Swarm Control for Distributed Construction: A Computational Complexity Perspective

The human computational effort in human-robot interaction has been investigated using the notion of cognitive complexity developed in [30, 31] (see also [28, Section III-A]). This scheme adapts the \(O(f(n))\) (“Big-Oh”) asymptotic worst case time complexity notation from computational complexity analysis to describe three categories of tasks in terms of human operator effort in swarm control² [28, page 13]:

These categories essentially characterize both the degree to which co-ordination among swarm members can be automated and whether or not a human-in-the-loop system can be scaled to larger numbers of robots [30, page 139]. Swarm control situations requiring \(O(1)\) cognitive effort are considered the most desirable, and guidelines for designing and dealing effectively with all three categories of systems are given on pages 162–163 of [30]. Though useful, this cognitive complexity scheme is incomplete—this is so because, as noted on pages 13–14 of [28], the primary purpose of this scheme is to emphasize the attention effort of the human operator in controlling a multi-robot system, and other factors (e.g., human-robot communication, swarm state estimation and prediction, design of swarm control inputs) may increase the overall required cognitive effort of swarm control greatly.

There are many tasks relative to which swarm robotics and control can be investigated [4, 5]. Given that some of the most impressive activities performed by insect swarms involve creating structures, e.g., wasp and termite nests, it is not surprising that distributed construction has been of key interest. From the start of such research over 25 years ago [48], the focus has by large been on construction by fully autonomous robot swarms (Collective Robotic Construction (CRC)), e.g., [2, 19, 39, 45, 47, 67]. Research on CRC has in the past been restricted to academia, in large part because of the conservatism of the real-world construction industry and its concern with safety issues when robots and humans work in shared environments [3, 43]. Recent research has however focused much more on Collective Human-Robot Construction (CHRC) [20, 49, 51]. CHRC is seen as both an important research challenge in CRC [39, Page 5] as well as a critical technology for fully integrating robots into real-world construction. Though many researchers see the role of robots in CHRC primarily as assistants to human beings in structure fabrication [39, Page 10], there is a growing number who foresee opportunities for humans and robots to collaborate in designing structures as well [20, 49].

Various work has been done on the computational complexity of both verifying if a given multi-entity system can perform a task and designing such systems for tasks [1, 8, 12, 13, 24, 46, 69]. Seven complexity-theoretic articles to date incorporate both autonomous robots and a suitably simple and explicit model of robot architecture and environment that allows investigation of possible restrictions that could yield tractability [50, 59, 60, 61, 62, 63, 64]. Three of these [59, 61, 63] consider navigation tasks performed by robots with deterministic Brooks-style subsumption [59, 61] and single-state finite-state [63] reactive controllers, respectively. The other four consider construction-related tasks performed by robots with deterministic finite-state controllers relative to the robot controller and environment design in a given environment where robot controllers are designed from scratch [62, 64], robot swarm design where swarm members are designed by selection from a provided library [60, 64], and robot swarm/environment co-design where swarm members are designed by selection from a provided library [50]. No complexity-theoretic analysis to date has considered the computational difficulty associated with swarm control.

In this article, we present the first computational complexity analyses of problems that are generalizations of the algorithm, environmental influence, and leader selection swarm control methods described above relative to the distributed construction task [39]. These problems can be stated informally as follows (and are described in more detail in Sections 2.2 and 2.3):

We consider the following types of efficient solvability (described in more detail in Section 3):

Our analyses are done in terms of the distributed construction model defined in [62] in which robots equipped with finite-state controllers move in a synchronous non-continuous manner in grid-based 2D environments and swarms are restricted to complete their construction tasks in a polynomial number of timesteps. Relative to the types of solvability listed above and various conjectures that are widely believed to be true within the Computer Science community, e.g., \(P \ne NP\) [16, 18], we prove the following results (Section 3):

As is discussed in Section 4.1, our use of simplified models, as well as our problem- rather than algorithm-oriented method of analysis, has exposed what we believe to be under-appreciated underpinnings of the computational difficulty of swarm control arising from the combinatorial choices in swarm control inputs. The implications of our results for swarm control assistance software tools and the human control of swarms are discussed in Sections 4.3 and 4.4, respectively; as part of the latter, we also describe how computational complexity analyses such as ours may allow a detailed and comprehensive assessment of the human cognitive effort associated with swarm control.

