A Distributed Algorithm for Constrained Multi-Robot Task Assignment for Grouped Tasks

Transcription

1 A Dstrbuted Algorthm for Constraned Mult-Robot Tas Assgnment for Grouped Tass Lngzh Luo Robotcs Insttute Carnege Mellon Unversty Pttsburgh, PA Nlanjan Charaborty Robotcs Insttute Carnege Mellon Unversty Pttsburgh, PA Kata Sycara Robotcs Insttue Carnege Mellon Unversty Pttsburgh, PA Abstract In ths paper, we present provably-good dstrbuted tas allocaton (assgnment) algorthms for a heterogeneous mult-robot system where the tass form dsjont groups and there are constrants on the number of tass a robot can do (both wthn the overall msson and wthn each tas group). Our problem s motvated by applcatons where multple robots wth heterogeneous capabltes have to wor together to accomplsh tass. Thus, for our purposes, a tas group s a compound tas composed of more than one atomc tass where one robot s requred for each atomc tas. Snce robots have lmted battery lfe, we assume that the number of (atomc) tass that a robot can do wthn a msson has an upper bound. Futhermore, each robot has a constrant on the number of tass t can do from each group (ths models the fact that multple robots may be needed to smultaneously perform the atomc tass that mae up the compound tas). Each robot obtans a payoff (or ncurs a cost) for each tas and the overall objectve for tas allocaton s to maxmze the total payoff (or mnmze the total cost) of all the robots. In general, exstng (centralzed or dstrbuted) algorthms for tas allocaton ether assume that (atomc) tass are ndependent, or do not provde performance guarantee for the stuaton where tas constrants exst. We show that our problem can be solved n polynomal tme by a centralzed algorthm by reducng t to a mnmum cost networ flow problem. We then present a decentralzed algorthm (that extends the aucton algorthm of Bertseas for lnear assgnment problems [1]) to provde an almost optmal soluton. We prove that our soluton s wthn a factor of O(n t ε) of the optmal soluton, where n t s the total number of tass and ε s a parameter that we choose (the guarantees are the same as that of the orgnal aucton algorthm for unconstraned tass). The decentralzed algorthm assumes a shared memory model of computaton that may be unrealstc for many mult-robot deployments. Therefore, we show that by usng a maxmum consensus algorthm along wth our algorthm, we can desgn a totally dstrbuted algorthm for tas allocaton wth group constrants. The ey aspect of our dstrbuted algorthm s that the overall objectve s (nearly) maxmzed by each robot maxmzng ts own objectve teratvely (usng a modfed payoff functon based on an auxlary varable, called prce of a tas). Our algorthm s polynomal n the number of tass as well as the number of robots. Index Terms Mult-robot assgnment, Tas allocaton, Aucton algorthm. I. INTRODUCTION For autonomous operatons of multple robot systems, tas allocaton s a basc problem that needs to be solved effcently [2], [3]. The basc verson of the tas allocaton problem (also nown as lnear assgnment problem n combnatoral optmzaton) s the followng: Gven a set of agents (or robots) and a set of tass, wth each robot obtanng some payoff (or ncurrng some cost) for each tas, fnd a one-toone assgnment of agents to tass so that the overall payoff of all the agents s maxmzed (or cost ncurred s mnmzed). The basc tas assgnment problem can be solved (near) optmally n polynomal tme by centralzed algorthms [4], [5] and decentralzed algorthms [1]. Generalzatons of the lnear assgnment problem where the number of tass and agents are dfferent and each agent s capable of dong multple tass can also be solved optmally by both centralzed and decentralzed algorthms [5], [6], [7]. However, n all of these wors, t s assumed that the tass are ndependent of each other and an agent can do any number of tass. In practce, robots have lmted battery lfe and thus there s a lmt on the number of tass that a robot can do. Furthermore, the tass may not be ndependent and may occur n groups, where there s a constrant on the number of tass that a robot can do from each group. Therefore, n ths paper, we ntroduce and study the mult-robot tas allocaton problem wth group constrants, where robots have constrants on the number of tass they can perform (both wthn the whole msson and wthn each tas group). More sepecfcally, the mult-robot (tas) assgnment problem for grouped tass (MAP-GT) that we study can be stated as follows: Gven robots and n t tass, where (a) the tass are organzed nto n s dsjont groups, (b) each robot has an upper bound on the number of tass that t can perform wthn the whole msson and also wthn a group, and (c) each robot, r, has a payoff, a j for each tas, t j, fnd the assgnment of the robots to tass such that the sum of the payoffs of all the robots s maxmzed. For