Shifting coresets: obtaining linear-time approximations for unit disk graphs and other geometric intersection graphs

Shifting coresets: obtaining linear-time approximations for
unit disk graphs and other geometric intersection graphs

Guilherme D. da Fonseca
IUT and LIMOS
Université d’Auvergne
fonseca@isima.fr
   Vinícius Gusmão Pereira de Sá
Instituto de Matemática
Universidade Federal do Rio de Janeiro
vigusmao@dcc.ufrj.br
   Celina Miraglia Herrera de Figueiredo
COPPE, Universidade Federal do Rio de Janeiro
celina@cos.ufrj.br
To appear at International Journal of Computational Geometry and Applications
Abstract

Numerous approximation algorithms for problems on unit disk graphs have been proposed in the literature, exhibiting a sharp trade-off between running times and approximation ratios. We introduce a variation of the known shifting strategy that allows us to obtain linear-time constant-factor approximation algorithms for such problems. To illustrate the applicability of the proposed variation, we obtain results for three well-known optimization problems. Among such results, the proposed method yields linear-time -approximations for the maximum-weight independent set and the minimum dominating set of unit disk graphs, thus bringing significant performance improvements when compared to previous algorithms that achieve the same approximation ratios. Finally, we use axis-aligned rectangles to illustrate that the same method may be used to derive linear-time approximations for problems on other geometric intersection graph classes.

1 Introduction

Linear- and near-linear-time approximation algorithms constitute an active topic of research, even for problems that can be solved exactly in polynomial time, such as maximum flow and maximum matching [4, 14, 19, 23, 28]. This paper employs a variation of the shifting strategy introduced by Hochbaum and Maass three decades ago [21]. While the shifting strategy usually leads to high running times, our variation, which we call the shifting coresets method, allows us to obtain linear-time constant-approximation algorithms for problems on unit disk graphs. Three algorithms for well-studied optimization problems on unit disk graphs are presented to illustrate the method. A fourth algorithm, for maximum-weight independent sets of axis-aligned rectangles in the plane, is also given, showing that the method applies to other geometric intersection graph classes as well.

Geometric intersection graphs are graphs whose vertices correspond to geometric objects and whose edges correspond to pairs of intersecting objects. Several classes of geometric intersection graphs are defined by restricting the shape of the objects: disks, unit disks, squares, rectangles, etc. Approximation algorithms for such classes are either graph-based, when they receive as input solely the adjacency representation of the graph, or geometric, when the input consists of a geometric description of the objects. Note that when the goal is to design -time algorithms, the geometric representation is required, since the number of edges in a geometric intersection graph can be as high as .

Polynomial-time approximation schemes (PTASs) have been developed for several optimization problems on multiple classes of intersection graphs. Among such problems, we focus on the following three in the present paper:

  • Maximum-weight independent set (WIS): Given a graph with weighted vertices, find the maximum-weight subset of mutually non-adjacent vertices.

  • Minimum dominating set (DS): Given a graph, find the minimum-cardinality vertex subset such that every vertex not in is adjacent to some vertex in .

  • Minimum vertex cover (VC): Given a graph, find the minimum-cardinality vertex subset such that every edge is incident on some vertex in .

The shifting strategy has given rise to geometric PTASs for several problems on geometric intersection graphs [21, 5, 13, 22]. A set of geometric objects has constant diameter if the Euclidean distance between any two points contained inside the objects is upper-bounded by a constant. Essentially, the shifting strategy reduces the original problem to a set of subproblems of constant diameter. Such a reduction takes time and yields a -approximation to the original problem once the solutions to the subproblems are known. Each subproblem is then solved exactly by exploiting the fact that it has constant diameter. For example, it is often possible to show by packing arguments that an input instance whose diameter is admits a solution with vertices, so that exhaustive enumeration can find the optimal solution in roughly time. As a consequence, the running times of such PTASs are high-degree polynomials. Other PTASs that are not based on the shifting strategy also exist, but their complexities are usually even higher [27].

Among countless classes of geometric intersection graphs, unit disk graphs are arguably the most studied one, owing largely to their applicability in wireless networks [27, 24]. A unit disk graph (UDG) is the intersection graph of unit disks in the plane, and its vertices are usually represented by the coordinates of the disk centers. With respect to this representation, two vertices are adjacent if the corresponding points (the disk centers) are within Euclidean distance at most from one another. Next, we review the state of the art of the three aforementioned optimization problems on the class of UDGs.

