Optimal Perimeter Guarding with Heterogeneous Robot Teams: Complexity Analysis and Effective Algorithms

Optimal Perimeter Guarding with Heterogeneous Robot Teams: Complexity Analysis and Effective Algorithms

Abstract

We perform structural and algorithmic studies of significantly generalized versions of the optimal perimeter guarding (OPG) problem [1]. As compared with the original OPG where robots are uniform, in this paper, many mobile robots with heterogeneous sensing capabilities are to be deployed to optimally guard a set of one-dimensional segments. Two complimentary formulations are investigated where one limits the number of available robots (OPG) and the other seeks to minimize the total deployment cost (OPG). In contrast to the original OPG which admits low-polynomial time solutions, both OPG and OPG are computationally intractable with OPG being strongly NP-hard. Nevertheless, we develop fairly scalable pseudo-polynomial time algorithms for practical, fixed-parameter subcase of OPG; we also develop pseudo-polynomial time algorithm for general OPG and polynomial time algorithm for the fixed-parameter OPG case. The applicability and effectiveness of selected algorithms are demonstrated through extensive numerical experiments.

1 Introduction

Consider the scenario where many mobile guards (or sensors) are to be deployed to patrol the perimeter of some 2D regions (Fig. 1) against intrusion, where each guard may effectively cover a continuous segment of a region’s boundary. When part of a boundary need not be secured, e.g., there may already be some existing barriers (the blue segments in Fig. 1), optimally distributing the robots so that each robot’s coverage is minimized becomes an interesting and non-trivial computational task [1]. It is established [1] that, when the guards have the same capabilities, the problem, called the optimal perimeter guarding (OPG), resides in the complexity class P (polynomial time class), even when the robots must be distributed across many different boundaries.

{overpic}

[width=4in ,tics=5]figures/opg-eps-converted-to.pdf

Figure 1: A scenario where boundaries of three (gray) regions must be secured. Zooming in on part of the boundary of one of the regions (the part inside the small circle), portions of the boundary (the red segments) must be guarded while the rest (the blue dotted segments) does not need guarding. For example, the zoomed-in part of the boundary may be monitored by two mobile robots, each patrolling along one of the green segments.

In this work, we investigate a significantly more general version of OPG where the mobile guards may be heterogeneous. More specifically, two formulations with different guarding/sensing models are addressed in our study. In the first, the number of available robots is fixed where robots of different types have a fixed ratio of capability (e.g., one type of robot may be able to run faster or may have better sensor). The guarding task must be evenly divided among the robots so that each robot, regardless of type, will not need to bear a too large coverage/capability ratio. This formulation is denoted as optimal perimeter guarding with limited resources or OPG. In the second, the number of robots is unlimited; instead, for each type, the sensing range is fixed with a fixed associated cost. The goal here is to find a deployment plan so as to fully cover the perimeter while minimizing the total cost. We call this the optimal perimeter guarding with minimum cost problem, or OPG.

Unlike the plain vanilla version of the OPG problem, we establish that both OPG and OPG are NP-hard when the number of robot types is part of the problem input. They are, however, at different hardness levels. OPG is shown to be NP-hard in the strong sense, thus reducing the likelihood of finding a fully polynomial time approximation scheme (FPTAS). Nevertheless, for the more practical case where the number of robot types is a constant, we show that OPG can be solved using a pseudo-polynomial time algorithm with reasonable scalability. On the other hand, we show that OPG is weakly NP-hard through the establishment of a pseudo-polynomial time algorithm for OPG with arbitrary number of robot types. We further show that, when the number of robot types is fixed, OPG can be solved in polynomial time through a fixed-parameter tractable (FPT) approach. This paragraph also summarizes the main contributions of this work.

