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Given an initial placement of a set of rectangles in the plane, we consider the problem of finding a disjoint placement of the rectangles that minimizes the area of the bounding box and preserves the orthogonal order i.e. maintains the sorted ordering of the rectangle centers along both $x$-axis and $y$-axis with respect to the initial placement. This problem is known as Layout Adjustment for Disjoint Rectangles(LADR). It was known that LADR is $mathbb{NP}$-hard, but only heuristics were known for it. We show that a certain decision version of LADR is $mathbb{APX}$-hard, and give a constant factor approximation for LADR.
We consider variants of the following multi-covering problem with disks. We are given two point sets $Y$ (servers) and $X$ (clients) in the plane, a coverage function $kappa :X rightarrow mathcal{N}$, and a constant $alpha geq 1$. Centered at each se
The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time $O(log n)$-approximation algorithm for placing as few guards as possible at vertices of a simple $n$-gon $P$, such that every
MAX CLIQUE problem (MCP) is an NPO problem, which asks to find the largest complete sub-graph in a graph $G, G = (V, E)$ (directed or undirected). MCP is well known to be $NP-Hard$ to approximate in polynomial time with an approximation ratio of $1 +
In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $Asubseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope $X$ on $k$ v
We show that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials. A two-dimensional packing problem is defined by the type of pieces, containers, and motions that are allowed.