The minimum dominating set problem (DS) on UDGs admits some PTASs, the fastest of which is geometric and provides a -approximation in time [22, 27]. For the sake of comparison with the linear-time -approximation algorithm introduced in Section 4, such a PTAS produces a -approximation in roughly time. The high running times of the existing PTASs have therefore motivated the study of faster constant-factor approximation algorithms. Examples of graph-based algorithms include a -approximation that runs in time and a -approximation that runs in time [15]. Among the geometric algorithms, we cite the original -approximation, which can be implemented in time if the floor function and constant-time hashing are available [24]; a -approximation that uses local improvements and runs in time [15]; a -approximation that uses grids and runs in time [18]; and an improved -approximation that uses hexagonal grids and runs in time [10].

The maximum-weight independent set problem (WIS) also admits PTASs for UDGs, the fastest of which is geometric and attains a -approximation in time [22, 27, 26]. Using a slab decomposition and the longest path in a DAG, Matsui [26] showed how to obtain an approximation ratio of in time.111After this article, the running time has been reduced to using advanced data structures [16]. Alternatively, a geometric -approximation can be obtained in time by a greedy approach that considers the vertices in decreasing order of weights, or in time in the graph-based version [24]. In comparison, the algorithm presented in Section 3 obtains a -approximation in linear time. Note that, for the unweighted version, a geometric greedy approach that considers the vertices from left to right can be implemented to give a -approximation in time with floor function and constant-time hashing [24]. The high running time of the existing PTASs also motivated the discovery of an -time -approximation algorithm for the unweighted version [17].

The minimum vertex cover problem (VC) for UDGs is a rare example for which a geometric linear-time approximation scheme is known [25]. Previously, two high-complexity PTASs (one geometric and one graph-based) had been proposed [22, 29].

Another important class of geometric intersection graphs is defined by axis-aligned rectangles in the plane. Maximum-weight independent sets in such graphs are widely studied due to their application in data mining, map labeling, and VLSI design [5, 3, 11, 12, 1]. However, the lack of polynomial constant-factor approximation algorithms has motivated the investigation of more restricted subclasses, for example the intersection graphs of squares and of unit-height rectangles [5, 12]. Also, PTASs for the WIS exist for the class of fat convex objects, which generalizes unit disk graphs and several intersection graphs of rectangles [11, 20]. The fastest PTAS for the WIS on fat convex objects takes time and space, or alternatively time with space [11].

Our results.

We present a method to solve the constant-diameter subproblems that appear when the shifting strategy is used. We call it the shifting coresets method (Section 2). By incorporating this method into the shifting strategy idea, we obtain linear-time approximation algorithms for problems on unit disk graphs and other geometric intersection graphs. The method is based on approximating the input set of geometric objects, which can be arbitrarily dense, by a sparse set of objects, that is, a set of objects such that any square of constant size contains at most a constant number of objects.

To obtain efficient algorithms through the application of our method, one needs to investigate the fundamental question of how well a sparse set—generated using only local information—can approximate a denser set for each considered problem. Thus, although the algorithms in this text share the same basic strategy, their analyses differ significantly. For example, the WIS analyses apply advanced graph-coloring theorems, whereas the DS analysis applies geometric packing arguments.

previous / new results WIS DS
previous ratio in time [24] [15]
our approximation ratio in time Sec. 3 Sec. 4
previous time for a ()-approximation [26] [10]
Table 1: Comparison of new and previous approximation algorithms for UDGs.

By using the shifting coresets method, we obtain linear-time -approximation algorithms on UDGs for the WIS (which is tackled first, in Section 3, owing to its greater simplicity) and for the DS (Section 4). The proposed algorithms provide significant improvements when compared to existing linear- and near-linear-time algorithms for these problems (see Table 1). The text also includes (Section 5) an alternative linear-time -approximation obtained for the VC. We reiterate that in that section we have independently derived a previous result from Marx, for the sake of illustrating an indirect application of our method [25]. Finally, our method is applied to obtain a linear-time -approximation algorithm for the WIS on the class of intersection graphs of axis-aligned rectangles with side lengths in , for an arbitrary constant .

