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The maximum k-colorable subgraph problem and orbitopes

Tim Januschowski, Marc E. Pfetsch
2011 Discrete Optimization  
Given an undirected node-weighted graph and a positive integer k, the maximum kcolorable subgraph problem is to select a k-colorable induced subgraph of largest weight.  ...  We study conditions under which facet-defining inequalities for the polytope associated with the maximum k-colorable subgraph problem and the orbitope remain facet-defining for the combined polytope or  ...  Research Council for Science, Technology and Engineering.  ... 
doi:10.1016/j.disopt.2011.04.002 fatcat:soakhwah2bhddf3ktkwnyrg7bq

Branch-Cut-and-Propagate for the Maximum k-Colorable Subgraph Problem with Symmetry [chapter]

Tim Januschowski, Marc E. Pfetsch
2011 Lecture Notes in Computer Science  
Given an undirected graph and a positive integer k, the maximum k-colorable subgraph problem consists of selecting a k-colorable induced subgraph of maximum cardinality.  ...  We investigate the integration of a branch-and-cut algorithm for solving the maximum k-colorable subgraph problem with constraint propagation techniques to handle the symmetry arising from the graph.  ...  Thus, the objective function represents the cardinality of the selected subgraph. The k-colored subgraph polytope corresponding to the maximum k-colorable subgraph problem is (2) and (3)}.  ... 
doi:10.1007/978-3-642-21311-3_11 fatcat:m5jmp46aezbjtj5h6d4xa7654a

Mini-Workshop: Exploiting Symmetry in Optimization

Volker Kaibel, Leo Liberti, Achill Schürmann, Renata Sotirov
2010 Oberwolfach Reports  
The exchange of state-of-the-art knowledge between these areas lead to identification of important directions for future research activities.  ...  The feasible regions of mathematical optimization models quite often exhibit a high degree of symmetry.  ...  The Maximum k-Colorable Subgraph Problem and Symmetry Marc E.  ... 
doi:10.4171/owr/2010/38 fatcat:n2zcqzooonex7d3kcknkfsxpxa

Mixed-integer programming techniques for the connected max-k-cut problem

Christopher Hojny, Imke Joormann, Hendrik Lüthen, Martin Schmidt
2020 Mathematical Programming Computation  
We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected.  ...  For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics  ...  This research has been performed as part of the Energie Campus Nürnberg and has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie  ... 
doi:10.1007/s12532-020-00186-3 fatcat:ispaycmmqbbunjfobn2e7iwzbi

Extended formulations in combinatorial optimization

Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli
2013 Annals of Operations Research  
formulation for orbitopes.  ...  In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set.  ...  An example is the formulation for graph coloring with binary variables x ij assuming value 1 if and only if color j is assigned to node i.  ... 
doi:10.1007/s10479-012-1269-0 fatcat:erkhgmmppfhh5dk4waqr5xtihe

The SCIP Optimization Suite 8.0 [article]

Ksenia Bestuzheva, Mathieu Besançon, Wei-Kun Chen, Antonia Chmiela, Tim Donkiewicz, Jasper van Doornmalen, Leon Eifler, Oliver Gaul, Gerald Gamrath, Ambros Gleixner, Leona Gottwald, Christoph Graczyk (+23 others)
2021 arXiv   pre-print
presolving methods were added to SCIP-SDP, and additional problem classes and major performance improvements are available in SCIP-Jack.  ...  This paper discusses enhancements and extensions contained in version 8.0 of the SCIP Optimization Suite.  ...  Acknowledgements The authors want to thank all previous developers and contributors to the SCIP Optimization Suite and all users that reported bugs and often also helped reproducing and fixing the bugs  ... 
arXiv:2112.08872v1 fatcat:fqkdhu4bhngcblzaaaasqneplm

Extended formulations in combinatorial optimization

Michele Conforti, Gérard Cornuéjols, Giacomo Zambelli
2010 4OR  
Natural formulations often have a number of constraints that is exponential in the size of the data needed to describe the problem.  ...  This survey is concerned with the size of formulations for combinatorial optimization problems.  ...  An example is the formulation for graph coloring with binary variables x ij assuming value 1 if and only if color j is assigned to node i.  ... 
doi:10.1007/s10288-010-0122-z fatcat:ieyjoo567bgkjkwjlfjkouxycm

On the combinatorics of the 2-class classification problem [article]

