scholarly journals Fault-tolerant metric dimension problem: A new integer linear programming formulation and exact formula for grid graphs

2018 ◽  
Vol 42 (4) ◽  
pp. 495-503 ◽  
Author(s):  
Ana Simić ◽  
Milena Bogdanović ◽  
Zoran Maksimović ◽  
Jelisavka Milošević
Keyword(s):  
Linear Programming ◽  
Integer Linear Programming ◽  
Fault Tolerant ◽  
Exact Formula ◽  
Metric Dimension ◽  
Programming Formulation ◽  
Grid Graphs ◽  
Linear Programming Formulation ◽  
Integer Linear Programming Formulation

scholarly journals Tournaments as Feedback Arc Sets

10.37236/1214 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Garth Isaak
Keyword(s):  
Linear Programming ◽  
Lower Bounds ◽  
Integer Linear Programming ◽  
Minimum Size ◽  
Programming Formulation ◽  
Binary Expansion ◽  
Feedback Arc Set ◽  
Disjoint Cycles ◽  
Linear Programming Formulation ◽  
Integer Linear Programming Formulation

We examine the size $s(n)$ of a smallest tournament having the arcs of an acyclic tournament on $n$ vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds $s(n) \geq 3n - 2 - \lfloor \log_2 n \rfloor$ or $s(n) \geq 3n - 1 - \lfloor \log_2 n \rfloor$, depending on the binary expansion of $n$. When $n = 2^k - 2^t$ we show that the bounds are tight with $s(n) = 3n - 2 - \lfloor \log_2 n \rfloor$. One view of this problem is that if the 'teams' in a tournament are ranked to minimize inconsistencies there is some tournament with $s(n)$ 'teams' in which $n$ are ranked wrong. We will also pose some questions about conditions on feedback arc sets, motivated by our proofs, which ensure equality between the maximum number of arc disjoint cycles and the minimum size of a feedback arc set in a tournament.

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