concreteness, a tas group can be thought of as a compound tas composed of more than one atomc tas where one robot s requred for each atomc tas. As an llustratve example, consder the problem of transportng objects from a start locaton to a goal locaton where an object needs to be carred by multple robots. Such pc and place tass are common n many applcaton scenaros le automated warehouse, automated ports, and factory floors. If three robots are requred to carry an object then the overall tas of carryng the object can be decomposed nto three atomc tass of robots holdng the object at three dfferent places and movng wth t. Thus, the three atomc tass form a tas group where each tas n a group has to be performed by one robot and the robots have to execute the tass smultaneously. The energy costs ncurred by the robots

2 n transportng an object may be dfferent because the weghts and load carryng capabltes of the robots may be dfferent and the force transmtted from the object to the robots may be dfferent dependng on the holdng locaton. Thus, the problem of assgnng robots to tass for pc and place operatons for object transport to mnmze total energy cost can be modeled as a MAP-GT wth each robot constraned to do at most one tas wthn each tas group (please see Secton III-A, for detaled dscusson on example applcaton scenaros). Our wor here focuses on the desgn and theoretcal analyss of algorthms (both centralzed and dstrbuted) for mult-robot tas assgnment for grouped tass. We frst show that the mult-robot assgnment problem for grouped tass can be reduced to a mnmum cost networ flow problem. Thus, MAP-GT can be solved optmally n polynomal tme by usng standard algorthms for solvng networ flow problems [5]. We then present a decentralzed teratve algorthm for solvng MAP-GT where t s assumed that the robots have access to a shared memory (or there s a centralzed auctoneer). Our algorthm s a generalzaton of the aucton algorthm developed by Bertseas [1] for solvng lnear assgnment problems. We prove that by approprately desgnng and updatng an auxlary varable for each tas, called the prce of each tas, each robot optmzng ts own objectve functon leads to a soluton where the overall objectve of all the robots s maxmzed. Mathematcally, the prce of a tas s the Lagrange multpler (or dual varable) correspondng to the constrant that each tas can be done by exactly one robot. The shared memory mantans the global values of the prce of each tas. However, assumpton of the avalablty of such a shared memory may be unrealstc for many deployments of mult-robot systems. Therefore, we also present a totally dstrbuted algorthm, where each robot mantans a local value of the global prce and updates t usng a maxmum consensus algorthm. In our dstrbuted algorthm, each robot teratvely assgns tself (and nforms ts neghbors) to the tass that s most valuable to t based on her payoff and local prce nformaton. We prove that ths algorthm converges to the same soluton as the algorthm wth the shared memory assumpton. Ths s analogous to the wor n [8], where the decentralzed algorthm of [1] for lnear assgnment problem was made totally dstrbuted by combnng t wth a maxmum consensus algorthm. Our algorthm for MAP-GT provdes a soluton that s nearoptmal, namely, wthn a factor of O(n t ε) of the optmal soluton where n t s the number of tass and ε s a parameter to be chosen. Ths approxmaton guarantee s called near-optmal, snce we can choose ε to mae the soluton arbtrarly close to the optmal soluton. The runnng tme of our algorthm for the shared memory model s O( n t log(n t ) max{a j} mn{a j } ε. For the totally dstrbuted model, we wll need to multply the complexty by the dameter of the communcaton networ of the robots, whch s at most. Thus, our algorthm s polynomal n the number of robots and number of tass. However, t s pseudo-polynomal n the payoff values. By approprately scalng the payoffs we can mae the algorthm polynomal n the payoffs. Ths paper s organzed as follows: In Sectonsec:rw, we dscuss the related lterature on mult-robot tas allocaton. In Secton III, we gve a formal defnton of the mult-robot assgnment problem for groups of tass wth constrants on the number of tass that a robot can do. In Secton IV, we present the assgnment algorthm wth shared-memory model and n Secton V, we brefly dscuss how to extend the algorthm to a totally dstrbuted algorthm wth consensus technques. In Secton VII, we demonstrate the performance of our algorthm wth some example smulatons. Fnally, n Secton VIII, we present our conclusons and outlne future avenues of research. II. RELATED WORK Tas allocaton s mportant n many applcatons of multrobot systems, e.g., mult-robot routng [9], mult-robot decson mang [10], and other mult-robot coordnaton problems (see [11], [12]). There are dfferent varatons of