A main motivation behind our study of the OPG formulations is to address a key missing element in executing autonomous, scalable, and optimal robot deployment tasks. Whereas much research has been devoted to multi-robot motion planning [2, 3] with great success, e.g., [4, 5, 6, 7, 8, 9], existing results in the robotics literature appear to generally assume that a target robot distribution is already provided; the problem of how to effectively generate optimal deployment patterns is largely left unaddressed. It should be noted that control-based solutions to the multi-agent deployment problem do exist, e.g.,[10, 11, 12, 13, 14, 15, 16], but the final solutions are obtained through many local iterations and generally do not come with global optimality guarantees. For example, in [12], Voronoi-based iterative methods compute locally optimal target formations for various useful tasks. In contrast, this work, as well as [1], targets the scalable computation of globally optimal solutions.

As a coverage problem, OPG may be characterized as a 1D version of the well-studied Art Gallery problems [17, 18], which commonly assume a sensing model based on line-of-sight visibility[19]; the goal is to ensure that every point in the interior of a given region is visible to at least one of the deployed guards. Depending on the exact formulation, guards may be placed on boundaries, corners, or the interior of the region. Not surprisingly, Art Gallery problems are typically NP-hard [20]. Other than Art Gallery, 2D coverage problems with other sensing models, e.g., disc-based, have also been considered [21, 22, 23, 12, 24, 25], where some formulations prevent the overlapping of individual sensing ranges [21, 22] while others seek to ensure a full coverage which often requires intersection of sensor ranges. In viewing of these studies, this study helps painting a broader landscape of sensor coverage research.

In terms of structural resemblance, OPG and OPG share many similarities with bin packing [26] and other related problems. In a bin packing problem, objects are to be selected to fit within bins of given sizes. Viewing the segments (the red ones in Fig. 1) as bins, OPG seeks to place guards so that the segments are fully contained in the union of the guards’ joint coverage span. In this regard, OPG is a dual problem to bin packing since the former must overfill the bins and the later cannot fully fill the bins. In the extreme, however, both bin packing and OPG converge to a Subset Sum [27] like problem where one seeks to partition objects into halves of equal total sizes, i.e., the objects should fit exactly within the bins. With an additional cost term, OPG has further similarities with the Knapsack problem [28], which is weakly NP-hard [29].

The rest of the paper is organized as follows. In Section 2, mathematical formulations of the two OPG variants are fully specified. In Section 3, both OPG and OPG are shown to be NP-hard. Despite the hardness hurdles, in Section 4, multiple algorithms are derived for OPG and OPG, including effective implementable solutions for both. In Section 5, we perform numerical evaluation of selected algorithms and demonstrate how they may be applied to address multi-robot deployment problems. We discuss and conclude our study in Section 6. Please see https://youtu.be/6gYL0_B3YTk for an illustration of the problems and selected instances/solutions.

2 Preliminaries

Let be a compact (closed and bounded) two-dimensional workspace. There are pairwise disjoint regions where each region is homeomorphic to the closed unit disc, i.e., there exists a continuous bijection for all . For a given region , let be its (closed) boundary (therefore, maps to the unit circle ). With a slight abuse of notation, define . Let be the part of that is accessible, specifially, not blocked by obstacles in . This means that each is either a single closed curve or formed by a finite number of (possibly curved) line segments. Define as the perimeter of which must be guarded. More formally, each is homeomorphic to a compact subset of the unit circle (i.e., it is assumed that the maximal connected components of are closed line segments). For a given , each one of its maximal connected component is called a perimeter segment or simply a segment, whereas each maximal connected component of is called a perimeter gap or simply a gap. An example setting is illustrated in Fig. 2 with two regions.

{overpic}

[width=4in ,tics=5] figures/example-boundaries-eps-converted-to.pdf

Figure 2: An example of a workspace with two regions . Due to three gaps on , marked as dotted lines within long rectangles, has three segments (or maximal connected components); has a single segment with no gap.

