Sangho Shim

Total embedding distributions for bouquets of circles

Crosscap-number distributions, the distribution of graph embeddings into nonorientable surfaces, have been known for only a few cases. Chen et al. (Discrete Math. 128 (1994) 73) calculated the crosscap-number distribution of necklaces, closed-end ladders and cobblestone paths. In this paper, we compute the total genus polynomials and the total embedding polynomials of bouquets of circles with an aid of edge-attaching surgery technique. It extends their genus distributions computed by Gross et al. (J. Combin. Theory (B) 47 (1989) 292). The same work is also done for dipoles.

Counterexamples to the uniform shortest path routing conjecture for vertex-transitive graphs

In this note we disprove the uniform shortest path routing conjecture for vertex-transitive graphs by constructing an infinite family of counterexamples.

A few strong knapsack facets

We perform a shooting experiment for the knapsack facets and observe that 1∕k-facets are strong for small k; in particular, k dividing 6 or 8. We also observe spikes of the size of 1∕k-facets when k = n or when k + 1 divides n + 1. We discuss the strength of the 1∕n-facets introduced by Araoz et al. (Math Program 96:377–408, 2003) and the knapsack facets given by Gomory’s homomorphic lifting.A general integer knapsack problem is a knapsack subproblem where a portion, often a significant majority, of the variables are missing from the master knapsack problem. The number of projections of 1∕k-facets on a knapsack subproblem of l variables is \(O(l^{\lceil k/2\rceil })\), note that this is independent of the size of the master problem. Since 1∕k-facets are strong for small k, we define the 1∕k-inequalities which include the 1∕d-facets with d dividing k and fix k to be a small constant such as k = 6 or k = 8. We develop an efficient way of enumerating violated valid 1∕k-inequalities. For each violated 1∕k-inequality, we determine its validity by solving a small integer programming problem, the size of which depends only on k.

Binary group facets with complete support and non-binary coefficients

We identify binary group facets with complete support and non-binary coefficients. These inequalities can be used to obtain new facets for larger problems using Gomorys homomorphic lifting.

An extended formulation of the convex recoloring problem on a tree

AbstractWe 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.

Primal-dual simplex method for shooting

Abstract Gomory [R.E. Gomory, Some polyhedra related to combinatorial problems, Linear Algebra and its Applications 2 (1969), 451–558] solved the cyclic group problem by a dynamic programming algorithm. We discuss its complexity and introduce a (fractional) cutting plane algorithm as an alternative algorithm. Each cutting plane is generated by solving a shooting linear programming problem. We implement primal-dual simplex method to solve the shooting linear programming problem. A computational result is given on a Wong-Coppersmith digraph.

Separation Algorithm for Tree Partitioning Inequalities

Abstract We consider the tree partition problem to partition the node set of a tree into subsets where the induced subgraph by each subset is connected and the total weight of nodes in a subset cannot exceed the capacity of the subset. We identify exponentially many valid inequalities for an integer programming formulation of the problem and develop a linear time separation algorithm for the valid inequalities.

Column Generation Approach to the Convex Recoloring Problem on a Tree

The convex recoloring (CR) problem is to recolor the nodes of a colored graph at minimum number of color changes such that each color induces a connected subgraph. We adjust to the convex recoloring problem the column generation framework developed by Johnson et al. (Math Program 62:133–151, 1993). For the convex recoloring problem on a tree, the subproblem to generate columns can be solved in polynomial time by a dynamic programming algorithm. The column generation framework solves the convex recoloring problem on a tree with a large number of colors extremely fast.