In Section 7, some open problems and lower bounds to the approximation ratios of the presented algorithms are discussed.

2 The shifting coresets method

The shifting strategy is the core of most of the existing geometric PTASs for problems on geometric intersection graphs [21]. Generally, the shifting strategy reduces the original problem with objects to a set of subproblems whose inputs have constant diameter and the sum of the input sizes is . Such a reduction is based on partitioning the objects according to a number of iteratively shifted grids and takes time (by using the floor function and constant-time hashing). Exploiting the inputs’ constant diameter, each subproblem is solved exactly in polynomial time. The solutions to the subproblems are then combined appropriately (normally in time) to yield feasible solutions to the original problem, the best of which is returned. The high complexities of these geometric PTASs are due to the exact algorithms that are employed to solve each subproblem.

The shifting coresets method we introduce is based on the shifting strategy. However, it presents a crucial new aspect. Rather than obtaining exact, costly solutions for the subproblems, it solves each subproblem approximately. To do that, it employs the coresets paradigm, which consists in considering only a constant-size subset of the input [2].   Let denote the maximum distance between two points contained inside objects of a set of objects . For a problem whose input is a set of objects and a parameter , the method can be briefly described as follows.

  1. Apply the shifting strategy to construct a set of subproblems with inputs such that and for all , is a constant that depends on .

  2. For each subproblem instance , obtain a coreset with , such that the optimal solution for instance is an -approximation to the optimal solution for instance .

  3. Solve the problem exactly for each .

  4. Combine the solutions into an -approximation for the original problem.

Coresets for different problems must be devised on a case-by-case basis. For the WIS on UDGs, we create a grid with cells of diameter and consider only one disk center of maximum weight inside each cell. For the DS on UDGs, we create a grid with cells of diameter and consider only the (at most four) disk centers, inside each cell, with minimum or maximum coordinate in either dimension (breaking ties arbitrarily). We solve the VC on UDGs by breaking each subproblem into two cases. In the first one, the number of input disks is already bounded by a constant, therefore already a coreset. In the second one, we use the same coreset as in the WIS. Finally, we solve the WIS on a class of rectangle intersection graphs by representing each rectangle as a -dimensional point and using a -dimensional grid with cells of side . Our coreset is then formed by the rectangle of maximum weight inside each -dimensional cell.

Assuming a real-RAM model of computation with floor function and constant-time hashing, it is possible to partition the input points into grid cells efficiently, yielding an overall running time for the algorithms [8]. Without these operations, the running time becomes . We also assume that is constant. Otherwise, the running time becomes for the WIS and the DS on UDGs, since their corresponding coresets contain points.222The dependency on may be conceivably improved to —for instance, by replacing brute-force search with a -time exact algorithm via separators. Such a dependency is not too important, though, since the approximation factor does not approach . We have therefore opted for the simplest approach. For the VC, the running time becomes . As for the WIS on axis-aligned rectangles, if one regards and (the maximum allowed side length) as asymptotic variables, then the running time becomes .

Definitions.

A grid is said to be rooted at a point if there is a grid cell with corner at . Given a grid cell and a real parameter , the square region , called the -contraction of , is formed by removing from the points within distance at most from the boundary of . Analogously, the square region , called the expansion of , is formed by and all points within distance at most from . Finally, we denote the collection of all -expansions of cells of a grid rooted at a point by .

3 WIS on UDGs

In this section, we show how to apply the shifting coresets method to obtain a linear-time -approximation to the WIS on unit disk graphs. We start by presenting a -approximation for point sets of constant diameter, and then we use the shifting strategy to obtain the desired -approximation.

Given a point and a set of points, let denote the weight of , and let . Two or more points are considered independent if their minimum distance is strictly greater than .

Theorem 1.

Given a set of points with real weights as input, with , the WIS for the corresponding UDG can be -approximated in time on the real-RAM.

Proof.

Our algorithm proceeds as follows. First, find the points of with minimum or maximum coordinates in either dimension. That defines a bounding box of constant size for . Within this bounding box, create a grid with cells of diameter and side (any value suffices). Note that the number of grid cells is constant, and therefore the points of can be partitioned among the grid cells in time (even without using the floor function or hashing). Then, build the subset as follows. For each non-empty grid cell , add to a point of maximum weight in . Afterwards, determine the maximum-weight independent set of . Since , this can be done in constant time. Return the solution .

