arXiv subjects

Irina Kostitsyna

Publications and source records attributed to Irina Kostitsyna.

44 records · Page 3Linked to original sources

arXiv2016

Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.

cs.CC↗

arXiv2016

Mapping polygons to the grid with small Hausdorff and Fréchet distance

We show how to represent a simple polygon $P$ by a grid (pixel-based) polygon $Q$ that is simple and whose Hausdorff or Fréchet distance to $P$ is small. For any simple polygon $P$, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.

cs.CG↗

arXiv2016

Computing Nonsimple Polygons of Minimum Perimeter

We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation. When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5% of the optimum.

cs.CG↗

arXiv2016

On the complexity of minimum-link path problems

We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several variants of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [1] despite a large body of work on the topic. We also resolve the open problem from [2] mentioned in the handbook [3] (see Chapter 27.5, Open problem 3) and The Open Problems Project [4] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

cs.CG↗

arXiv2015

On Minimizing Crossings in Storyline Visualizations

In a storyline visualization, we visualize a collection of interacting characters (e.g., in a movie, play, etc.) by $x$-monotone curves that converge for each interaction, and diverge otherwise. Given a storyline with $n$ characters, we show tight lower and upper bounds on the number of crossings required in any storyline visualization for a restricted case. In particular, we show that if (1) each meeting consists of exactly two characters and (2) the meetings can be modeled as a tree, then we can always find a storyline visualization with $O(n\log n)$ crossings. Furthermore, we show that there exist storylines in this restricted case that require $Ω(n\log n)$ crossings. Lastly, we show that, in the general case, minimizing the number of crossings in a storyline visualization is fixed-parameter tractable, when parameterized on the number of characters $k$. Our algorithm runs in time $O(k!^2k\log k + k!^2m)$, where $m$ is the number of meetings.

cs.DS↗

arXiv2014

Region-based approximation of probability distributions (for visibility between imprecise points among obstacles)

Let $p$ and $q$ be two imprecise points, given as probability density functions on $\mathbb R^2$, and let $\cal R$ be a set of $n$ line segments (obstacles) in $\mathbb R^2$. We study the problem of approximating the probability that $p$ and $q$ can see each other; that is, that the segment connecting $p$ and $q$ does not cross any segment of $\cal R$. To solve this problem, we approximate each density function by a weighted set of polygons; a novel approach to dealing with probability density functions in computational geometry.

cs.CG↗

arXiv2013

Local Redesigning of Airspace Sectors

In this paper we study the Airspace Sectorization Problem (ASP) where the goal is to find an optimal partition (sectorization) of the airspace into a certain number of sectors, each managed by an air traffic controller. The objective of the ASP is to find a "well-balanced" sectorization that distributes the workload evenly among the controllers. We formulate the ASP as a partitioning problem of a set of moving points in a polygonal domain. In addition to the requirement of balancing the workload, we introduce restrictions on the geometry of the sectorization which come from the Air Traffic Management aspects. We investigate several versions of the problem that arise from different definitions of the notion of the workload and various choices of geometric restrictions on the sectorization. We conclude that most of the formulations of the problem, except maybe in some trivial cases, are NP-hard. Finally, we propose a Local Redesigning Method (LRM), a heuristic algorithm that rebalances a given sectorization by adjusting the boundaries of the sectors. We evaluate LRM experimentally on synthetically generated scenarios as well as on the real historical traffic data. We demonstrate that the sectorizations produced by our method are superior in comparison with the current sectorizations of the US airspace.

cs.CG↗

arXiv2010

Simple Wriggling is Hard unless You Are a Fat Hippo

We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length/width ratio is small, the shortest path can be computed in polynomial time.

cs.CG↗