs. ... They capture*a**number**of*specific clustering problems as special cases, including sparsest cut*and*cluster deletion. ...*The**new*approximate function has*a*smaller*number**of*linear pieces, since we have selected*a*strict sub-family*of*clusterings,*and*upper*bounds**the*function OPT. ...##
###
Higher-order segmentation via multicuts

2016
*
Computer Vision and Image Understanding
*

*The*basic Potts model

*and*natural extensions thereof to higherorder models provide

*a*prominent class

*of*representatives, that cover

*a*broad range

*of*segmentation problems relevant to image analysis

*and*... We show how to take into account such higher-order terms systematically in view

*of*computational inference,

*and*present results

*of*

*a*comprehensive

*and*competitive numerical evaluation

*of*

*a*variety

*of*dedicated ... This work has been supported by

*the*German Research Foundation (DFG) within

*the*program "Spatio-/Temporal Graphical Models

*and*Applications in Image Analysis", grant GRK 1653. ...

##
###
A Comparative Study of Modern Inference Techniques for Structured Discrete Energy Minimization Problems
[article]

2014
*
arXiv
*
pre-print

However,

arXiv:1404.0533v1
fatcat:eh2mhtz2dngnjh4uhzscw2w7c4
*on**new**and*challenging types*of*models our findings disagree*and*suggest that polyhedral methods*and**integer*programming solvers are competitive in terms*of*runtime*and*solution quality over*a*... We present an empirical comparison*of*32 state-*of*-*the*-art optimization techniques*on**a*corpus*of*2,453 energy minimization instances from diverse applications in computer vision. ... This work has been supported by*the*German Research Foundation (DFG) within*the*program "Spatio-/ Temporal Graphical Models*and*Applications in Image Analysis", grant GRK 1653. ...##
###
Exact Algorithms for Combinatorial Optimization Problems with Submodular Objective Functions
[chapter]

2013
*
Facets of Combinatorial Optimization
*

We propose two

doi:10.1007/978-3-642-38189-8_12
fatcat:atqmavcja5c77mvs7tk5uzwjem
*new*approaches for constrained submodular minimization problems.*The*first is*a*linearization approach that requires only*a*small*number**of*additional variables. ... We exploit*a*tight polyhedral description*of*this*new*model*and*an efficient separation algorithm. ...*The*expected returns µ*and**the*costs*a*are independent random*numbers**between*0*and*100.*The*variances σ are chosen as*the*expected returns multiplied by an independent random*number**between*0*and*1. ...##
###
Integer Programming in Parameterized Complexity: Three Miniatures
[article]

2018
*
arXiv
*
pre-print

Specifically, we consider graphs

arXiv:1711.02032v3
fatcat:zzn6cumyqrcifpk7kevo5kaafm
*of**bounded*neighborhood diversity which are in*a*sense*the*simplest*of*dense graphs,*and*we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring,*and*... This is understandable: it is often difficult to infer exact runtimes or even*the*distinction*between*FPT*and*XP algorithms,*and*some knowledge is simply unwritten folklore in*a*different community. ... Provided*bounds**on**the*encoding length*and**number**of*inequalities, there is*a*good*bound**on**the**number**of**vertices**of**the**integer*hull*of**a*polyhedron: Theorem 22 (Cook et al. [14] (*A*), tw P (*A*), td ...##
###
Higher-order Segmentation via Multicuts
[article]

2015
*
arXiv
*
pre-print

*The*basic Potts model

*and*natural extensions thereof to higher-order models provide

*a*prominent class

*of*such objectives, that cover

*a*broad range

*of*segmentation problems relevant to image analysis

*and*... Furthermore, we present results

*of*

*a*comprehensive

*and*competitive numerical evaluation

*of*

*a*variety

*of*dedicated cutting-plane algorithms. ... This work has been supported by

*the*German Research Foundation (DFG) within

*the*program "Spatio-/Temporal Graphical Models

*and*Applications in Image Analysis", grant GRK 1653. ...

##
###
A Message Passing Algorithm for the Minimum Cost Multicut Problem
[article]

2017
*
arXiv
*
pre-print

We propose

arXiv:1612.05441v2
fatcat:nqnac7jo65bk5kfjru4eahccbu
*a*dual decomposition*and*linear program relaxation*of**the*NP -hard minimum cost multicut problem. ... We define an algorithm that alternates*between*message passing*and*efficient separation*of*cycle-*and*odd-wheel inequalities. ... #I means*number**of*instances in dataset, #V*and*#E mean*number**of**vertices**and*edges in multicut instances. † signifies method did not finish*one*iteration after*one*hour, so was excluded from comparison ...##
###
Graph Spanners: A Tutorial Review
[article]

2020
*
arXiv
*
pre-print

As an additional contribution, we offer

arXiv:1909.03152v2
fatcat:vqme4ewuvbbd5ac5erinyykwsy
*a*list*of*open problems*on*graph spanners. ... This tutorial review provides*a*guiding reference to researchers who want to have an overview*of**the*large body*of*literature about graph spanners. ...*The*integrality gap is defined as*the*maximum ratio*between**the**ILP*solution*and**the*solution*of**the*corresponding relaxed*LP*. ...##
###
CROSS cyclic resource-constrained scheduling solver

2014
*
Artificial Intelligence
*

Our approach has been tested

doi:10.1016/j.artint.2013.09.006
fatcat:rpbdeh7jrvf2ddz5ysctbj3xrm
*on**a**number**of*non-trivial synthetic instances*and**on**a*set*of*realistic industrial instances. ...*The*first*one*(referred to as CROSS) models*a*pure cyclic scheduling problem*and*makes use*of*both our novel constraints. ... Acknowledgements We thank Maria Ayala*and*Christian Artigues for*the*useful discussions*and**the*industrial benchmarks. ...##
###
Applications of reformulations in mathematical programming

2012
*
4OR
*

*One*

*of*

*the*reasons why

*the*problem is hard to solve is

*the*presence

*of*several symmetric optimal solutions,

*and*consequently

*a*very large Branch-

*and*-

*Bound*tree. ... Both computational time

*and*size

*of*

*the*Branch-

*and*-

*Bound*tree outperform

*the*

*ones*provided by

*the*original formulation. ... In

*a*

*simple*graph, there is at most

*one*edge

*between*any pair

*of*

*vertices*,

*and*no loops. ...

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