Two issues with respect to the results listed above and their derivation should be noted. First, to aid readability, various definitions and lemmas stated in previous articles are repeated here. In the interests of concision and avoiding self-plagiarism, abbreviated forms are often used (most notably, in Section 2.1, the “short form” of definitions associated with our distributed construction model as given in [50]). Second, as some proofs of results presented here are related to proofs given previously in [60, 62], there are potential concerns about the originality of results presented in the current article. These relationships can be categorized as follows:

The results derived relative to the proofs in the first category (parts of Results A.2, A.4, and A.5 and all of Results A.1, B.SA.1, B.SA.2, B.SE.1–4, C.SE.1, C.SE.2, and C.SA.7) are outnumbered two to one by the results derived relative to the second (Results B.SA.4, B.SL.4, C.SA.2, and C.SL.3) and third (parts of Results A.2, A.4, and A.5 and all of Results X.1, A.3, B.SA.3, B.SA.5–9, B.SE.5, B.SL.1–3, B.SL.5, C.SA.1–6, B.SE.3, C.SL.1–4, C.SE.4. and C.SE.5) categories. Hence, though some of the results presented here resemble those presented previously in [60, 62], the majority of the results in the current article are in fact new.

This article is organized as follows. In Section 2, we describe our robot controller, environment, target structure, and swarm control models and use these to formalize the swarm control problems for distributed construction that we will analyze. In Section 3, we consider viable algorithmic options for these problems relative to several popular types of exact unrestricted, approximate unrestricted, and exact restricted efficient solvability. All proofs of stated results are given in the appendix. In Section 4, we discuss a variety of issues raised by our results, including the completeness and generality of these results (Sections 4.1 and 4.2, respectively) and the implications of these results for both swarm control assistance software tools (Section 4.3) and human control of swarms in general (Section 4.4); the last of these also describes how computational complexity analysis (and in particular parameterized complexity analysis) can be used to create a detailed and comprehensive assessment of human cognitive effort for swarm control. Finally, our conclusions and directions for future work are given in Section 5.

We here review the basic entities in our model of distributed construction by robot swarms — namely, environments, target structures, individual robots, and robot swarms. In the interests of concision and avoiding self-plagiarism, this review is the “short form” originally given in [50]; readers wishing more details and examples should consult [60, 62].

The basic entities in our model are as follows:

Notions of deterministic and time-bounded robot and swarm operation were introduced in [62] to ensure that requested structures are created by swarms reliably and quickly. We use the notion of deterministic robot and swarm operation introduced in [62] as extended in [50] (i.e., requiring that at any time as the swarm operates in an environment, all transitions enabled in a robot relative to the current state of that robot perform the same environment modifications and progress to the same next state).⁴

We also retain the second notion, encapsulated as \((c_1,c_2)\)-completability, which requires that each swarm complete its task within \(c_1|E|^{c_2}\) timesteps for constants \(c_1\) and \(c_2\). However, we here make two modifications. First, as was done in [50], we broaden the timestep bound to \(c_1(|E| + |Q|)^{c_2}\), in order to accommodate FSR that makes a number of internal state-changes without moving. Second, we fix a technical flaw in our previous articles. In those articles, \(c_1\) and \(c_2\) were part of the inputs to our analyzed problems. Following standard practice in encoding problem inputs [18, Page 10], base-10 decimal values \(c_1\) and \(c_2\) were encoded in base-2 binary, e.g., \(c_1 = 7_{dec} = 4 + 2 + 1 = 2^2 + 2^1 + 2^0 = 111_{bin}\). Note that binary notation is of length logarithmic in the value, e.g., \(|111_{bin}| = 3 \approx \log _2 7_{dec}\). This has the unfortunate effect of rendering the task completion bound exponential rather than polynomial in the input size, as \(c_1(|E| + |Q|)^{c_2} = 2^{\log _2 c_1}(|E| + |Q|)^{2^{\log _2 c_2}} = 2^{|c_{1,bin}|}(|E| + |Q|)^{2^{|c_{2,bin}|}}\). One can get around this by assuming that \(c_1\) and \(c_2\) are encoded in the input in base-1 unary, e.g., \(7_{dec} = 1111111_{un}\). However, this has the equally unfortunate effect of allowing exponential-time computations to be artificially reduced to polynomial time by “padding” the input length with large unary encoded values [18, Pages 9–10]. To avoid both of these difficulties, we chose here to remove \(c_1\) and \(c_2\) from the problem input and assume that suitable constant values of \(c_1\) and \(c_2\) are specified beforehand. To ensure generous but still low-order polynomial swarm runtime bounds, we will assume that \(c_1 = 10\) and \(c_2 = 3\).

Given this model of distributed construction, we say that the task of constructing structure \(X\) at position \(p_X\) in \(E\) is \((c_1,c_2)\)-completable by a swarm \(T\) relative to a positioning \(p_I\) of \(T\) in \(E\) if the members of \(T\), when started at \(p_I\), operate as dictated by their control algorithms and succeed in constructing \(X\) at \(p_X\) within at most \(c_1(|E| + |Q|)^{c_2}\) timesteps. In the next subsection, we will similarly formalize how the operation of swarms during this construction process can be modified by the swarm control methods described in Section 1.