the multrobot tas assgnment problem that have been studed n the lterature dependng on the assumptons about the tass and the robots (see [2] for a taxonomy of tas allocaton problems). One axs of dvdng the tas assgnment problem s as onlne versus offlne. In offlne tas allocaton the set of tass are nown beforehand, whereas n onlne problems the tass arse dynamcally. In ths paper, we wll consder the offlne tas allocaton problem and therefore we wll dvde our dscusson of the relevant lterature here nto the offlne and onlne tas allocaton problems. Moreover, our objectve s to desgn algorthms for tas allocaton wth provable performance guarantees. Therefore, we wll elaborate on algorthms that provde performance guarantees. Offlne Tas Allocaton: In offlne tas allocaton, the payoff s of a robot for each tas s assumed to be nown beforehand. In the smplest verson of the offlne tas allocaton problem (also nown as the lnear assgnment problem), each robot can perform at most one tas and the robots are to be assgned to tass such that the overall payoff s maxmzed. The lnear assgnment problem s essentally a maxmum weghted matchng problem for bpartte graphs. Ths problem can be solved n a centralzed manner usng the Hungaran algorthm [4], [5]. Bertseas [1] gave a decentralzed algorthm (assumng a shared memory model of computaton,.e., each processor can access a common memory) that can solve the lnear assgnment problem almost optmally. In subsequent papers, the basc aucton algorthm was extended to more general tas assgnment problems wth dfferent number of tass and robots and each robot capable of dong multple tass [1], [7]. Recently, [8], [12] have combned the aucton algorthm wth consensus algorthms n order to remove the shared memory assumpton and obtan a totally dstrbuted algorthm for the basc tas assgnment problem. However all of ths wor assume that the tass are ndependent of each other. For the more general case, where the tass are organzed nto dsjont groups such that each robot can be assgned to at most one tas from each group and there s a bound on the number of tass that a robot can do, [13] generalzed the aucton algorthm of [1] to gve an algorthm wth near optmal soluton.

3 In the above dscusson, the total payoff of a robot depends on the ndvdual tass assgned to a robot, but t does not depend on the sequence n whch the tass should be done or the combnaton of tass that the robots perform. For mult-robot routng problems, where the ndvdual robot payoffs depend on the sequence n whch the tass are performed, [9] has gven dfferent aucton algorthms wth performance guarantees for dfferent team objectves. When the objectve s to mnmze the total dstance traveled by all the robots they provde a 2-approxmaton algorthm. For all other objectves the performance guarantees are lnear n the number of robots and/or tass. For example, when allocatng m spatally dstrbuted tass to obots, for mnmzng the maxmum dstance traveled by a robot, ther algorthm gves a performance guarantee of O(n). Onlne Tas Allocaton: Even the smplest verson of the onlne tas allocaton problem, whch s (a varaton of) the onlne lnear assgnment problem s NP-hard [2]. As stated before, ths s the onlne MWBMP where the edge weghts are revealed randomly one at a tme,.e., the tass arrve randomly and a robot already assgned to a tas cannot be reassgned. Greedy algorthms for tas allocaton, wheren the tas s assgned to the best avalable robot has been used n a number of mult-robot tas allocaton systems (e.g., MURDOCH [14], ALLIANCE [15]) and therefore, have the same compettve rato of 1 3 as [16], f the payoff s are non-negatve and satsfy some techncal assumptons. Note that the greedy algorthm gves a soluton that s exponentally worse n the number of robots, when the objectve s to mnmze the total payoff [16]. Ths s dfferent from the offlne lnear assgnment problem where both the maxmzaton and mnmzaton problems can be solved optmally n polynomal tme. There are other varatons of the tas allocaton problem studed n the mult-robot tas allocaton communty, as well as operatoesearch communty that have been shown to be NPhard, and for many of them there are no algorthms wth worst case approxmaton guarantees [2]. Therefore, a substantal amount of effort has been nvested n developng and testng heurstcs for dynamc tas allocaton [17], [18], [19]. These algorthms are based on dstrbuted constrant optmzaton (DCOP). Aucton-based heurstcs for mult-robot tas allocaton n dynamc envronments have also been proposed, where the robots may fal durng tas executon and the tass need to be reassgned [20], [21]. Tas allocaton s mportant n many applcatons of multrobot systems, e.g., mult-robot routng [9], mult-robot decson mang [10], and other mult-robot coordnaton problems (see [11], [12]). There are dfferent varatons of the multrobot