After deployment, some number of robots are to cover the perimeter such that a robot is assigned a continuous closed subset of some . All of must be covered by , i.e., , which implies that elements of need not intersect on their interiors. Hence, it is assumed that any two elements of may share at most their endpoints. Such a is called a cover of . Given a cover , for a , let denote its length (more formally, measure).

To model heterogeneity of the robots, two models are explored in this study. In either model, there are types of robots. In the first model, the number of robots of each type is fixed to be with . For a robot , let denote its type. Each type of robots has some level of capability or ability . We wish to balance the load among all robots based on their capabilities, i.e., the goal is to find cover for all robots such that the quantity

which represents the largest coverage-capacity ratio, is minimized. We note that when all capacities are the same, e.g., for all robots, this becomes the standard OPG problem studied in [1]. We call this version of the perimeter guarding problem optimal perimeter guarding with limited resources or OPG. The formal definition is as follows.

Problem 2.1 (Optimal Perimeter Guarding with Limited Resources (Opg)).

Let there be types of robots. For each type , there are such robots, each having the same capability parameter . Let . Given the perimeter set of a set of 2D regions , find a set of continuous line segments such that covers , i.e.,

(1)

such that a is covered by robot of type , and such that, among all covers satisfying (1),

(2)

Whereas the first model caps the number of robots, the second model fixes the maximum coverage of each type of robot. That is, for each robot type , , the number of robots of type , is unlimited as long as it is non-negative, but each such robot can only cover a maximum length of . At the same time, using each such robot incurs a cost of . The goal here is to guard the perimeters with the minimum total cost. We denote this problem optimal perimeter guarding with minimum cost or OPG.

Problem 2.2 (Optimal Perimeter Guarding with Minimum Cost (Opg)).

Let there be types of robots of unlimited quantities. For each robot of type , it can guard a length of with a cost of . Given the perimeter set of a set of 2D regions , find a set of continuous line segments where such segments are guarded by type robots, such that covers , i.e.,

(3)

such that a is covered by robot of type , i.e., , and such that, among all covers satisfying (3),

(4)

3 Computational Complexity for Variable Number of Robot Types

We explore in this section the computational complexity of OPG and OPG. Both problems are shown to be NP-hard with OPG being strongly NP-hard. We later confirm that OPG is weakly NP-hard (in Section 4).

3.1 Strong NP-hardness of Opg

When the number of types is a variable, i.e., is not a constant and may be arbitrarily large, OPG is shown to be NP-hard via the reduction from -Partition [30]:

PROBLEM: -Partition
INSTANCE: A finite set A of elements, a bound , and a “size” for each , such that each satisfies and .
QUESTION: Is there a partition of into disjoint subsets such that for , ?

-Partition is shown to be NP-complete in the strong sense[31], i.e., it is NP-complete even when all numeric inputs are bounded by a polynomial of the input size.

For the reduction, it is more convenient to work with a decision version of the OPG problem, denoted as D-OPG. In the D-OPG problem, is the actual length robot type covers. That is, the coverage length of a robot is fixed. The D-OPG problem is specified as follows.

PROBLEM: D-OPG
INSTANCE: types of robots where there are robots for each type ; . A robot of type has a coverage capacity . A set of perimeters of a set of 2D regions .
QUESTION: Is there a deployment of disjoint subsets of such that , where is a continuous segment for all , and for each , there is a unique robot whose type , satisfies ?

Theorem 3.1.

OPG is strongly NP-hard.

Proof.

A polynomial reduction from -Partition to D-OPG is constructed by a restriction of D-OPG. Given a -Partition instance with former notations, we apply several restrictions on D-OPG: (i) there are types of robot and there is a single robot for each type, i.e., for , so (ii) the capacities are set to be equal to for each of the elements , and (iii) there are perimeters and each perimeter is continuous and for all .

