On the shortest separating cycle

Abstract

According to a result of Arkin et al. (2016), given n point pairs in the plane, there exists a simple polygonal cycle that separates the two points in each pair to different sides; moreover, a $O(\sqrt{n})$-factor approximation with respect to the minimum length can be computed in polynomial time. Here the following results are obtained:

(I) We extend the problem to geometric hypergraphs and obtain the following characterization of feasibility. Given a geometric hypergraph on points in the plane with hyperedges of size at least 2, there exists a simple polygonal cycle that separates each hyperedge if and only if the hypergraph is 2-colorable.

(II) We extend the $O(\sqrt{n})$-factor approximation in the length measure as follows: Given a geometric graph $G=(V,E)$, a separating cycle (if it exists) can be computed in $O(m+n\log n)$ time, where $|V|=n$, $|E|=m$. Moreover, a $O(\sqrt{n})$-approximation of the shortest separating cycle can be found in polynomial time. Given a geometric graph $G=(V,E)$ in ${\mathbb{R}}^3$, a separating polyhedron (if it exists) can be found in $O(m+n\log n)$ time, where $|V|=n$, $|E|=m$. Moreover, a $O(n^{2/3})$-approximation of a separating polyhedron of minimum perimeter can be found in polynomial time.

(III) Given a set of n point pairs in convex position in the plane, we show that a $(1+\epsilon )$-approximation of a shortest separating cycle can be computed in time $n^{O({\epsilon }^{-1/2})}$. In this regard, we prove a lemma on convex polygon approximation that is of independent interest.

Introduction

Given a set of n pairs of points in the plane with no common elements, $\{(p_i,q_i)|i=1,\dots ,n\}$, a shortest separating cycle is a plane cycle (a closed curve, a.k.a. tour) of minimum length that contains inside exactly one point from each of the n pairs. The problem Shortest Separating Cycle is that of finding such a cycle, given the input pairs. It was introduced by Arkin et al. [3] motivated by applications in data storage and retrieval in distributed sensor networks. The authors gave a $O(\sqrt{n})$-factor approximation for the general case and better approximations for some special cases. On the other hand, using a reduction from Vertex Cover, they showed that the problem is hard to approximate for a factor of 1.36 unless $\mathrm{P}=\mathrm{NP}$, and is hard to approximate for a factor of 2 assuming the Unique Games Conjecture; see, e.g., [24, Ch. 16] for technical background.

The assumption that no point appears more than once, i.e., $|\{p_1,\dots ,p_n\}\cup \{q_1,\dots ,q_n\}|=2n$, is sometimes necessary for the existence of a separating cycle; i.e., there are instances of sets of pairs with common elements and no separating cycle; see for instance Fig. 2 (the edges in these graphs represent pairs of input points). For convenience, points on the boundary of the cycle are considered inside; it is easy to see that requiring points to lie strictly in the interior or also on the boundary are equivalent variants in regards to the existence of a separating cycle. Moreover, the equivalence is almost preserved in the length measure: given any positive $\epsilon >0$, and a separating cycle C for n pairs, enclosing $P=\{p_1,\dots ,p_n\}$ (after relabeling each pair, if needed), with some of the points of P on its boundary, a separating cycle of length at most $(1+\epsilon )\mathrm{len}(C)$ can be constructed, having all points of P in its interior.

In this paper we study the extension of the concept of separating cycle to arbitrary graphs and hypergraphs, and to higher dimensions; in the original version introduced by Arkin et al. [3], the input graph is a matching, i.e., it consists of n edges with no common endpoints; see Fig. 1 for an example. Two instances with 8 and respectively 3 point pairs that do not admit separating cycles are illustrated in Fig. 2; the common reason is that both graphs contain odd cycles and odd cycles do not admit separating cycles.

We observe that for arbitrary input graphs, one cannot use the algorithm from [3]. That algorithm (in [3, Subsec. 3.5]) first computes a minimum-size square Q containing at least one point from each pair, and then computes a constant-factor approximation of a shortest cycle (tour) of the points contained in Q, in the form of a simple polygon. In the end, this tour is refined to a separating cycle of the given set of point pairs with only a small increase in length. Here we note that there exist instances, such as that in Fig. 2 (right), for which there is no separating cycle confined to Q; moreover, the length of a shortest separating cycle can be arbitrarily larger than any function of $\mathrm{diam}(Q)$ and n, and so a new approach is needed for the general version with arbitrary input graphs, or its extension to hypergraphs; i.e., the current $O(\sqrt{n})$-factor approximation does not carry through to these settings.