assgnment problem that have been studed n the lterature dependng on the assumptons about the tass and the robots (see [2], [11], [22] for surveys), and there also exsts mult-robot tas allocaton systems (e.g., Traderbot [23], [24], Hopltes [25], MURDOCH [14], ALLIANCE [15]) that buld on dfferent algorthms. III. PROBLEM STATEMENT In ths secton, we gve the formal defnton of our multrobot tas assgnment problem wth tas group constrants. We wll frst ntroduce some notatons. Suppose that there are robots, R = {r 1,...,r nr }, and n t tass, T = {t 1,...,t nt }, for the robots. Let a j R be the payoff for the assgnment par (r,t j ),.e., for assgnng robot r to tas t j. Wthout loss of generalty, we assume that any robot can be assgned to any tas. Each tas can be performed by exactly one robot. Each robot can perform at most N tass (we call, N, the budget of robot r ). Snce, performng each tas needs a sngle robot, we should have N n t, for all tass to be performed. Let f j be the varable that taes a value 1 f tas, t j, s assgned to robot, r, and 0 otherwse. The tas set T forms n s dsjont groups/subsets {T 1,...,T ns } so that n s =1 T = T. We assume that each robot, r, can perform at most N, tass from tas group T, whch we call the tas group constrants. matehmatcally, the tas group constrants can be wrtten as j: t j T f j N,, = 1,...,, = 1,...,n s (1) The overall objectve s to assgn all tass to robots so that the total payoff from the assgnment s maxmzed. The multtobot tas assgnment problem wth grouped tass can be formally stated as follows: Problem 1. Gven robots and n t tass wth the tass formng n s dsjont groups, maxmze the total payoffs of robot-tas assgnment such that each tas s performed by exactly one robot, each robot r performs at most N tass n the overall msson and at most N, tass from a tas group T. Problem 1 can be wrtten as an nteger lnear program (ILP) gven below s.t. max n t j=1 a j f j f j = 1, j = 1,...,n t, (2) n t f j N, = 1,...,, (3) j=1 f j j: t j T N,, = 1,...,, = 1,...,n s, (4) f j {0,1},, j. (5) In the above formulaton, the optmzaton varables are f j. Equaton (2) states that each tas can be assgned to exactly one robot and also mples that all tass should be assgned. Equaton (3) gves the budget constrants of the robot. Note that the above problem s a generalzaton of the lnear assgnment problem (LAP). In LAP, Equaton (4) s not present and n Equaton (3), N = 1. Remar 1. Generally speang, the assgnment payoff a j can be consdered as the dfference between assgnment beneft b j and the assgnment cost c j,.e., a j = b j c j. Thus, f cost c j s the only component to be consdered, (.e., b j = 0), Problem 1 would become an assgnment problem n the form of cost mnmzaton. Note that some papers use the term payoff for the beneft b j and the term utlty for a j. In the

4 context of ths paper, we wll use the terms payoff and utlty nterchangbly. The MAP-GT problem defned above can be solved n polynomal tme n the number of tass and number of robots by a centralzed algorthm by reducng t to a networ flow problem. We wll then use a dual decomposton-based method to desgn a decentralzed algorthm for MAP-GT and also show that the algorthm can be made totally dstrbuted. For clarty of exposton, we wll frst present the solutons to MAP-GT under the followng assumptons: (a) N, = 1 for all tas groups,.e., each robot can do at most 1 tas from each group and (b) each robot has to perform exactly N tass durng the msson. In Secton VI, we wll show how these assumptons can be removed. Thus MAP-GT problem wth assumptons (a) and (b) above can be wrtten as: s.t. max n t j=1 a j f j f j = 1, j = 1,...,n t (6) n t f j = N, = 1,..., (7) j=1 f j j: t j T 1, = 1,...,, = 1,...,n s (8) f j 0,, j (9) Note that the constrants above mplctly mply that (a) the number of tass n any subset must be no more than the number of robots (otherwse at least one tas n the subset cannot be performed),.e., max n s =1 T, and (b) the number of subsets must be no less than any N (otherwse r cannot be assgned to N tass),.e., n s max N. A. Motvaton TGC arse n two dfferent nds of scenaros: (a) each tas group conssts of tghtly-coupled tass,.e., tass whch robots must perform smultaneously, and thus each robot can only be assgned to one of them; (b) there exst group precedence constrants among tass,.e., only after all tass n one group are fnshed by robots, the subsequent group of tass can get started. To fully explore the parallelsm and ncrease the effcency, each robot can be assgned to at most one tas n each group. These constrants were motvated by a combnaton of the followng tass n mult-robot systems: Go-and-return tass: In such tass, the robots have to repeatedly vst a gven ste and return to base locaton. Such tass arse n a varety of applcaton scenaros ncludng transportaton of pacages n