With the setup, the reduction proof is straightforward. Clearly, the -Partition instance admits a partition of into such that for all if and only if a valid depolyment exists in the corresponding D-OPG instance. It is clear that the reduction from -Partition to D-OPG is polynomial (in fact, linear). Based on the reduction and because -Partition is strongly NP-hard, so is D-OPG and OPG. ∎

Remark.

One may also reduce weakly NP-hard problems, e.g., Partition [27], to OPG for variable number of robot types . Being strongly NP-hard, OPG is unlikely to admit pseudo-polynomial time solutions for variable . This contrasts with a later result which provides a pseudo-polynomial time algorithm for OPG for constant , as one might expect in practice where robots have limited number of types. We also note that Theorem 3.1 continues to hold for a single perimeter with multiple segments, each having a length in previous notation, separated by “long” gaps. Obviously, D-OPG is in NP, thus rendering it NP-complete.

3.2 NP-hardness of Opg

The minimum cost OPG variant, OPG, is also NP-hard, which may be established through reduction from the Subset Sum problem [27]:

PROBLEM: Subset Sum
INSTANCE: A set with and a weight function , and an integer .
QUESTION: Is there a subset such that ?

Theorem 3.2.

OPG is NP-hard.

Proof.

Given a Subset Sum instance, we construct an OPG instance with a single perimeter containing a single segment with length to be specified shortly. Let there be types of robots. For , let robot type have and let robot type have . Here, can be any integer number no less than . Set . We ask the “yes” or “no’ decision question of whether there are robots that can be allocated to have a total cost no more than (equivalently, equal to , as the cost density is always ).

Suppose the Subset Sum instance has a yes answer that uses a subset . Then, the OPG instance has a solution with cost that can be constructed as follows. For each , a single robot of type is taken if . Otherwise, a single robot of type is taken. This allocation of robots yields a total length and cost of .

For the other direction, we first show that if the OPG instance is to be satisfied, it can only use a single robot from type or for all . First, if more than robots are used, then the total cost exceeds as . Similarly, if less than robots are used, the total length is at most . Also, to match the part of the cost, exactly one robot from type or for all must be taken. Now, if the OPG decision instance has a yes answer, if a robot of type is used, let be part of , which constructs a that gives a yes answer to the Subset Sum instance. ∎

Remark.

It is also clear that the decision version of the OPG problem is NP-complete. The Subset Sum is a weakly NP-hard problem that admits a pseudo-polynomial time algorithm [29]. As it turns out, OPG, which shares similarities with Subset Sum and Knapsack (in particular, Unbounded Knapsack [28]), though NP-hard, does admit a pseudo-polynomial time algorithm as well.

4 Exact Algorithms for Opg and Opg

In this section, we describe three exact algorithms for solving the two variations of the OPG problem. First, we present a pseudo-polynomial time algorithm for OPG when the number of robot types, , is a fixed constant. Given that OPG is strongly NP-hard, this is in a sense a best possible solution. For OPG, in addition to providing a pseudo-polynomial algorithm for arbitrary , which confirms that OPG is weakly NP-hard, we also provide a polynomial time approximation scheme (PTAS). We then further show the possibility of solving OPG in polynomial time when is a fixed constant. We mention that our development in this section focuses on the single perimeter case, i.e., , as the generalization to arbitrary is straightforward using techniques described in [1]. With this in mind, we also provide the running times for the general setting with arbitrary but refer the readers to [1] on how these running times can be derived.

For presenting the analysis and results, for the a perimeter that we work with, assume that it has perimeter segments that need to be guarded; these segments are separated by gaps . For , define where may be smaller than (i.e., may wrap around ), For the general case with perimeters, assume that a perimeter has segments.

4.1 Pseudo-Polynomial Time Algorithm for Opg with Fixed Number of Robot Types

We set to develop an algorithm for OPG for arbitrary , the number of robot types; the algorithm runs in pseudo-polynomial time when is a constant. At a higher level, our proposed algorithm works as follows. First, our main effort here goes into deriving a feasibility test for D-OPG as defined in Section 3.1. With such a feasibility test, we can then find the optimal in (2) via binary search. Let us denote the optimal value of as .