In this article, we shall focus on three of the four methods of swarm control described in Section III-D of [28]—namely, algorithm selection, environment influence selection, and leader selection. In the literature, variants of each of these methods have been investigated:

Rather than analyze each of these variants separately, we will instead analyze the following generalizations of these methods⁵:

Note that these generalized methods have all studied variants as special cases (with transient environmental changes simulated by later environmental changes that undo earlier ones and explicit indication of leaders simulated via leader-made environmental modifications). Hence, by the logic of problem generalization laid out in Section 1, these methods can (under the appropriate restrictions) be used to analyze those variants. When combined with additional restrictions (see parameters \(t_{\max }\), \(k_a\), \(k_e\), and \(k_l\)), these methods will also allow particularly fine-grained analyses of the computational complexity associated with the time-periods in which controller algorithm- and environment-changes take place and the maximum numbers of allowable changes.

Given the above, consider the following formalizations of these three generalized methods of swarm control:

All changes are assumed to occur at or before timestep \(t_{\max }\), where \(1 \le t_{\max } \le c_1(|E| + |Q|)^{c_2}\) and \(c_1\) and \(c_2\) are the constants used to specify \((c_1,c_2)\)-completability. There may be multiple changes occurring in a particular timestep. In the cases of algorithm and environmental influence selection, all changes are assumed to occur simultaneously at the beginning of a timestep prior to swarm member environment sensing and action. In the case of leader selection, changes are assumed to occur simultaneously during the swarm member action phase, such that only leaders move with respect to the specified changes and all other swarm members move as specified by their controller algorithms. Given this, no two changes in a timestep can affect the same entity. If \(t_{\max } = 1\), all changes occur in the first timestep and the relative timing of changes is not an issue; however, this is most certainly not the case when \(t_{\max } \gt 1\), as factors like necessary swarm stabilization time after certain operator commands and degradation in swarm member performance between operator commands mean that the same changes under different timings can have radically different and sometimes even adverse effects on swarm behavior [28, page 19].

We can now fuse the model of distributed construction in Section 2.1 with our formalizations of swarm control methods in Section 2.2 to define the computational problems which will analyze in this article:

Robot Algorithm Selection (SelAlg)

Input: An environment \(E\) based on square-type set \(E_T\), an FSR swarm \(T\) based on controllers from library \(L\), an initial positioning \(p_I\) of \(T\) in \(E\), a structure \(X\), a position \(p_X\) of \(X\) in \(E\), and positive integers \(t_{\max }\) and \(k_a \le |T| \times t_{\max }\).

Output: A set \(C_A\) of \(k_a\) algorithm changes to \(T\) relative to \(L\) and \(t_{\max }\) such that the task of constructing \(X\) at \(p_X\) when \(T\) is initially positioned at \(p_I\) and the changes in \(C_A\) are subsequently applied is \((c_1,c_2)\)-completable, if such a \(C_A\) exists, and special symbol \(\bot\) otherwise.

Environmental Influence Selection (SelEnvInf)

Input: An environment \(E\) based on square-type set \(E_T\), an FSR swarm \(T\), a structure \(X\), an initial positioning \(p_I\) of \(T\) in \(E\), a position \(p_X\) of \(X\) in \(E\), and positive integers \(t_{\max }\) and \(k_e \le |E| \times t_{\max }\).

Output: A set \(C_E\) of \(k_e\) environmental changes to \(E\) relative to \(E_T - \lbrace e_X\rbrace\) and \(t_{\max }\) such that the task of constructing \(X\) at \(p_X\) in \(E\) when \(T\) is initially positioned at \(p_I\) and the changes in \(C_E\) are subsequently applied is \((c_1,c_2)\)-completable, if such a \(C_E\) exists, and special symbol \(\bot\) otherwise.

Robot Leader Selection (SelLead)

Input: An environment \(E\) based on square-type set \(E_T\), an FSR swarm \(T\), a structure \(X\), an initial positioning \(p_I\) of \(T\) in \(E\), a position \(p_X\) of \(X\) in \(E\), and positive integers \(t_{\max }\) and \(k_l \le |T| \times t_{\max }\).

Output: A set \(C_L\) of \(k_l\) changes to the positions of members of \(T\) relative to \(t_{\max }\) such that the task of constructing \(X\) at \(p_X\) in \(E\) when \(T\) is initially positioned at \(p_I\) and the changes in \(C_L\) are subsequently applied is \((c_1,c_2)\)-completable, if such a \(C_L\) exists, and special symbol \(\bot\) otherwise.