automated warehouse, collecton of sensng nformaton usng moble sensors, where the locatons to be vsted are spatally clustered. The spatal clusterng gves a natural groupng of the tass. Each robot has to return to some base locaton to unload the products (e.g., a pacage the robot has pced up or collected sensng nformaton) before movng to another tas locaton. Thus each robot can be assumed to be dong at most one tas at a tme from a group. The costs of dfferent tass to one robot are ndependent of each other, and can be defned as twce the dstance from the robot base locaton to the tas locaton. The objectve s to mnmze the total costs (travelng dstance) of the assgnments whle satsfyng all the constrants. Tghtly-coupled tass: In such tass, multple robots must smultaneously wor on a gven tas to perform t successfully. Examples of such tas nclude multrobot collaboratve manpulaton/assembly tass. Snce, for any tas, robots must smultaneously perform the atomc tass, each robot can only be assgned to at most one atomc tas from each tas set. If we assume that the robots are desgned to be heterogeneous and each robot has a certan degree of generalty and specalty for tass, the payoffs for the dfferent robots for a tas wll be dfferent. The objectve here s to maxmze the overall payoff of the assgnment. One example scenaro of Problem 1 s the sensng nformaton collecton by mult-robot systems. Consder the msson of sendng robots equpped wth sensors to collect sensng nformaton from spatally dstrbuted regons. Insde each regon, there exst dfferent tas locatons where robots must collect sensng nformaton smultaneously wth (almost) the same tme stamp. In ths scenaro, tass are naturally formng groups due to the spatal dstrbuton of regons and each robot can be assgned to at most one tas locaton nsde each regon. If one robot was assgned to more than one tas n one regon, t can only collect sensng nformaton from dfferent locatons wth dfferent tme stamp, whch volates the msson requrement. Assume that the sensng nformaton collecton tass are go-and-return style, and the payoff of assgnng one robot to one tas locatons depends on the travelng dstance as well as the value of the sensng nformaton. The objectve here s to assgobots to all tas locatons n dfferent regons so that the total payoffs are maxmzed whle the msson requrements are met. IV. ALGORITHM DESIGN AND PERFORMANCE ANALYSIS A. Overvew In Secton IV, we desgn an algorthm to get the optmal (or almost-optmal) soluton for mult-robot tas assgnment wth tas group constrants. Frst, we show how to reduce Problem 1 to a mn-cost networ flow problem, whch can be solved n polynomal tme usng centralzed networ flow algorthm (Secton IV-B). Second, we loo at a dstrbuted way to fnd the optmal soluton, where a centralzed controller s not requred, and nstead each robot can mae decsons on ts own n a dstrbuted way. In Secton IV-C, we desgn an algorthm, whch extends the basc aucton algorthm n [1], and prove that the algorthm can acheve an almost-optmal soluton. The algorthm s mplemented n each sngle robot, so the decson-mang process s dstrbuted. However, each robot does not only need to now ts local nformaton, such as ts budget, payoffs between each tas and tself, but also need a shared memory (.e., a centralzed component) to access

5 some global nformaton of each tas,.e., the hghest bddng prce of each tas from all robots, whch are auxlary varables created and mantaned durng the algorthm mplementaton. In Secton V, we modfy the algorthm by addng consensus technques among networed mult-robot system. So robots do not need to now the global prce nformaton of each tas, nstead, each robot just needs to get the local tas nformaton through local peer-to-peer communcaton wth ts neghbors. In ths way, we remove the shared memory requrement, whch maes the algorthm totally dstrbuted. Meanwhle, the dstrbuted algorthm can stll acheve the almost-optmal soluton qualty. B. Centralzed Soluton: Reducton to networ flow problem For any MAP-GT problem mentoned above, we can construct a mn-cost networ flow problem. A mn-cost networ flow problem s defned as follows: [26] The MAP-GT problem can be reduced to a networ flow problem by the followng constructon (shown n Fgure 1). Consder a drected graph G = (V,E), wth a set of nodes V = R T S, and edges E = E 1 E2, where Nodes: R = {r = 1,..., } represent robots, T = {t j j = 1,...,n t } represent tass, S = {T, = 1,...,, = 1,...,n s } s ntroduced to represent each tas subset T for each robot r. Edges: E 1 = {(r,t, ) = 1,...,, = 1,...,n s }, and E 2 = {(T,,t j ), j,, s.t., t j T }. Source and sn nodes: All nodes n R are source nodes wth supply N, and all nodes n T are sn nodes wth demand 1. Capacty and cost of edges: The capacty of all edges n E s 1. The cost for edges n E 1 s 0, whle for edges (T,,t j ) n E 2 