The optimization of container ship and depot operations embeds the k-shift problem, in which containers must be stowed in stacks such that at most k containers must be removed in order to reach containers below them. We first solve an open problem introduced by Avriel et al. (2000) by showing that changing from uncapacitated to capacitated stacks reduces the complexity of this problem from NP-c...

In the d-dimensional Generalized Minimum Manhattan Network (d-GMMN) problem one is interested in finding a minimum cost rectilinear network N connecting a given set R of n pairs of points in R such that each pair is connected in N via a shortest Manhattan path. The problem is known to be NP-complete and does not admit a FPTAS, the best known upper bound is an O(log n)-approximation for d > 2 an...

We solve for the SO(3)-invariant Kähler–Einstein metric on P2 with cone singularities along a smooth quadric curve using a numerical approach. The numerical results show the sharp range of angles ((π/2, 2π ]) for the solvability of equations, and the correct limit metric space (P(1, 1, 4)). These results exactly match our theoretical conclusion. We also point out the cause of incomplete classif...

We investigate a new optimization problem involving minimizing the Ratio of two Submodular (RS) functions. We argue that this problem occurs naturally in several real world applications. We then show the connection between this problem and several related problems including minimizing the difference between submodular functions (Iyer & Bilmes, 2012b; Narasimhan & Bilmes, 2005), and to submodula...

We study several variations of line segment covering problem with axis-parallel unit squares in IR. A set S of n line segments is given. The objective is to find the minimum number of axis-parallel unit squares which cover at least one end-point of each segment. The variations depend on the orientation and length of the input segments. We prove some of these problems to be NP-complete, and give...

The known facts about solvability of equations over groups are considered from a more general point of view. A generalized version of the theorem about solvability of unimodular equations over torsion-free groups is proved. In a special case, this generalized version become a multivariable variant of this theorem. For unimodular equations over torsion free groups, we prove an analogue of Magnus...

We consider global solvability of complex vector fields on noncompact manifolds. The case of real vector fields had been considered by Malgrange, and Hörmander studied the complex case, assuming that the real and imaginary parts are everywhere linearly independent. 1. We consider a smooth complex vector field L = X + iY without zeros on a smooth, paracompact, noncompact manifold M. The real vec...

By definition, a P2-graph Γ is an undirected graph in which every vertex is contained in a path of length two. For such a graph, pc(Γ) denotes the minimum number of paths of length two that cover all n vertices of Γ. We prove that ⌈n/3⌉ ≤ pc(Γ) ≤ ⌊n/2⌋ and show that these upper and lower bounds are tight. Furthermore we show that every connected P2-graph Γ contains a spanning tree T such that p...

Solving equations in algebraic structures is a fundamental task in mathematics. Here we tackle this problem for free partially commutative monoids with involution and for graph groups, which are free groups with a partial commutation relation between generators. Basic algebraic structures involving partial commutations are free partially commutative monoids (also known as trace monoids). They w...

assigning facilities to locations is one of the important problems, which significantly is influence in transportation cost reduction. in this study, we solve quadratic assignment problem (qap), using a meta-heuristic algorithm with deterministic tasks and equality in facilities and location number. it should be noted that any facility must be assign to only one location. in this paper, first o...