Unless stated otherwise, all swarm operations in this article will be synchronous. Without loss of generality, we will assume that each member of \(T\) starts operating in the initial state \(q_0\) of its associated controller, either at the start of the construction task or whenever that controller is changed. Moreover, in the case of problem SelLead, if the execution of a change causes two robots in a swarm to attempt to occupy the same space or an individual robot to occupy the same space as an obstacle then the execution of the task terminates and is considered unsuccessful. Note that problems SelAlg and SelEnvInf when \(t_{\max } = 1\) (i.e., all algorithm- and environment-changes are done prior to the first timestep) are for all intents and purposes problems DesCon and EnvDes investigated previously in [60] and [62], respectively. This similarity will allow us to easily adapt several previously-published result proofs for DesCon and EnvDes to derive analogous results for SelAlg and SelEnvInf (most notably, Results B.SA.1, B.SA.2, and B.SE.1–B.SE.4) in Section 3 (see Section 1.2 for further discussion of this issue).

At this point, the reader may be concerned that the problems we analyze are overly abstract, both in the synchronous non-continuous manner in which our robots operate in 2D grid-based environments and their ignoring of known issues in the implementation of swarm control such as communication latency and bandwidth between human operators and swarm members [28, Section III-B] and swarm state estimation and prediction by human operators in the face of incomplete observations [28, Section III-C] (courtesy of our assumptions of complete swarm observability, deterministic swarm operation, and \((c_1, c_2)\)-completability of tasks). We do not deny that these issues are critical to real-world swarm control. However, as we shall show in the remainder of this article, it is such simplifications in tandem with our problem- rather than algorithm-oriented method of analysis that will allow us to focus on and investigate what we believe to be under-appreciated underpinnings of the computational difficulty of swarm control, in particular those arising from the combinatorial choices in swarm control inputs.

Let us first consider the completeness of our results ordered by type of solvability. Our results with respect to exact polynomial-time solvability give us our first surprise. Though SelAlg is polynomial-time tractable when \(h = t_{\max } = 1\), i.e., the final swarm is homogeneous and all controller changes required to make it so happen in the first timestep (and hence there are no issues related to change the timing) (Result A.1), SelAlg is not polynomial-time tractable when \(h = 2\) and \(t_{\max } = 1\), and neither are SelEnvInf or SelLead when \(h = t_{\max } = 1\) (Result A.2). That this intractability holds at the lowest possible values of \(h\) and \(t_{\max }\) emphasizes that swarm heterogeneity and timing of swarm inputs, both of which have been assumed to be central to the difficulty of swarm control ([28, page 16] and [28, Section III-F], respectively), are in fact not — intractability remains even if these (and, as noted at the end of Section 2.3, issues related to human-swarm communication and swarm state estimation and prediction by human operators in the face of incomplete observations) are factored out. Moreover, this intractability is not a product of “bad actor” instances of our problems, which have no solutions and hence force needless full searches of the solution space, but holds even for instances where solutions are known to exist (Result A.3). Our results thus expose what we believe to be an under-appreciated factor in the intractability of swarm control—namely, the combinatorial choice in swarm control inputs. This is perhaps not surprising, given that the swarm- and environment-design problems DesCon and EnvDes from [60, 62] (in which combinatorial choice is arguably more obvious) are intractable and special cases of problems SelAlg and SelEnvInf analyzed here. This combinatorial choice is also powerful enough to rule out both frequently deterministic and frequently probabilistic types of polynomial-time approximate solvability (Results A.4 and A.5, respectively).¹²

The situation is more optimistic with respect to fixed-parameter and \(|x|^k\) solvability. Our results show that all three of our problems are fp-intractable, both relative to each of the parameters in Table 1 and in many combinations of these parameters (Results B.SA1–B.SA.7, B.SE.1–B.SE.4, and B.SL.1–B.SL.4); moreover, this fp-intractability holds when many of the parameters have constant values (see Tables 2–5). That being said, we do have several combinations of parameters that yield fp-tractability for each of our problems (Results B.SA.8, B.SA.9, B.SE.5, and B.SL.5). In the case of \(|x|^k\)-tractability, intractability holds relative to a number of combinations of the parameters in Table 1 (Results C.SA.1–C.SA.5, C.SE.1, C.SE.2, and C.SL.1–C.SL.3). Though we do have tractability relative to each of \(k_a\), \(k_e\), and \(k_l\) (Results C.SA.6, C.SE.3, and C.SL.4), tractability in the most useful case when the exponent functions in the algorithm runtimes are sublinear seems unlikely (Results C.SA.7, C.SE.4, and C.SL.5).