s a j. Flow: f j, assocated wth each edge between T, and t j, represents the flow from node T, to node t j, where t j T. Fg. 1. Reducton to the mn-cost networ flow problem. For dsplay purpose, just robot r 1, ts correspondng nodes T 1, and edges are shown. For each other robot r, there are another set of nodes {T, = 1,...,n s }, edges {(r,t, ) = 1,...,n s } and {(T,,t j ) t j T }, whch are omtted. +N 1 and 1 represent nodes supply and demand; [0,1] shows that the capacty of flow along the edges s 1. Solvng the constructed mn-cost networ flow problem above, wll lead to the optmal soluton for Problem 1 n Secton III due to the followng facts: the demand and supply constrants are equal to the constrant (1) and (2); the capacty constrants of flow f j are equal to constrants n (3) and (4); the objectve functon mn j c j f j here s equal to the objectve functon max j a j f j, snce c j = a j for edges n E 2 and the cost of edges n E 1 s 0. So after solvng the mn-cost networ flow problem, the non-zero (value 1) flow n E 2 corresponds to the optmal assgnment of Problem 1 n Secton III. The mn-cost networ flow problem s a classcal problem that has been studed extensvely. Centralzed polynomal-tme algorthms exst that can be used to compute the optmal soluton [26]. So we can drectly use the off-the-shelf algorthms to solve Problem 1 n a centralzed way. Usng ths method, a centralzed controller s requred so that all robots nput the nformaton of payoffs and budgets to the controller, the controller solves the whole problem, and then t sends bac commands to robots for ther tas assgnments. However, n some applcatons, there s often need for decentralzed/dstrbuted algorthms so that robots can mae decsons by themselves n the feld accordng to the nformaton they possess. C. Decentralzed Soluton: Aucton-based Algorthm Desgn In ths secton, we extend the basc aucton algorthm [1] to provde a decentralzed and almost-optmal soluton for Problem 1. The outlne of ths secton s as follows: Frst, we dscuss the basc dea of aucton algorthm and several mportant concepts (ntroduced n [7]), e.g., robot s (almost) happy, and the assgnment s (almost) at equlbrum; second, we desgn a decentralzed aucton-based algorthm for Problem 1, where each robot can bd on ts own for tass, and prove the algorthm can acheve an almost-optmal soluton. 1) Basc Idea and Concepts of Aucton Algorthm: Aucton algorthm matches robots and n t tass wth constrants (1)- (4) through a maret aucton mechansm, where each robot s an economc agent actng n ts own best nterest. Although each robot r wants to be assgned to ts favorte N tass, the dfferent nterest of robots wll probably cause conflcts. Ths can be resolved through ntroducng a prce varable to each tas, and an aucton mechansm of robots bddng for tass. Suppose the prce for tas t j at teraton τ s p j (τ), and the robot assgned to the tas must pay p j (τ). So the net value of tas t j to robot r at teraton τ becomes v j (τ) = a j p j (τ) nstead of just a j. The teratve bddng from robots leads to the evoluton of p j (τ), whch can gradually resolve the nterest conflcts among robots (as shown later n ths secton). Every robot r wants to be assgned to a tas set T J = {t j j J } wth maxmum net values whle satsfyng ts constrants J = N and T J T 1, = 1,...,n s : j J (a j p j (τ)) = ( (N) max) =1,...,ns max j T (a j p j (τ)) (10)

6 where (max (N ) ) s used to get the sum of the N bggest values. When (10) s satsfed, we say robot r s happy. If all robots are happy, we say the whole assgnment and the prces at teraton τ are at equlbrum. Suppose we fx a postve scalar ε. When each assgned tas for robot r s wthn ε of beng n the set of r s maxmum values, that s, {a j p j (τ) j J } ( (N ) max) =1,...,ns (max(a j p j (τ)) ε) j T (11) (after sortng both the left and rght sets of (11) above, any value n the left set s no less than ts correspondng value n the rght set), we say robot r s almost happy. If all robots are almost happy, we say the whole assgnment and the prces at teraton τ are almost at equlbrum. 2) Aucton-based Algorthm Desgn: Appendx B dscusses two methods of drectly applyng the basc aucton algorthm to Problem 1, but they are ether not decentralzed or cannot acheve good soluton qualty. In ths subsecton, we provde a decentralzed algorthm for Problem 1, whch drectly modfes the bddng procedure of aucton algorthm. A sngle teraton of our aucton algorthm for each robot r at teraton τ s descrbed n Algorthm 1. We can defne the aucton-based algorthm for our assgnment problem by settng all robots to run copes of Algorthm 1 sequentally. The algorthm termnates when all robots have been assgned to ther tass (.e., N = N for all tass). The sequental aucton s nown as one-at-a-tme or Gauss-Sedel mplementaton. One alternatve s to let all robots bd smultaneously and assgn tass