We first show that a planar geometric graph $G=(V,E)$ admits a separating cycle (for all its edge-pairs) if and only if it is bipartite. This result can be extended to hypergraphs in ${\mathbb{R}}^d$. Given a geometric hypergraph on points in ${\mathbb{R}}^d$ with no singleton edges,¹ there exists a simple polyhedron that separates each hyperedge if and only if the hypergraph is 2-colorable.

Definitions and notations  A hypergraph is a pair $H=(V,E)$, where V is finite set of vertices, and E is a family of subsets of V, called edges. H is said to be 2-colorable if there is a 2-coloring of V such that no edge is monochromatic; see, e.g., [2, Ch. 1.3].

For a polygonal cycle C, let C̊ and $\bar{C}$ denote the interior and exterior of C, respectively; and ∂C denote its boundary. Consider a geometric hypergraph $H=(V,E)$ on points in the plane with no singleton edges. A polygonal cycle C is said to be a separating cycle for H if (i) C is simple (i.e., with no self-intersections); and (ii) each edge of H has points inside C (in its interior or on its boundary) and points in the exterior of C; that is, for each edge $A\in E$, both $A\cap (\mathrm{C̊}\cup \partial C)$ and $A\cap \bar{C}$ are nonempty.

A simple polygonal cycle is said to have zero area, if $\mathrm{Area}(C)\le \epsilon$, for a sufficiently small given $\epsilon >0$. Similarly, a polyhedron P is said to have zero volume, if $\mathrm{Vol}(P)\le \epsilon$, for a sufficiently small given $\epsilon >0$.

For $r>0$, let $B(r)$ denote the ball (i.e., disk in the plane) of radius r. For two convex bodies, A and B let $A+B$ denote their Minkowski sum, namely $A+B=\{a+b|a\in A,b\in B\}$.

Preliminaries and related work  Let S be a finite set of points in the plane. According to an old result of Few [11], the length of a minimum spanning path (resp., minimum spanning tree) of any n points in the unit square is at most $\sqrt{2n}+7/4$ (resp., $\sqrt{n}+7/4$). Both upper bounds are constructive; for example, the construction of a short spanning path works as follows. Lay out about $\sqrt{n}$ equidistant horizontal lines, and then visit the points layer by layer, with the path alternating directions along the horizontal strips. In particular, the length of the minimum spanning tree of any n points in the unit square is bounded from above by the same expression. An upper bound with a slightly better multiplicative constant for a path was derived by Karloff [19]. Fejes Tóth [10] had observed earlier that for n points of a regular hexagonal lattice in the unit square, the length of the minimum spanning path is asymptotically equal to ${(4/3)}^{1/4}\sqrt{n}$, where ${(4/3)}^{1/4}=1.0745\dots$. As such, the maximum length of the minimum spanning tree of any n points in the unit square is $\mathrm{\Theta }(\sqrt{n})$, for a small constant (close to 1). The $O(\sqrt{n})$ upper bound also holds for points in a convex polygon of diameter $O(1)$, in particular for n points in a rectangle of diameter $O(1)$. In every dimension $d\ge 3$, Few showed that the maximum length of a shortest path (or tree) through n points in the unit cube is $\mathrm{\Theta }(n^{1-1/d})$; the $O(n^{1-1/d})$ upper bound is again constructive and extends to rectangular boxes of diameter $O(1)$.

The topic of “separation” has appeared in multiple interpretations; here we only give a few examples: [1], [6], [7], [13], [15], [16], [17]. Some results on watchman tours relying on Few's bounds can be found in [8]; others can be found in [4]. For instance, in the problem of finding a separating cycle for a given set of segment pairs, that we study here, it is clear that the edges of the cycle must hit all of the given segments. As such, this problem is related to the classic problem of hitting a set of segments by straight lines [16]. In a broader context, coloring of geometric hypergraphs has been studied, e.g., in [23].