Though a number of additional fp- and \(|x|^k\) tractability and intractability results can be derived from our given results courtesy of Lemmas 3.1–3.4, our results are incomplete in two senses:

However, even these incomplete results allow a much more nuanced view of the computational difficulty of swarm control for distributed construction. For now, this is via fp-tractability results that are minimal, i.e., fp-tractability results \(\langle K \rangle\)-X such that X is fp-intractable relative to all proper subsets \(K^{\prime }\) of \(K\). Such minimal fp-tractability results are useful because they indicate problem mechanisms that interact with each other (such that restrictions in one can be tradedoff against not restricting others to maintain intractability, and tractability only occurs if all are restricted together). At present, we have two minimal fp-tractability results:

Choice in swarm control input timing is encoded in parameter \(t_{\max }\) and choice in swarm control inputs is encoded in parameters \(|T|\) and \(|L|\) (observe that in problem SelLead, there are four choices of movement of a selected robot (north, south, east, and west) and hence \(|L|\) is effectively four). Our results show that intractability in SelAlg and SelLead arises not from choices of either timing or swarm control inputs but rather a combination of the two; hence, attempts to get tractability by restricting only one of these aspects (see Section 4.3) are doomed to failure. Hints of other mechanism interactions may also be implicit in certain pairs of intractability results (as visualized in Tables 2–5), where restriction-tradeoffs between pairs of parameter-sets that maintain intractability seem to be occurring, e.g., \(\lbrace |T|, h\rbrace\) vs. \(\lbrace |Q|, d, |f|, r, |L|\rbrace\) for SelAlg (Results B.SA.2 and B.SA.3), \(\lbrace |Q| \rbrace\) vs. \(\lbrace r \rbrace\) for SelEnvInf (Results B.SE.1 and B.SE.2), \(\lbrace |Q|, d\rbrace\) vs. \(\lbrace |f| \rbrace\) for SelLead (Results B.SL1. and B.SL.2). It would be most interesting to see if this interaction can be verified, either via minimal fp-tractability or some other recognizable tractability or intractability result-pattern.

These issues suggest that it would be most useful to first complete our analyses by determining the fp-status of our problems relative to all combinations of parameters in Table 1, i.e., performing a systematic parameterized complexity analysis (SPCA) [58] for each of our problems. Experience has shown that the effort involved in performing such SPCA can be minimized courtesy of Lemmas 3.1 and 3.2 by deriving fp-tractability and fp-intractability results relative to the smallest and largest possible sets of parameters, respectively. The benefits in turn would be many, including not only resolution of mechanism interaction issues mentioned above (the most notable of which is the fp-minimality of \(\langle |E|, |E_T|, t_{\max }\rangle\)-SelAlg (Result B.SE.5)) but also determining if other parameter-restricted situations are fp-tractable in addition to those we know of now, as they would yield new and possibly much better algorithms for our problems. Once this SPCA is done, we can then more profitably address the issue of sharpening the frontiers of polynomial-time and fp-tractability for our problems.

Let us now consider the generality of our tractability and intractability results. A valid objection is that these results are of limited use as they were derived relative to simplified models. However, as noted previously in [60], it turns out that these results have a broad applicability because our models are special cases of a number of more realistic models (see [60, Section 5] for details), e.g.,

More realistic versions of our problems can be created by replacing any combination of simple special-case models with the more general models above. Courtesy of this special-case relationship, any automated system that solves such a more realistic problem \(\Pi ^{\prime }\) can also solve the original problem \(\Pi\) defined relative to the simple special-case models analyzed here. Intractability results for \(\Pi\) then also apply to \(\Pi ^{\prime }\) as well as the operation of any automated system solving \(\Pi ^{\prime }\) (see [60, Section 5] for details).¹³ Tractability results typically do not propagate from special cases to more general problems, as algorithms often exploit particular details of the inputs and outputs in their associated problems to attain efficiency or even work at all. That being said, some of our tractability results also have surprisingly broad applicability because the algorithms described in their proofs depend only on the combinatorics of the number of possible swarm control input choices.

Though all of this applicability noted here is for now limited to simple systems in which swarms move in a non-continuous deterministic manner in grid-based environments and perform construction tasks, there is no reason to expect that our analyses cannot be extended to more realistic swarm models and swarm control in the service of non-construction tasks, e.g., foraging, consensus, navigation (for further discussion on this point, see Section 5.2 in [60]). The latter is particularly important, as such tasks are typically the focus of swarm control research to date. Our results have already made the following first steps in this:

This is not to say that the full pattern of intractability results we have observed relative to the distributed construction task and the navigation task described above necessarily holds for simpler task such as consensus or foraging; indeed, we suspect it does not. However, this can and should be investigated using computational complexity analysis, to establish the frontiers of polynomial-time tractability for swarm control with respect to the types of tasks being performed. It is our hope that the analysis framework we have used and at least some of the proof techniques we have developed will be of use in this endeavor.

