scholarly journals Uniquely $K_r$-Saturated Graphs

10.37236/2162 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Stephen G. Hartke ◽  
Derrick Stolee
Keyword(s):  
Cayley Graphs ◽  
Linear Programs ◽  
Number Of Generators ◽  
Integer Linear Programs ◽  
Saturated Graphs ◽  
Orbital Branching

A graph $G$ is uniquely $K_r$-saturated if it contains no clique with $r$ vertices and if for all edges $e$ in the complement, $G+e$ has a unique clique with $r$ vertices. Previously, few examples of uniquely $K_r$-saturated graphs were known, and little was known about their properties. We search for these graphs by adapting orbital branching, a technique originally developed for symmetric integer linear programs.  We find several new uniquely $K_r$-saturated graphs with $4 \leq r \leq 7$, as well as two new infinite families based on Cayley graphs for $\mathbb{Z}_n$ with a small number of generators.

scholarly journals Empowering the configuration-IP: new PTAS results for scheduling with setup times

2021 ◽  
Author(s):  
Klaus Jansen ◽  
Kim-Manuel Klein ◽  
Marten Maack ◽  
Malin Rau
Keyword(s):  
Approximation Scheme ◽  
Constant Factor ◽  
Setup Times ◽  
Linear Programs ◽  
Scheduling Problems ◽  
Design Of Algorithms ◽  
Packing Problems ◽  
Integer Linear Programs ◽  
One Machine ◽  
Target Locations

AbstractInteger linear programs of configurations, or configuration IPs, are a classical tool in the design of algorithms for scheduling and packing problems where a set of items has to be placed in multiple target locations. Herein, a configuration describes a possible placement on one of the target locations, and the IP is used to choose suitable configurations covering the items. We give an augmented IP formulation, which we call the module configuration IP. It can be described within the framework of n-fold integer programming and, therefore, be solved efficiently. As an application, we consider scheduling problems with setup times in which a set of jobs has to be scheduled on a set of identical machines with the objective of minimizing the makespan. For instance, we investigate the case that jobs can be split and scheduled on multiple machines. However, before a part of a job can be processed, an uninterrupted setup depending on the job has to be paid. For both of the variants that jobs can be executed in parallel or not, we obtain an efficient polynomial time approximation scheme (EPTAS) of running time $$f(1/\varepsilon )\cdot \mathrm {poly}(|I|)$$ f ( 1 / ε ) · poly ( | I | ) . Previously, only constant factor approximations of 5/3 and $$4/3 + \varepsilon $$ 4 / 3 + ε , respectively, were known. Furthermore, we present an EPTAS for a problem where classes of (non-splittable) jobs are given, and a setup has to be paid for each class of jobs being executed on one machine.

scholarly journals Distributionally robust mixed integer linear programs: Persistency models with applications

2014 ◽  
Vol 233 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Xiaobo Li ◽  
Karthik Natarajan ◽  
Chung-Piaw Teo ◽  
Zhichao Zheng
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs ◽  
Distributionally Robust

An optimality cut for mixed integer linear programs

1999 ◽  
Vol 119 (3) ◽  
pp. 671-677
Author(s):  
Gilbert Laporte ◽  
Frédéric Semet
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs

scholarly journals Analyzing Infeasible Mixed-Integer and Integer Linear Programs

1999 ◽  
Vol 11 (1) ◽  
pp. 63-77 ◽  
Author(s):  
Olivier Guieu ◽  
John W. Chinneck
Keyword(s):  
Mixed Integer ◽  
Linear Programs ◽  
Integer Linear Programs

Three algorithms for bicriteria integer linear programs

OR Spectrum ◽  
1994 ◽  
Vol 16 (4) ◽  
pp. 267-276 ◽  
Author(s):  
P. Neumayer ◽  
D. Schweigert
Keyword(s):  
Linear Programs ◽  
Integer Linear Programs
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