Feasibility Test for D-Opg

The feasibility test for D-OPG essentially tries different candidate to find . Our implementation uses ideas similar to the pseudo-polynomial time algorithm for the Knapsack problem which is based on dynamic programming (DP). In the test, we work with a fixed starting point on , which is set to be the counterclockwise end point of a segment , . Essentially, we maintain a dimensional array where dimension has a size of . An element of the array, , holds the maximal distance starting from that can be covered by type 1 robots, type 2 robots, and so on. The DP procedure OPG-lr-Feasible (), outlined in Algorithm 1, incrementally builds this array . For convenience, in the pseudo code, denotes an element of with being a dimensional integer vector.

Data: , , ,
Result: true or false, indicating whether can be covered
1 Initialize as a dimensional array with dimension having a size of ; for all ; for  do
2       ; for  to  do
3             if  then continue; ; ; (, Inc ());
4       end for
5      
6 end for
return ;
Algorithm 1 OPG-lr-Feasible ()

In Algorithm 1, the procedure Inc () checks how much of the perimeter can be covered when an additional coverage length is added, assuming that a distance of (starting from some ) is already covered. An illustration of how Inc () works is given in Fig. 3.

{overpic}

[width=4in ,tics=5] figures/inc-eps-converted-to.pdf Inc(, )

Figure 3: Suppose starting from the fixed left point, a length of on the boundary is successfully guarded by a group of robots. Then, a robot with coverage capacity is appended to the end of the group of robots to increase the total guarded distance. In the figure, the added additional capacity can fully cover the third red segment plus part of the third (dashed) gap. Because there is no need to cover the rest of the third gap, Inc(, ) extends to the end of the gap.

By simple counting, the complexity of the algorithm is . However, the amortized complexity of Inc () for each is ; the algorithm thus runs in , which is pseudo-polynomial for fixed . After trying every possible starting position with OPG-lr-Feasible (), for a fixed candidate , D-OPG is solved in .

Solving Opg using Feasibility Test for D-Opg

Using OPG-lr-Feasible () as a subroutine to check feasibility for a given , bisection can be applied over candidate to obtain . For completing the algorithm, one needs to establish when the bisection will stop (notice that, even though we assume that , for each , need not be an integer).

To derive the stop criterion, we note that given the optimal , there must exist some that is “exactly” spanned by the allocated robots. That is, assume that is covered by of type robots and of type robots, and so on, then

(5)

(5) must hold for some because if not, the solution is not tight and can be further improved. Therefore, the bisection process for locating does not need to go on further after reaching a certain granularity[1]. With this established, using similar techniques from [1] (we omit the technical detail as it is quite complex but without additional new ideas beyond beside what is already covered in [1]), we could prove that the full algorithm needs no more than calls to OPG-lr-Feasible (). This directly implies that OPG also admits a pseudo-polynomial algorithm for fixed .

Multiple Perimeters

Also using techniques developed in [1], the single perimeter result can be readily generalized to multiple perimeters. We omit the mechanical details of the derivation and point out that the computational complexity in this case becomes .

4.2 Polynomial Time Algorithm for Opg with Fixed Number of Robot Types

The solution to OPG will be discussed here. A method based on DP will be provided first, which leads to a polynomial time algorithm for a fixed number of robot types and a pseudo-polynomial time algorithm when the number of robot types is not fixed. For the latter case, a polynomial time approximation scheme (PTAS) will also be briefly described.