In any case, as was pointed out in Section 4.1, our simplified model has already proven most useful in allowing us (as was promised at the end of Section 2.3) to explore the sources of computational difficulty in swarm control for distributed construction. In the following two subsections, we will see how such explorations in turn have implications for both the design and deployment of software assistance tools for swarm control and the investigation of feasible options for the human control of swarms.

Modulo concerns about the simplifications in our models relative to real-world robotics, our tractability results have implications for the design and deployment of software tools to assist human operators in swarm control. In order to maintain user attention, it is known that the time taken by a software system to provide requested advice to humans users should be under 10 seconds [38]. As all of our problems are polynomial-time intractable in general (Result A.2) and do not even allow fast heuristic algorithms that are frequently correct in the most desirable deterministic or probabilistic senses (Results A.4 and A.5, respectively), we need to look at restricted swarm control situations for algorithm runtime efficiency. At present, our fixed-parameter and \(|x|^k\) tractability results suggest several such options:

Options 1–3 would work best either for small overall swarms or designated subswarms of larger swarms performing temporally (in the case of Algorithm and Leader Selection) or spatially (in the case of Environmental Influence Selection) limited tasks. Indeed, such algorithms would be very useful in implementing control of large swarms by instead controlling collections of independent subswarms [28, pages 20–22]. Regardless of other factors, if it is known that very few swarm control inputs need to be suggested (as may be the case in temporally or spatially limited tasks), option 4 may be best.

Our analyses can also be used to analyze and mitigate encountered difficulties with existing software tools for swarm control. For example, consider the tool designed in [32, 36] to address the following problem [36, page 2,674]:

This tool actually solves the simpler version of this problem in which the set \({\bf T}\) of times at which behaviors can be switched are given as part of the input. Though this tool gives optimal behavior sequences relative to the given performance criterion [36, Theorem 2], it was too slow for real-time supervisory control of a real-world swarm [36, page 2,681]. This problem was just barely mitigated (i.e., tool latency times were reduced to 2–8 seconds) by extensive parallelization and GPU programming when \(|{\bf B}|\) \(= 7\) [32, page 3,808], and even then, human participants in experiments were mostly unwilling to use the sequencing algorithm and found it less usable then generating the sequences manually [32, page 3,811]. To us, these difficulties are not surprising, as the proof of Result B.SA.4 can be easily modified to show that the problem solved by the tool in [32, 36] is polynomial-time intractable (albeit for a distributed construction task under our simplified swarm-operation model). This suggests that a complement to algorithm optimization efforts such as those described in [32] would be to perform a computational complexity analysis like those in this article to establish other possibly more useful algorithm options. Such algorithms for restricted swarm control situations might be a valuable part of proposed research involving incremental construction of behavior change sequences [32, page 3811]. This analysis could also include looking at speeding up the bounded sub-optimal algorithm sketched on page 2,677 of [36] under parameter restrictions using techniques described in [33].

A final note is in order here regarding the admittedly impractical runtimes of the algorithms underlying our presented fixed-parameter and \(|x|^k\) tractability results. As noted previously in [60, Section 5], this is an artifact of the goal of the analyses in our article, which is to provide a general overview of the options for restricting problems to obtain practical algorithms; it is the job of future research to derive the best possible algorithms relative to those sets of restrictions for which we know algorithms exist. There are a number of established techniques for deriving such algorithms [7, 15, 37], and it has been observed multiple times within the parameterized complexity community that once fixed-parameter tractability is established, these techniques are applied by different groups of researchers in “FPT Races” to produce increasingly (and, on occasion, spectacularly) more efficient algorithms [29]. For example, once it was realized in 1992 that the polynomial-time intractable problem Vertex Cover is fixed-parameter tractable relative to the parameter \(k\) encoding the size of the wanted vertex cover, the \(2^kk^{2k+2} + k|V|\) runtime of the initial fixed-parameter algorithm was bettered over the next 20 years to \(1.2738^k + k|V|\). It seems reasonable to conjecture that, if there is sufficient interest, such an improvement could also happen with respect to the fixed-parameter and \(|x|^k\) tractability results for swarm control given in this article.

In the previous section we considered the implications of our complexity results for swarm control assistance software tools and observed that such tools face the intractability inherent in controlling multiple autonomous and interacting robots. Instead of trying to fully automate and offload swarm control to a computer program, might it be more realistic for a human controller to tackle these problems effectively and efficiently in real-time? Our intractability results suggests this is unlikely to be possible.