to ts hghest bdder, whch s nown as all-at-once or Jacob mplementaton. The Jacob mplementaton s convenent for parallel mplementaton, but tends to termnate slower as dscussed n [7]. Algorthm 1 can be summarzed as follows. I Durng the frst part of Algorthm 1 (from Lne 2 to 7), robot r needs to update ts assgnment nformaton from ts prevous teraton, snce other robots may bd hgher prce for ts assgned tass after ts prevous teraton. If that s the case, some prevous assgnments of tass for r wll be broen and r needs to gve new bds. II Durng the bddng part of Algorthm 1 (from Lne 10 to 21), robot r eeps the N assgned tass snce ts prevous teraton, and bds for N N tass wth the best values from dfferent subsets (whch do not contan any of N assgned tass). Ths part guarantees that after the teraton, all constrants for robot r are satsfed: (a) robot r s assgned to exactly N tass (N prevously assgned tass plus N N newly assgned tass); (b) r s assgned to at most one tas n each subset. Meanwhle each tas s assgned to at most one robot, because each tas ether does not change assgnment status (assgned to prevous robot or remans unassgned) or swtch from the prevous assgned robot to robot r. The bddng prce for each tas s at least ε bgger than ts prevous prce: snce j s the best canddate tas n T and s among the N N best from { j IT }, j s the second best n T, Algorthm 1 Bddng Procedure For Robot r 1: Input: a j, p j (τ), T for all j,, < I t,i T,P > // I t : ndces of tass assgned to r durng // r s prevous teraton; I T : ther correspondng subset // ndces; P: ther correspondng bddng prces from r 2: // Update the assgnment nformaton: 3: m {1,..., I t } // m-th prevously assgned tas 4: f P(m) < p I t (m)(τ) then 5: // another robot has bd hgher tha s prevous bd 6: remove I t (m), correspondng I T (m), P(m) from I t, I T, and P, respectvely 7: end f 8: Denote N = It // number of tass stll assgned to r 9: // Collect nformaton for new bds 10: Denote v j (τ) = a j p j (τ) // value of t j to r 11: Select the best canddate tas from each subset T, where I T : j = argmax j T v j (τ) 12: Store the ndex of second best canddate from each T : j = argmax j T, j j v j(τ) 13: Select the N N best canddate tass from { j IT }: 14: K = arg(max (N N ) ) I T v j (τ) // arg(max (N N ) ) s the //operator to get ndces of the N N bggest values 15: Store the ndex of (N N +1)-th best canddate tas from { j IT }: = argmax (I T K ) v j (τ) 16: // Start new bds 17: Bd for t K = {t j K } wth prce: 18: b j = p j (τ)+v j (τ) max{v j (τ),v j (τ)}+ε 19: // Update assgnment nformaton and prce nformaton: 20: Add { j K } to I t, K to I T, and {b j K } to P 21: Set p j (τ + 1) = b j for K and set p j (τ + 1) = p j (τ) for j { j K } j s the (N N + 1)-th best from { j IT }, v j (τ) max{v j (τ),v j (τ)} b j p j (τ) = v j (τ) max{v j (τ),v j (τ)}+ε ε So the tass recevng r s bds must be assgned to r at the end of the teraton. The bddng value of b j s related to the proof of the optmalty of the algorthm, whch wll be dscussed n Secton IV-C3. 3) Algorthm Performance Analyss: In ths subsecton, we wll answer the followng questons about Algorthm 1:(a) Wll Algorthm 1 termnate wth a feasble assgnment soluton n a fnte number of teratons? (b) How good s the soluton when Algorthm 1 termnates? Lemma 1. When Algorthm 1 termnates for all robots, the acheved assgnment must be a feasble soluton for Problem 1,.e., (1)-(4) are satsfed. Proof: When Algorthm 1 for robot r termnates, t means that r has already been assgned to N tass and no other robot would bd hgher for r s assgned tass. Snce the algorthm termnates for all robots, accordng to summary (II) of Algorthm 1, all the constrants have been satsfed for

7 all robots. So the acheved assgnment s a feasble soluton satsfyng (1)-(4). Lemma 1 mples Algorthm 1 s sound,.e., when t outputs a soluton, the soluton s feasble. The next result asserts that Algorthm 1 always termnates n fnte number of teratons assumng the exstence of at least one feasble assgnment for the problem. The proof reles on the observatons below: (a) When a tas s assgned, t wll reman assgned durng the whole process of the algorthm. The reason s that durng the bddng and assgnment process, the assgnment status of a tas can ether transfer from unassgned to assgned, or be reassgned from one robot to another, but cannot become unassgned from assgned. (b) Each tme when a tas receves a bd, ts new prce wll ncrease by at least ε accordng to the algorthm,.e., p j (τ+1)=b j = p j (τ)+v j (τ) max{v j (τ),v j (τ)}+ε p j (τ)+ε So f one tas receves nfnte number of bds, ts prce wll become +. (c) If a robot r bds for nfnte number of tmes, all tass n the subsets, where r does not have fxed assgned tass, wll receve nfnte number of bds. The reason s that there are fnte number of tass, and thus there must be at least one