Dynamic Programming Procedure for Opg

When no gaps exist, the optimization problem becomes a covering problem as follows. Let , , correspond to the cost, coverage length, and quantity of robot type , respectively, and let total length to cover be . We are to solve the optimization problem

(6)

Let the solution to the above integer programming problem be Sol(). Notice that, for , the minimum cost cover is by either: (i) covering the total boundary without skipping any gaps, or (ii) skipping or partially covering some gap, for example . In the first case, the minimum cost is exactly Sol. In the second case, the optimal structure for the two subsets of perimeter segments and still holds. This means that the continuous perimeter segments can be divided into two parts, each of which can be treated separately. This leads to a DP approach for OPG. With denoting the minimum cost to cover , the DP recursion is given by

The DP procedure is outlined in Algorithm 2. In the pseudo code, it is assumed that indices of are modulo , e.g., . is a temporary variable.

Data: , , ,
Result: , the minimum covering cost
1 a matrix; ; for  to  do
2       for  to  do
3             Sol; for  to  do
4                   ;
5             end for
6            ; if  then ;
7       end for
8      
9 end for
return
Algorithm 2 OPG-mc-DP

A Polynomial Time Algorithm for Opg for a Fixed Number of Robot Types

We mention briefly that, by a result of Lenstra [32], the optimization problem  (6) is in P (i.e., polynomial time) when is a constant. The running time of the algorithm [32] is however exponential in .

A Pseudo-polynomial Time Algorithm for Arbitrary

As demonstrated in the hardness proof, similarities exist between OPG and the Knapsack problem. The connection actually allows the derivation of a pseudo-polynomial time algorithm for arbitrary . To achieve this, we use a routine to pre-compute Sol(), called PreSolve(), which is itself a DP procedure similar to that for the Knapsack problem. The pseudo code of PreSolve() is given in Algorithm 3. PreSolve() runs in time . Overall, Algorithm 2 then runs in time .

Data: ,
Result: A lookup table for retrieving Sol()
; % is an integer.
1 an array of length ; ; for  to  do
2       ; for  to  do
3             ; ;
4       end for
5      
6 end for
return
Algorithm 3 PreSolve

With the establishment of a pseudo-polynomial time algorithm for OPG, we have the following corollary.

Corollary 4.1.

OPG is weakly NP-hard.

FPTAS for Arbitrary

When the number of robot types is not fixed, Lenstra’s algorithm[32] or its variants no longer run in polynomial time. We briefly mention that, by slight modifications of a FPTAS for Unbounded Knapsack problem from [33], a FPTAS for OPG can be obtained that runs in time , where is the approximation ratio for both OPG and  (6).

Multiple Perimeters

For OPG, when there are multiple perimeters, e.g., , a optimal solution can be obtained by optimally solving OPG for each perimeter individually and then put together the solutions.

5 Performance Evaluation and Applications

In this section, we provide examples illustrating the typical optimal solution structures of OPG and OPG computed by our DP algorithms. Using an application scenario, solutions to OPG and OPG are also compared. Then, computational results from extensive numerical evaluations are presented, confirming the effectiveness of these algorithms. The implementation is done using the Python and all computations are performed on an Intel(R) Core(TM) i7-7700 CPU@3.6GHz with 16GB RAM.

5.1 Basic Optimal Solution Structure

Fig. 4 shows the typical outcome of solving an OPG instance with two perimeters () for two types of robots with , and . In the figure, the red segments are parts of the two perimeters that must be guarded. The three orange (resp., five green) segments across the two perimeters indicate the desired coverage regions of the three (resp., five) type (resp., type ) robots. These coverage regions correspond to the optimal solution returned by the DP algorithm. As may be observed, the optimal solution is somewhat complex with robots of both types on each of the two perimeters; a gap on the second boundary also gets covered. The coverage lengths for a robot type are generally different; this is due to adjustments that shrink some robots’ coverage. For example, the first perimeter has a very short orange cover because the corresponding perimeter segment is short and gaps around it need not be covered (The adjustment procedure is also shown in the video).

Figure 4: An OPG problem and an associated optimal solution. The problem has two perimeters and with =3, =5, =5, =8. The boundaries are shown as circles for ease of illustration.