This is so because arguably humans, like any resource-bounded system, cannot solve intractable problems efficiently [54]. While it is commonly believed that humans may be able to solve such problems approximately or “good enough”, such claims turn out to be unfounded: of course humans do use various heuristics or other efficient strategies for problem-solving, but if a well-defined computational problem such as those analyzed here is computationally intractable (i.e., \(NP\)-hard (see the Appendix)), then no such efficient strategies, either deterministically or probabilistically, can yield provably good bounds on the degree of approximation [54, 56, 57] (see also Results A.4 and A.5).

Previous work has already considered the cognitive complexity of human control of autonomous swarms [30]. While these analyses are informative about how the amount of global attention needed to be distributed over the different controlled robots, these analyses are also in a sense limited. This is because those analyses effectively assume that deciding how to control is cognitively for free, in that Lewis’ big-Oh scheme (see Section 1.1) only expressed the attentional resources demands for executing chosen control actions, and not the computational complexity of the swarm control input selection problem itself. However, we have shown in this article that the bulk of the computational complexity of swarm control is hidden in this selection problem.

So is human swarm control hopeless? Most certainly not. This is because, as proposed in [52], humans can, at least in principle and under ideal conditions, efficiently solve fixed-parameter tractable parameterized versions of problems such as those listed in Section 4.3. This does, however, impose strong constraints on swarm design and the design of their control interfaces, as they will need to meet exactly those constraints that render the swarm control problem tractable for humans. Analogous to Lewis’ guidelines on how to structure swarms to meet attentional resource limits given on pages 162–163 of [30], we propose to design swarms and the control interfaces to meet the cognitive resource limits on swarm control input selection noted in Section 4.3. For the simple scenarios modeled here this means that our swarm control input selection problems need to be constrained such that the human controller selects

This also demonstrates how computational complexity analysis, and in particular parameterized complexity analysis, can be used to create the novel notion of cognitive complexity for swarm control requested on page 23 of [28] that takes into account not only problem aspects such as system size, task difficulty, and levels of automation but also their respective interactions (along the lines sketched in Section 4.1).

The above poses new and most interesting challenges for swarm control interface design. It would also be interesting to explore the links between our computational complexity results and human performance measures in empirical research, both past, and future. We think this latter research will enrich the human factors literature on this topic because as noted by [28, page 23], this literature has to date focused on performance aspects such as fatigue, attention, and task switching costs, whereas the types of complexity results discussed here address more fundamental competence limitations [17].

All of our intractability results will be derived using polynomial-time and parameterized reductions. Given two problems \(\Pi\) and \(\Pi ^{\prime }\), a polynomial-time reduction from \(\Pi\) to \(\Pi ^{\prime }\) [18] is a polynomial-time algorithm for transforming instances of \(\Pi\) into instances of \(\Pi ^{\prime }\) such that any polynomial-time algorithm for \(\Pi ^{\prime }\) can be used in conjunction with this instance transformation algorithm to create a polynomial-time algorithm for \(\Pi\). Analogously, a parameterized reduction from \(\langle K \rangle\)-\(\Pi\) to \(\langle K^{\prime }\rangle\)-\(\Pi ^{\prime }\) [11] allows the instance transformation algorithm to run in fp-time relative to \(K\) and requires that (1) for each \(k^{\prime } \in K^{\prime }\) that there is a function \(g_{k^{\prime }}()\) such that \(k^{\prime } = g_{k^{\prime }}(K)\) and (2) the instance transformation algorithm can be used in conjunction with any fp-algorithm for \(\langle K^{\prime }\rangle\)-\(\Pi ^{\prime }\) to create an fp-algorithm for \(\langle K\rangle\)-\(\Pi\).¹⁵ The power of such reductions comes from the following observation: given a reduction from a problem \(\Pi\) to a problem \(\Pi ^{\prime }\) that preserves a particular type of tractability, if \(\Pi\) is not tractable then neither is \(\Pi ^{\prime }\) (otherwise, if \(\Pi ^{\prime }\) was tractable by some algorithm \(A\), \(A\) could be combined with the instance transformation algorithm in the reduction to prove \(\Pi\) tractable, which is a contradiction). If necessary, the “seed” intractable problem \(\Pi\) used in such a proof may only be known to be intractable if a particular conjecture holds, e.g., \(P \ne NP\), \(FPT \ne W[1]\); in those cases, the reduction-derived intractability result for \(\Pi ^{\prime }\) holds relative to that conjecture.

All of our reductions are from the following problems:

3-Satisfiability (3SAT) [18, Problem LO2]

Input: A set \(U\) of variables and a set \(C\) of disjunctive clauses over \(U\) such that each clause \(c \in C\) has \(|c| = 3\).

Question: Is there a satisfying truth assignment for \(C\)?

