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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]

D. V. Gribanov, D. S. Malyshev, I. A. Shumilov
2022 arXiv   pre-print
Using this fact, we prove 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]

D. V. Gribanov
2021 arXiv   pre-print
In this note, we show that 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]

D.V. Gribanov, I.A. Shumilov, D.S. Malyshev, P.M. Pardalos
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]

Thang N. Dinh, My T. Thai
2011 arXiv   pre-print
Our experiments show that 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

Thang N. Dinh, My T. Thai
2014 Internet Mathematics
By exploiting 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

Junhao Gan, David F. Gleich, Nate Veldt, Anthony Wirth, Xin Zhang, Daniel Kráľ, Javier Esparza
2020 International Symposium on Mathematical Foundations of Computer Science
This 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

Jörg Hendrik Kappes, Markus Speth, Gerhard Reinelt, Christoph Schnörr
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]

Jörg H. Kappes, Bjoern Andres, Fred A. Hamprecht, Christoph Schnörr, Sebastian Nowozin, Dhruv Batra, Sungwoong Kim, Bernhard X. Kausler, Thorben Kröger, Jan Lellmann, Nikos Komodakis, Bogdan Savchynskyy, Carsten Rother
2014 arXiv   pre-print
However, 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]

Frank Baumann, Sebastian Berckey, Christoph Buchheim
2013 Facets of Combinatorial Optimization
We propose two 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]

Tomáš Gavenčiak and Dušan Knop and Martin Koutecký
2018 arXiv   pre-print
Specifically, we consider graphs 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.  (A), tw P (A), td  ...

### Higher-order Segmentation via Multicuts [article]

Joerg Hendrik Kappes and Markus Speth and Gerhard Reinelt and Christoph Schnoerr
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]

Paul Swoboda, Bjoern Andres
2017 arXiv   pre-print
We propose 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]

Reyan Ahmed, Greg Bodwin, Faryad Darabi Sahneh, Keaton Hamm, Mohammad Javad Latifi Jebelli, Stephen Kobourov, Richard Spence
2020 arXiv   pre-print
As an additional contribution, we offer 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

Alessio Bonfietti, Michele Lombardi, Luca Benini, Michela Milano
2014 Artificial Intelligence
Our approach has been tested 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

Alberto Costa
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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