Shifting our attention to OPG, Fig. 5 illustrates the structure of an optimal solution to a problem with three types of robots with capacities and costs being , , and , respectively. In this case, the majority of the deployed robots are of type with . Only one type and one type robots are used. The four perimeter segments are covered by three robot groups. The only type robot guards (the purple segment) across two different perimeter segments. Coverage length adjustment is also performed to avoid the unnecessary coverage of some gaps.

Figure 5: An OPG problem and an associated optimal solution. The problem has four (red) perimeter segments and three types of robots with (orange), (green), and (purple), respectively.

5.2 A Robotic Guarding and Patrolling Application

In this subsection, as a potential application, the DP algorithms for OPG and OPG are employed to solve the problem of securing the perimeter of the Edinburgh castle, an example used in [1]. As shown in Fig. 6 (minus the orange and green segments showing the solutions), the central region of the Edinburgh castle has tall buildings on its boundary (the blocks in brick red); these parts of the boundary are the gaps that do not need guarding. In the figure, the top sub-figure shows the optimal solution for an OPG instance and an OPG instance with a total of robots. The bottom sub-figure is a slightly updated OPG instance with slightly higher .

Figure 6: [left] OPG solution with and OPG solution with , and total boundary . Cost of OPG solution is . [right] OPG solution with . Cost of solution ( type , type ) is . In both solutions, covers by type (resp., type ) robots are shown in orange (resp., green).

It can be observed that the results, while having non-trivial structures, make intuitive sense. For the top sub-figure, solutions to both OPG and OPG (because robot with larger capacity is slightly lower in relative cost) use mainly higher capacity robots to cover longer perimeter segments and use the lower capacity robots mostly fillers. The solution covers a small gap at the bottom. For the bottom sub-figure, while only small changes are made to the cost, because the longer segment is more expensive to use now, the first type of robot is used mainly.

5.3 Computational Performance

With Section 4 fully establishing the correctness and asymptotic complexity of the pseudo-polynomial time algorithms, here, the running time of these algorithms are experimentally evaluated. In doing so, the main goal is demonstrating that, despite the hardness of OPG and OPG, the proposed algorithms could solve the target problems under reasonably broad settings in a scalable way. For results presented in this subsection, each data point is an average over 10 randomly generated instances.

The first two numerical evaluations (Table 1 and Table 2) focus on the running times of the pseudo-polynomial time algorithms for OPG over single and multiple perimeters, respectively. In these two tables, and are the number of types and the number of segments, respectively. For each type , a capacity () is randomly sampled as an integer between and , inclusive. The number of robots available for each type () is sampled uniformly between and , inclusive. For the multiple perimeters case, the parameter represents the number of perimeters for a given instance.

For the single perimeter case (Table 1), the results show that the pseudo-polynomial time algorithm is effective for up to five types of robots, for dozens of robots. We expect a more efficient (e.g., C++ based) implementation should be able to effectively handle up to five types of robots with the total number of robots being around a hundred, on a typical PC. This is likely sufficient for many practical applications which have limited types and numbers of robots. Since the algorithm has exponential dependency on , it becomes less efficient for larger as expected.

\diagbox  5  10  20  30  40  50
2 0.022 0.044 0.131 0.208 0.326 0.516
3 0.281 0.714 1.670 2.577 4.107 4.708
4 5.504 16.07 41.68 71.55 109.9 138.9
5 29.53 75.60 243.6 443.4 528.0 725.0
Table 1: Running time in seconds used by the DP algorithm for OPG over a single perimeter.

Table 2 illustrates the running time of the DP algorithm for OPG over multiple perimeters. As can be readily observed, the impact of the number of perimeters on the running time is relatively small; the number of robot types is still the determining factor for running time. In this case, our proposed solution is effective for up to and starts to slow down a robot types become larger than .