Weighted 3-satisfiability (W3SAT) [10]

Input: A set \(U\) of variables, a set \(C\) of disjunctive clauses over \(U\) such that each clause \(c \in C\) has \(|c| = 3\), and a positive integer \(k\).

Question: Is there a satisfying truth assignment for \(C\) of weight \(k\), i.e., a satisfying truth assignment in which exactly \(k\) variables are set to \(True\)?

Clique [18, Problem GT19]

Input: An undirected graph \(G = (V, E)\) and a positive integer \(k\).

Question: Does \(G\) contain a clique of size \(k\), i.e., is there a subset \(V^{\prime } \subseteq V\), \(|V^{\prime }| = k\), such that for all \(u, v \in V^{\prime }\), \((u,v) \in E\)?

Dominating set [18, Problem GT2]

Input: An undirected graph \(G = (V, E)\) and a positive integer \(k\).

Question: Does \(G\) contain a dominating set of size \(k\), i.e., is there a subset \(V^{\prime } \subseteq V\), \(|V^{\prime }| = k\), such that for all \(v \in V\), either \(v \in V^{\prime }\) or there is at least one \(v^{\prime } \in V^{\prime }\) such that \((v, v^{\prime }) \in E\)?

Let Dominating set\(^{PD3}\) be the version of Dominating set in which \(G\) is planar and each vertex has degree at most 3. The problems 3SAT, Clique, Dominating Set, and Dominating set\(^{PD3}\) are \(NP\)-hard in general (see problem references in [18] above); moreover problem Clique is \(W[1]\)-hard and problems W3SAT and Dominating set are \(W[1]\)-hard and \(W[2]\)-hard, respectively, relative to aspect-set \(\lbrace k\rbrace\) [10]. For each vertex \(v \in V\), let the complete neighborhood \(N_C(v)\) of \(v\) be the set composed of \(v\) and the set of all vertices in \(G\) that are adjacent to \(v\) by a single edge, i.e., \(v \cup \lbrace u ~ | ~ u ~ \in V ~ \rm {and} ~ (u,v) \in E\rbrace\). We assume below for each instance of Clique and Dominating set an arbitrary ordering on the vertices of \(V\) such that \(V = \lbrace v_1, v_2, \ldots , v_{|V|}\rbrace\); we also assume analogous orderings for the variables and clauses of each instance of 3SAT and W3SAT such that \(U = \lbrace u_1, u_2, \ldots , u_{|U|}\rbrace\) and \(C = \lbrace c_1, c_2, \ldots , c_{|C|}\rbrace\).

For technical reasons, all intractability results are proved relative to decision versions of problems, i.e., problems whose solutions are either “yes” or “no”. Though none of our problems defined in Section 2 are decision problems, each can be made into a decision problem by asking if that problem’s requested output exists; let that decision version for a problem X be denoted by X\({}_D\). The following four easily-proven lemmas will be useful below in transferring results from decision problems to their associated non-decision and parameterized problems; these lemmas follow from the observation that any algorithm for non-decision problem X can be used to solve X\({}_D\) and the definitions of fp- and \(|x|^k\)-tractability.

This subsection contains the proofs of all results cited in the main text. As noted in Section 1.2 in the main text, proofs of certain Lemmas are either straightforward adaptations of (Lemmas A.5, A.6, and A.13–A.16) or build on (Lemmas A.9 and A.20) proofs given previously in [60, 62]. In both cases, to avoid repeating the full proof and committing self-plagiarism, we first describe in general how the previous proof operated and then give the changes required for this article (which in the cases of Lemmas A.9 and A.20 are necessarily more detailed). All other proofs are given here in full detail.

We first address the issue of absolute versus operational FSR determinism discussed in Footnote 4 in the main text. Consider the following problem:

Input: An FSR \(X\) with state-set \(Q\) and sensory radius \(r\), a 2D grid \(G\), and environment square-type set \(E_T\).

Question: Is the operation of \(X\) not deterministic in some possible environment based on \(G\) and \(E\), in the sense that there is an environment \(E\) based on \(G\) and \(E_T\) and a state \(q \in Q\) such that \(X\) in state \(q\) can be positioned in \(E\) and at least two outgoing transitions from \(q\) are enabled that do not perform the same environment modifications and progress to the same next state?

We next consider the swarm control problems defined in Section 2.3.

This useful observation follows from the definition of \(|x|^k\)-tractability.

Results C.SA.1–C.SA.5, C.SE.1, C.SE.2, and C.SL.1–C.SL.3 in turn follow from this observation and the reductions in the proofs of Results B.SA.3–B.SA.7, B.SE.1, B.SE.2, and B.SL.1, B.SL.3, and B.SL.4, respectively, which show the \(NP\)-hardness of their associated problems when the listed aspects are of constant value.

Consider the following restricted form of parameterized reducibility.