Ricardo C. Corrêa and Diego Delle Donne and Javier Marenco
2017 arXiv   pre-print
We are able to find facet-inducing inequalities coming from the stable set polytope, hence showing that this classification problem has exploitable combinatorial properties.  ...  Based on these ideas, Bertsimas and Shioda proposed the classification and regression by integer optimization (CRIO) method in 2007.  ...  Acknowledgment We are very grateful for the invaluable remarks by the anonymous referee, in special for pointing out some mistakes in the original version of Lemma 1 and Proposition 5, as well as bringing  ... 
arXiv:1706.06214v2 fatcat:4bhtwjor3fgqvbgiematv5lk6m

Semidefinite Optimization [chapter]

2014 Computer Vision  
The coloring (or chromatic) number is the smallest integer k for which G admits a k-coloring, it is denoted as χpGq. Again it is an N P -complete problem to decide whether a graph is k-colorable.  ...  polynomial time maximum stable sets and minimum colorings in perfect graphs.  ...  The sum is finite and the maximum value of |α| " ř n i"1 α i for which p α ‰ 0 is the degree of p.  ... 
doi:10.1007/978-0-387-31439-6_100253 fatcat:ooxdgxvxnrfubhmopagjvtpnim

An Asymptotic PTAS for Batch Scheduling with Nonidentical Job Sizes to Minimize Makespan [chapter]

Yuzhong Zhang, Zhigang Cao
Lecture Notes in Computer Science  
We thank all the people who made this meeting possible: the authors for submitting papers, the Program Committee members and external referees (listed in the proceedings) for their excellent work, and  ...  Finally, we thank Xi'an Jiaotong University and NSF of China for the support and local organizers and colleagues for their assistance.  ...  The running time of our algorithm is O((k u + k l )|G| + 1.1892 ku+k l ), compared to the previous best algorithm for the problem of running time O((k u + k l )|G| + 1.26 ku+k l ).  ... 
doi:10.1007/978-3-540-73556-4_7 fatcat:qundndqypvgupglzb3iv4yhe7m

TWO APPLICATIONS OF COMBINATORIAL OPTIMIZATION Two applications of combinatorial optimization

Matthew Oster, Matthew Oster
We model the problem as a symmetric clustering problem, or a variant of minimum k-partition we call capacitated k-partition.  ...  We model the problem as a resource allocation problem and prove that it is N P-hard to solve.  ...  For instance, Guruswami and Sinop [50] proved the problem to be N P-hard even when the input graph G is known to be k-colorable-a problem sometimes called maximum k-colorable subgraph.  ... 

An Algorithmic Walk from Static to Dynamic Graph Clustering

Robert Görke
Distractions and Curiosities  ...  In all three drawings, k-cores are used for decomposition and color indicates a node's degree.  ...  .: 3 Add to Problem 5: with maximum quality. quality 4 Add to Problem 5: with quality no less than α times the value a re-clustering from scratch would yield and otherwise with minimum running time. 5  ... 
doi:10.5445/ir/1000018288 fatcat:xh4f3cmwozhs3pfhfb6ekgnjhu

Sizes of linear descriptions in combinatorial optimization [article]

Stefan Weltge, Universitäts- Und Landesbibliothek Sachsen-Anhalt, Martin-Luther Universität, Volker Kaibel
We consider the applicability and limitations of several known techniques with respect to relevant open questions, such as questions referring to the extension complexities of spanning-tree polytopes and  ...  For instance, we show that every relaxation of the set of characteristic vectors of Hamilton-cycles of the complete graph (which canonically describe the feasible objects of the traveling-salesman problem  ...  Acknowledgements I am very thankful for the support that I received from the German Research Foundation (DFG) and the German Academic Exchange Service (DAAD) who funded part of this research 1 and provided  ... 
doi:10.25673/4350 fatcat:codojo2hqnewjenun4m6bqcoam

Extended formulations for combinatorial polytopes [article]

Kanstantsin Pashkovich, Universitäts- Und Landesbibliothek Sachsen-Anhalt, Martin-Luther Universität, Volker Kaible
The second part deals with extended formulations which use special structures of graphs involved in combinatorial problems.  ...  In the first part, some known techniques for constructions of extended formulations are reviewed and the new framework of polyhedral relations (see Kaibel and Pashkovich [2011] ) is presented.  ...  It is easy to verify that the function k2 m−k achieves maximum 2 m−1 at k = 1 or k = 2.  ... 
doi:10.25673/3850 fatcat:d7xqqoifvjbihbgryxusubvu6q