Next, we show that is indeed a -approximation. We argue that, given an independent set , there is an independent set with . Given a point , let denote the point from that is contained in the same grid cell as . Consider the set . Note that and . The set may not be independent, but since is independent, the minimum distance in is at least . We claim that the unit disk graph formed by is a planar graph. To prove the claim, we show that a planar drawing can be obtained by connecting the points of within distance at most by straight line segments. Given a pair of points with distance , the Pythagorean Theorem shows that a unit disk centered within distance greater than from both and cannot intersect the segment . This implies no two edges in the drawing can cross. By the Four-Color Theorem [6], admits a partition into four independent sets . The set of maximum weight among must have weight at least .333Note that the Four-Color Theorem is only used in the argument. No coloring is ever computed by the algorithm.

Since is the maximum-weight independent set of , it follows that is a 4-approximation for the WIS. ∎

The following theorem uses the shifting strategy to obtain a -approximation for point sets of arbitrary diameter. The proof borrows ideas from Hunt III et al [22]. We present them in a different manner, though, including details about an efficient implementation of the strategy.

Theorem 2.

Given a set of points in the plane as input, the WIS for the corresponding UDG can be -approximated in time on the real-RAM with constant-time hashing and the floor function. Without these operations, it can be done in time.

Proof.

Let be the smallest integer such that

(1)

The algorithm proceeds as follows. For from to , create a grid with cells of side rooted at . For each cell in the grid, run the WIS -approximation algorithm from Theorem 1 with point set (the points in that belong to a -contraction of , see Fig.1), obtaining a solution . Then, the independent set is constructed as the union of the independent sets for all grid cells . Return the maximum-weight set that is found, call it .

Figure 1: Grid rooted at with cells of side and the -contraction of a cell.

To implement the algorithm efficiently, create a subgrid of subcells of side , assigning each point to the subcell that contains it. In order to partition the points into subcells efficiently, one should use the floor function and constant-time hashing, taking time. If these operations are not available, it is possible to determine the connected components of the graph (using the Delaunay triangulation, for example) and to partition the points of each component into subcells (through sorting by -coordinate, separating into columns, and sorting inside each column by -coordinate). The non-empty subcells are stored in a balanced binary search tree. This process takes time due to sorting, Delaunay triangulation, and binary search tree operations. Given the partitioning of the point set into subcells, each input to the WIS algorithm can be constructed as the union of a constant number of subcells. Finally, the total size of the constant-diameter WIS instances is , since each point from the original point sets appears in a constant number of such instances, namely .

To prove that the returned solution is indeed a -approximation, we use a probabilistic argument. Let be picked uniformly at random from and let denote the optimal solution. For every cell , we have

Consequently, by summing over all grid cells,

where denotes the probability that a given point is contained in some contracted cell. Because such probability is the same for all points, is bounded by

Note that, for all , corresponds to the ratio between the areas of and , namely

Therefore, by using inequality (1), we obtain

Since has maximum weight among the independent sets , it follows that is at least as large as their average weight. Therefore, satisfies

closing the proof. ∎

4 DS for UDGs

In this section, we show how to apply the shifting coresets method to obtain a linear-time -approximation to the DS (in fact, a generalization of it) on unit disk graphs. We start by presenting a -approximation for point sets of constant diameter, and then we use the shifting strategy to obtain the desired -approximation. A point dominates a point if . Given two sets of points and , the set is a -dominating set if every point in is dominated by some point in .

We now define a more general version of the DS, the minimum partial dominating set problem (PDS). Such a generalization is necessary to properly apply the shifting strategy. In the PDS, given a set of points and also a subset , the goal is to find the smallest -dominating subset .

In order to analyze our algorithm, we prove a geometric lemma that shows that the difference between a unit circle and two unit disks that are sufficiently close to it and form a sufficiently big angle consists of one or two “small” arcs. Given a point , let denote the unit disk centered at , and denote its boundary circle.

Lemma 3.

Given and three points with (i) , (ii) , and (iii) the smallest angle is greater than or equal to , it follows that:

  1. the portion of the boundary consists of one or two circular arcs;

  2. if consists of one circular arc, then the arc length is less than or equal to ; and

  3. if consists of two circular arcs, then each arc length is less than .

Proof.

