DSpace at Robert Morris University
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The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.2020-03-29T13:04:38ZA steepest edge rule for a column generation approach to the convex recoloring problem
http://hdl.handle.net/11347/352
A steepest edge rule for a column generation approach to the convex recoloring problem
Erdem, Ergin; Gahler, Kenneth; Kim, Eunseok; Shim, Sang Ho
The convex recoloring problem is a clustering problem to partition nodes of a network into connected subnetworks. We develop a hybrid rule combining the Dantzig’s Rule and the Steepest Edge Rule to produce columns which enter into the basis of the master problem in the column generation framework introduced by Chopra et al. [2]. The hybrid
rule leads to only a small number of iterations and makes it possible to perform the column generation approach in an undergraduate class using Microsoft Excel. We perform a large scale computational experiment and show that the hybrid rule is effective.
2018-01-01T00:00:00ZOn the Problem of Determining which (n, k)-Star Graphs are Cayley Graphs
http://hdl.handle.net/11347/351
On the Problem of Determining which (n, k)-Star Graphs are Cayley Graphs
Cheng, Eddie; Li, Li; Liptak, Laszlo; Shim, Sang Ho; Steffy, Daniel E.
In this paper we work to classify which of the (n, k)-star graphs, denoted by Sn,k , are Cayley graphs. Although the complete classification is left open, we derive infinite and non-trivial classes of both Cayley and non-Cayley graphs. We give a complete classification of the case when k=2 , showing that Sn,2 is Cayley if and only if n is a prime power. We also give a sufficient condition for Sn,3 to be Cayley and study other structural properties, such as demonstrating that Sn,k always has a uniform shortest path routing.
2017-01-01T00:00:00ZThe worst case analysis of strong knapsack facets
http://hdl.handle.net/11347/350
The worst case analysis of strong knapsack facets
Shim, Sang Ho; Chopra, Sunil; Cao, Wenwei
In this paper we identify strong facet defining inequalities for the master knapsack polytope. Our computational experiments for small master knapsack problems show that 1 / k-facets for small values of k ( k≤4 ) are strong facets for the knapsack polytope. We show that this finding is robust by proving that the removal of these facets from the master knapsack polytope significantly weakens the resulting relaxation in the worst case. We show that the 1 / k-facets for k=1 are the strongest in that their removal from the master knapsack polytope weakens the relaxation by a factor of 3 / 2 in the worst case. We then show that the 1 / k-facets with k=3 or 4 are the next strongest. We also show that the strength of the 1 / k-facets weakens as k grows and that the 1 / k-facets with k even are stronger than the 1 / k-facets with k odd.
2017-01-01T00:00:00ZAn extended formulation of the convex recoloring problem on a tree
http://hdl.handle.net/11347/349
An extended formulation of the convex recoloring problem on a tree
Chopra, Sunil; Filipecki, Bartosz; Lee, Kangbok; Ryu, Minseok; Shim, Sang Ho; Vyve, Mathieu Van
We introduce a strong extended formulation of the convex recoloring problem on a tree, which has an application in analyzing phylogenetic trees. The extended formulation has only a polynomial number of constraints, but dominates the conventional formulation and the exponentially many valid inequalities introduced by Campêlo et al. (Math Progr 156:303–330, 2016). We show that all valid inequalities introduced by Campêlo et al. can be derived from the extended formulation. We also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree. The solution time using the extended formulation is much smaller than that with the conventional formulation. Moreover the extended formulation solves all the problem instances attempted in Campêlo et al. (2016) and larger sized instances at the root node of the branch-and-bound tree without branching.
2017-01-01T00:00:00Z