\diagbox 10 20 30
   =3 =4 =3 =4    =3    =4
2 3.148 133.2 7.077 198.4 10.33 260.0
3 4.828 194.1 10.125 290.6 15.52 376.7
4 6.131 256.8 12.485 381.3 19.75 514.3
5 7.622 321.7 15.355 476.2 24.31 605.8
Table 2: Running time in seconds used by the DP algorithm for OPG over multiple perimeters.

Table 3 provides performance evaluation of OPG-mc-DP. Since there is no difference between single and multiple perimeters for OPG, only problems with single perimeters are attempted. Here, for each robot type, the cost is an integer randomly sampled between and , and the capacity is computed as five times the cost plus a random integer between and . In the table, , the total length of the entire boundary. Given OPG’s lower computational complexity, the DP algorithm, OPG-mc-DP, can effectively deal with over a few hundred types of robots with ease.

\diagbox
=20 =50 =20 =50 =20 =50
3 0.006 0.064 0.041 0.098 3.040 3.144
10 0.005 0.066 0.094 0.155 9.423 9.409
30 0.009 0.070 0.261 0.320 26.10 28.59
100 0.014 0.077 0.910 0.969 91.28 93.20
300 0.030 0.091 2.652 2.938 275.6 270.7
Table 3: Running time in seconds used by OPG-mc-DP algorithm.

6 Conclusion and Discussions

In this paper, we investigate two natural models of optimal perimeter guarding using heterogeneous robots, where one model (OPG) limits the number of available robots and the second (OPG) seeks to optimize the total cost of coverage.

These formulations have many potential applications. One application scenario we envision is the deployment of multiple agents or robots as “emergency responders” that are constrained to travel on the boundary. An optimal coverage solution will then translate to minimizing the maximum response time anywhere on the perimeter (the part that needs guarding). The scenario applies to OPG, OPG, and OPG.

Another application scenario is the monitoring of the perimeter using robots with different sensing capabilities. A simple heterogeneous sensing model here would be robots equipped with cameras with different resolutions, which may also be approximated as discs of different radii. The model makes sense provided that the region to be covered is much larger than the sensing range of individual robots and assuming that the boundary has relatively small curvature as compared to the inverse of the radius of the smallest sensing disc of the robots. For boundary with relatively small curvature, our solutions would apply well to the sensing model by using the diameter of the sensing disc as the 1D sensing range. As the region to be covered is large, covering the boundary will require much fewer sensors than covering the interior.

On the computational complexity side, we prove that both OPG and OPG are NP-hard, with OPG directly shown to be strongly NP-hard. This is in stark contrast to the homogeneous case, which admits highly efficient low polynomial time solutions [1]. The complexity study also establishes structural similarities between these problems and classical NP-hard problems including -Partition, Knapsack, and Subset Sum.

On the algorithmic side, we provide methods for solving both OPG and OPG exactly. For OPG, the algorithm runs in pseudo-polynomial time in practical settings with limited types of robots. In this case, the approach is shown to be computationally effective. For OPG, a pseudo-polynomial time algorithm is derived for the general problem, which implies that OPG is weakly NP-hard. In practice, this allows us to solve large instances of OPG. We further show that a polynomial time algorithm is possible for OPG when the types of robots are fixed.

With the study of OPG [1] for homogeneous and heterogeneous cases, some preliminary understanding has been obtained on how to approach complex 1D guarding problems. Nevertheless, the study so far is limited to one-shot settings where the perimeters do not change. In future research, we would like to explore the more challenging case where the perimeters evolve over time, which requires the solution to be dynamic as well. Given the results on the one-shot settings, we expect the dynamic setting to be generally intractable if global optimal solutions are desired, potentially calling for iterative and/or approximate solutions.

We recognize that our work does not readily apply to a visibility-based sensing model, which is also of interest. Currently, we are also exploring covering of the interior using range-based sensing. As with the OPG work, we want to push for optimal or near-optimal solutions when possible.

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