Statement (1) is clearly true. We start by proving statement (2). The arc length is maximized as the angle decreases while the distances are kept constant, therefore it suffices to consider the case in which . The arc centered at can be decomposed into three arcs by rays in directions and , as shown in Fig. 2(a). The central arc measures , while each of the other two arcs measures , proving statement (2).

Figure 2: Proof of Lemma 3.

Next, we prove statement (3). Let denote the two arcs that form with . The arc length is maximized in the limit when , as shown in Fig. 2(b). The rays connecting and to the two extremes of are parallel, and therefore . ∎

We are now able to prove the following theorem, which presents our -approximation algorithm for point sets of constant diameter.

Theorem 4.

Given two sets of points and as input, with , , and , the PDS can be -approximated in time on the real-RAM.

Proof.

First, determine a bounding box of constant size for , as in the algorithm for the WIS. Within this bounding box, create a grid with cells of diameter and side (any positive satisfying

suffices). Note that the number of grid cells is constant, and therefore the points of can be partitioned among the grid cells in time (even without using the floor function or hashing). Then, build the subset as follows. For each non-empty grid cell, add to the (at most four) extreme points inside the cell, i.e., those presenting minimum or maximum coordinate in either dimension. Ties are broken arbitrarily. Since there is a constant number of grid cells and contains at most four points per cell, it follows that . Now determine the smallest -dominating subset . To do that, examine the subsets of , from smallest to largest, verifying if all points of are dominated, until the dominating set if found and returned as the approximate solution. Since has a constant number of points, this procedure takes time.

Now we show that the returned solution is indeed a -approximation. We argue that, given a -dominating set , there is a -dominating set with . To build the set from , proceed as follows. For each point , if , add to . Otherwise, since the set contains points of extreme coordinates in both and axes, in the cell of , there are two points such that (i) , (ii) , and (iii) the smallest angle is at least . Add these two points to .

By Lemma 3, the portion of consists of one or two circular arcs. First consider the case in which consists of one circular arc. Let be the set of points from which are dominated by , but not by or . If is empty, then no extra point needs to be added to . Otherwise, the line which contains and bisects separates into two (possibly empty) sets . If , let be an arbitrary point of . Since contains a point in the same cell as , there is a point with . Add the point to . Analogously, if , let be an arbitrary point of and let be a point with . Add the point to .

We now show that the four points dominate all points dominated by . Consider a point that is dominated by but not by or . The point must be inside the circular crown sector depicted in Fig. 3(a) and described as follows. Because is dominated by , we have . By Lemma 3, the arc length . Also, , because otherwise the unit circles centered at and would intersect forming an arc of length at least , which is greater than , in which case is dominated by or . Finally, since is closer to than it is to or , it follows that must be between the lines that connect to the endpoints of . This circular crown sector is bisected by the line . By the law of cosines, the diameter of each circular crown sector is . Therefore, for any point inside the circular crown sector, the point (or , analogously) that is within distance at most from a point inside the same sector dominates , as .

Figure 3: Proof of Theorem 4.

Finally, if consists of two circular arcs centered in , then start by adding those same points to , as if consisted of only one arc. Then, if necessary, add new points to as follows. The points that are dominated by but not by or must be within distance of either or . Let be arbitrary points that are within distance of or , respectively, but are not dominated by or . If such points exist, then there are two points in that are within distance at most from respectively . By Lemma 3, the largest arc among measures at most . The proof that all points dominated by are dominated by , or is analogous to the case in which consists of a single arc, using the circular crown sector illustrated in Fig. 3(b).

Since is minimum among all subsets of that are -dominating sets, is a 4-approximation for the PDS. ∎

The following theorem uses the shifting strategy to obtain a -approximation for point sets of arbitrary diameter.

Theorem 5.

Given two sets of points and as input, with and , the PDS can be -approximated in time on the real-RAM with constant-time hashing and the floor function. Without these operations, it can be done in time.

Proof.

Let be the smallest integer such that

(2)

The algorithm proceeds as follows. For from to , create a grid with cells of side rooted at and, for each cell in the grid, use Theorem 4 to -approximate the PDS with point sets (the points of that belong to the -expansion of , see Fig. 4) and , obtaining a solution . The dominating set is constructed as the union of the dominating sets for all grid cells . Return the smallest dominating set that is found, call it .

Figure 4: Grid rooted at with cells of side and the -expansion of a cell.

To prove that the returned solution is indeed a -approximation, we use again a probabilistic argument. Let be picked uniformly at random from and let denote the optimal solution. For every cell , we have

Consequently, by summing over all grid cells,

We can now bound , since

by the linearity of expectation. Note that since the expected size of is the same for all points, it corresponds to the ratio between the areas of and , namely

Therefore, by using inequality (2), we obtain

Since the smallest among the dominating sets has no more than their average number of elements, the set returned by the algorithm satisfies

closing the proof. ∎

The DS is the special case of the PDS in which , and thus it can be -approximated in linear time by the same algorithm.

In the minimum distance -dominating set problem (MDDS), the input consists of a graph and an integer , and the goal is to find a minimum subset of vertices such that all graph vertices are within distance at most from a vertex in the subset (here is the graph distance, that is, the number of edges in a shortest path). The DS is a special case in which . Our algorithm for the DS can be generalized to -approximate the MDDS in linear time for constant . In contrast, the greedy heuristic that gives a -approximation to the DS gives a -approximation to the MDDS.

5 VC for UDGs

In this section, we show how to obtain a linear-time approximation scheme to the VC on unit disk graphs. We start by presenting an approximation scheme for point sets of constant diameter, and then we use the shifting strategy to generalize the result to arbitrary diameter. Differently from the two previous problems, the size of a minimum vertex cover for a point set of constant diameter is not bounded by a constant. Therefore, strictly speaking, a coreset for the problem does not exist. Nevertheless, it is possible to use coresets to approach the problem indirectly. We remark that a different linear-time approximation scheme to the VC was presented by Marx [25].

Given a graph with vertices, it is well known that is an independent set if and only if is a vertex cover. While a maximum independent set corresponds to a minimum vertex cover, a constant approximation to the maximum independent set does not necessarily correspond to a constant approximation to the minimum vertex cover. However, in certain cases, an even stronger correspondence holds, as we show in the following proof.

Theorem 6.

Given a set of points as input, with , the VC can be -approximated in time on the real-RAM, for constant .

Proof.

Our algorithm considers two cases, depending on the value of . If

then is constant, and the VC can be solved optimally in constant time.

Otherwise, use Theorem 1 to obtain a -approximation to the maximum independent set (other constant factor approximations can be used, adjusting the threshold accordingly). We now show that is a -approximation to the minimum vertex cover. Let respectively be the maximum independent set and the minimum vertex cover. Note that and . By a simple packing argument, dividing the area of a disk of diameter by the area of a unit disk,

and consequently

Manipulating the previous inequality, we obtain

(3)

Since is a -approximation to ,

(4)

Combining (3) and (4), we can write as desired. ∎

Using the shifting strategy the following result ensues.

Theorem 7.

Given a set of points in the plane as input, the VC for the corresponding UDG can be -approximated in time on the real-RAM with constant-time hashing and the floor function, for constant . Without these operations, it can be done in time.

Proof.

Let be the smallest integer such that

(5)

The algorithm proceeds as follows. For from to , create a grid with cells of side rooted at and, for each cell in the grid, use Theorem 6 to -approximate the VC for , obtaining a solution . The vertex cover is constructed as the union of the vertex covers for all grid cells . Return the smallest vertex cover that is found, call it .

To prove that the returned solution is indeed a -approximation, once again we use a probabilistic argument. Let be picked uniformly at random from and let denote the optimal solution. For every cell , we have

Consequently, by summing over all grid cells,

We can now bound , since

by the linearity of expectation. Note that since the expected size of is the same for all points, it corresponds to the ratio between the areas of and , namely

Therefore, by using inequality (5), we obtain

Since the smallest among the vertex covers has no more than their average number of elements, the set returned by the algorithm satisfies

closing the proof. ∎

6 WIS for Rectangles of Bounded Size

In this section, we consider the case in which the input is no longer a set of points, but a set of rectangles instead. Let be a constant and a set of axis-aligned rectangles in the plane, such that each rectangle , for , has width and height between and , and weight . Let be the intersection graph of . The shifting coresets method is applied to obtain a linear-time -approximation algorithm to the maximum-weight independent set of .

The overlap of two rectangles is defined as the minimum (horizontal or vertical) translation distance of a single rectangle necessary to make the interiors of and disjoint. The following lemma bounds the chromatic number of the intersection graph of rectangles with a small overlap.

Lemma 8.

If is a set of axis-aligned rectangles such that

  1. the width and height of each rectangle is at least , and

  2. for every two distinct rectangles ,

then the intersection graph of is -colorable.

Proof.

Let be a set as required and its intersection graph. A graph is -planar if it can be drawn on the plane in a way that each edge intersects at most one other edge. Borodin showed that -planar graphs are -colorable [9]. We prove the lemma by providing such a -planar drawing for .

For each rectangle , draw vertex on the center of . Given two intersecting rectangles , the edge is drawn as two straight line segments, connecting to the center of the rectangle and then to . We show that at most one other edge may cross the edge . Note that the edge is completely inside the region .

When two rectangles intersect one another, there are two possible types of edges corresponding to the relative positions of the rectangles (Fig. 5):

  1. contains two corners of (or vice versa); or

  2. contains one corner of , and vice versa.

Figure 5: Examples of (a) type-1 and (b) type-2 edges.

We show that a type- edge cannot possibly be crossed by any other edge. Indeed, if contains two corners of , then the straight segment from to the center of belongs to an axis-aligned line that bisects . This segment cannot be crossed by any edge . Otherwise, the centers of the intersecting rectangles would belong to distinct halfplanes defined by , and at least one of these rectangles, say , should cross . Since, by the maximum overlap allowed in , the length of the segment of which can be contained in measures at most (i.e., inside plus inside ), a contradiction ensues, because measures at least on both dimensions. The straight segment from to the center of , on its turn, cannot be crossed by any other edge because, since the edge is drawn completely inside the region , one of the rectangles, say , should intersect that segment, yet it should not contain any of its points whose distance to the border of that intersects is greater than . But now, since both sides of are greater than , it follows that , a contradiction.

It remains to show that a type- edge can be crossed by at most one other edge (which must also be a type- edge). Suppose, for the sake of contradiction, that is crossed by two other edges, and , as illustrated in Fig. 6. Now, without loss of generality, the maximum allowed overlap implies that:

Figure 6: Crossing twice a type-2 edge.
  • the width of satisfies ;

  • the width of satisfies ;

  • the width and height of satisfy either or .

If , then the width of is , a contradiction. If, on the other hand, , then an analogous contradiction ensues on the direction. ∎

Given a set of rectangles, the diameter is the maximum distance between two vertices of the rectangles in . We are now able to prove the following theorem, which presents our -approximation algorithm for sets of constant diameter.

Theorem 9.

Given a set of axis-aligned weighted rectangles as input, such that and each rectangle in has width and height at least , the WIS on the intersection graph of can be -approximated in time on the real-RAM.

Proof.

Represent each rectangle by four real values , corresponding to the and coordinates of its center, its width, and its height. The set can thus be seen as a constant-diameter set of points in . Create a -dimensional grid with cells of side (any positive suffices), and define the set by choosing the element of with maximum weight inside each non-empty grid cell. Note that since , we have , so the maximum-weight independent set among the points of can be computed in constant-time by brute force. Return such a set.

To prove the returned solution is indeed a -approximation, we show that, given an independent set , there is an independent set such that . Let be the set of rectangles obtained by selecting, for each rectangle , the rectangle whose corresponding -dimensional point lies inside the same grid cell as that of . Note that since contains the maximum weight rectangle inside each grid cell, we have , even though may not be an independent set. We claim that the rectangles in overlap by less than , hence Lemma 8 can be employed to partition into independent sets. Let be set of maximum weight among the partitions. Since , it follows that , proving the theorem.

To prove the claim, consider two disjoint rectangles , . Let be the corresponding rectangles in . Because the grid cells have side , we have that all the following quantities are less than , for : . The possible horizontal overlap between and may come from two sources: a displacement (by less than for each rectangle) and a change in size, which moves the boundary of each rectangle by less than . Therefore, the maximum horizontal overlap is at most . The same argument bounds the maximum vertical overlap. ∎

Using the shifting strategy, we extend the result for sets of arbitrary diameter.

Theorem 10.

Let be a constant. Given a set of axis-aligned weighted rectangles as input, such that each rectangle in has width and height between and , the WIS can be -approximated in time on the real-RAM with constant-time hashing and the floor function. Without these operations, it can be done in time.

Proof.

Let be the smallest multiple of such that

(6)

and let .

Throughout this proof, consider grids with square cells of side and use the same strategy as in the proof of Theorem 2, with some small modifications. To make sure that the union of two independent sets, each belonging to a different contraction of a cell, is still an independent set, we employ the contraction of .

For from to , create a grid with cells of side rooted at . For each cell in the grid, use Theorem 9 to obtain a -approximation for the WIS whose input consists of all rectangles whose centers belong to . The independent set is the union of all . Return the maximum-weight set that is found, call it .

We now prove that the returned solution is indeed a -approximation. Let be picked uniformly at random from and let denote the optimal solution. For every cell , we have

Consequently, by summing over all grid cells,

where denotes the probability that the center of a given rectangle is contained in some contracted cell. Because such probability is the same for all rectangles, can be bounded by

Note that, for all , corresponds to the ratio between the areas of and , namely

Thus, inequality (6) yields

Since has maximum weight among the independent sets , it follows that is at least as large as their average weight. Therefore, satisfies

closing the proof. ∎

7 Conclusion

This paper introduced the method of shifting coresets, which, combining the shifting strategy and the coresets paradigm, has allowed us to obtain improved linear-time approximations for problems on unit disk graphs. The method is applicable to other geometric intersection graph classes as well. The central idea of the method is the creation of coresets to obtain approximate solutions when the inputs are point sets of constant diameter. For the WIS and the DS on UDGs, the proposed algorithms provide improved approximation ratios when compared to existing linear-time algorithms, as shown in Table 1 in the Introduction.

Figure 7: (a) Example in which the approximation ratio for the DS is exactly . (b) Coin graph used in the example in which the approximation ratio for the WIS is .

While the approximation ratio for the WIS and the DS on UDGs is no greater than  (for constant-diameter inputs), we only know that the analysis is tight for the DS. Indeed, Fig. 7(a) shows a DS instance in which our algorithm does not achieve an approximation ratio better than , even if we reduce the grid size and search for extreme points in a larger number of directions. In contrast, for the WIS, the best lower bound we are aware of is , as shown in the following example. Let be the weighted point set from Fig. 7(b), in which all adjacent vertices are at distance exactly . Create another set by multiplying the coordinates of the points in by , while multiplying their weights by , for arbitrarily small . The set forms an independent set of weight just smaller than , while the maximum independent set in has weight . Since each vertex in has a smaller weight and is arbitrarily close to a vertex of , the vertices of will be disregarded by the algorithm for the input instance .

The analysis of the -approximation ratio for the WIS on rectangle intersection graphs leaves an even bigger gap. The best lower bound we are aware of is , since the graph illustrated in Fig. 8 (with vertices and maximum independent set with size ) is in the graph class used in Lemma 8. In fact, it is possible that such class is -colorable (the same graph in Fig. 8 shows it is not -colorable, though). We remark that the need for colors does not mean that the ratio between the total weight of the vertices and the maximum weight of an independent set can be as high as .

Figure 8: A -chromatic graph in the class from Lemma 8, and its representation by slightly-overlapping rectangles.

Several open problems remain. Is it possible to obtain an approximation ratio better than in time for the WIS on UDGs, or at least for the unweighted version? Can the linear-time approximation scheme for VC be generalized for the weighted version? Are the point coordinates really necessary, or is it possible to devise similar graph-based algorithms? Also, can our method be used to obtain better linear-time approximations to related problems on unit disk graphs such as finding the minimum-weight dominating set or the minimum connected dominating set? Is it possible to use the shifting coresets method to obtain a constant approximation for the WIS on disk graphs (of arbitrary radii) in linear time? Finally, is it possible to use similar ideas to derive improved linear-time approximations for problems on other classes of graphs such as planar graphs, bounded treewidth graphs, etc.?

Acknowledgements

The authors would like to thank Raphael Machado, Mickael Montassier, Petru Valicov and Yann Vaxès for the insightful discussions. The proofs of Theorems 2 and 7 use the same techniques as in Hunt III et al., which were in turn motivated by Baker’s technique [22, 7].

This research was partially supported by the Brazilian agencies CAPES, CNPq, and FAPERJ. An extended abstract of this paper appeared in the 12th Workshop on Approximation and Online Algorithms (WAOA 2014).

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