tas recevng nfnte number of bds. If there exsts one tas (from such subsets), whch does not receve nfnte number of bds, ts prce would be fnte, and ts value for r must be bgger than those tass recevng nfnte number of bds. So t has to receve more bds, whch leads to the contradcton. So all tass n those subsets receve nfnte number of bds and thus have the prce of + (accordng to (b)). Theorem 1. If there s at lest one feasble soluton for Problem 1, Algorthm 1 for all robots wll termnate n a fnte number of teratons. Proof: If the algorthm contnues nfntely, there must be some subsets {T K } where all tass have + prce accordng to (c) above. Denote T = K T. Suppose some robots {r I } already get N tass from T \ T, and are stll bddng for ts remanng N tass from T. (Please note, here N = N N does not necessarly equal to N n Algorthm 1 snce all those tass n T are not stably assgned to any robot.) Denote R = {r I }. Each tas t T remans assgned (accordng to (a) above). Each robot r R needs to be stably assgned to N more tass, but all tass n T cannot fll up all I N postons. So T < I N Please note that the above nequalty s strct, snce there must be at least one robot r R that has remanng tass unassgned (otherwse the algorthm termnates). On the other hand, each robot must already be assgned to exactly one tas n each subset T, K (accordng to (c) above). We have N = N + N I I I Suppose n any feasble assgnment, Nˆ and Nˆ are the number of assgned tass for r n T \ T and T, respectvely. N = Nˆ + Nˆ. It s easy to see that each N ( I ) has reached the bggest possble value, I N I Nˆ. So ˆ N I I N > T It means n any feasble assgnment, the number of assgned tass n T for R s bgger than the number of tass n T. By contradcton, we now that Algorthm 1 must termnate n a fnte number of teratons f there exsts a feasble soluton for Problem 1. Lemma 1 and Theorem 1 together prove that Algorthm 1 s both sound and complete, and also gve a postve answer to the frst queston (at the begnnng of Secton IV-C3), when there exsts at least one feasble soluton for the problem. Infeasblty chec: In the case when there does not exst any feasble soluton, the robots can detect that stuaton n a dstrbuted way durng the bddng procedure. The bddng procedure tself would guarantee that tas group constrant (8) s always satsfed snce each robot would bd for at most one tas from each group. Constrant (6) mght be volated due to the fact that N < n t. In that case, Algorthm 1 would output an almost-optmal soluton gven the budget constrants of robots, and leaves some tass unassgned. Moreover, the robots can detect that stuaton after the algorthm termnates by checng whether there stll exst tass wth ntal zero prce. The nfeasblty caused by budget constrant (7) can be detected whenever a robot start contnung bddng for a tas wth negatve values to t. At that tme, the robot can chec the prce of other tass: f all tass have non-zero prce, the robot can detect that there does not exst any feasble soluton snce t mples that N < n t ; f the number of tass wth zero prce (tass whch have not receved any bds) s n p0, the robot can detect the nfeasblty f t contnues bddng for tass wth negatve values for n p0 rounds snce t mples that the structure of tas groups prevents a feasble soluton satsfyng tas group constrant as well as budget constrant. In ths case, the robot detectng the nfeasblty could send out a message to ts neghbors to stop the bddng procedure. Please note that ths nfeasblty manly comes from the strct budget constrant that each robot r must be assgned to exactly N tass. When we relax ths budget constrant n Secton VI so that each robot can perform at most N tass, ths nfeasblty would not exst. Next we want to prove the performance of Algorthm 1. The result reles on the followng theorem. Theorem 2. After each teraton τ of robot r, r s newly assgned tass together wth the tas prces p j (τ + 1) eep r almost happy,.e., (11) s satsfed. Proof. Frst, let us prove t holds true for the frst teraton. At the begnnng of the frst teraton, r does not have any assgned

8 tass. Accordng to the bddng part of Algorthm 1, the bdden tass t K = {t j K } wth the prce before the teraton can mae r happy: {a j p j (τ) K } = ( (N ) max) =1,...,ns (max j T (a j p j (τ))) p j (τ + 1) = b j = p j (τ)+v j (τ) max{v j (τ),v j (τ)}+ε, and v j (τ + 1) = v j (τ), j { j K }, so a j p j (τ + 1) = max{v j (τ),v j (τ)} ε = max{v j (τ + 1),v j (τ + 1)} ε So the value of any tas n t K to robot r s wthn ε of the maxmum value of any tas n ts own subset and other subsets {T K }, so {a j p j (τ+1) K } ( (N ) max) =1,...,ns (max j T (a j p j (τ)) ε) whch means (6) s satsfed. Second, we prove that the unchanged tass assgned to r snce r s prevous teraton, must stll be n the new assgnment of r. That s, those tass are stll among tass, whch mae r almost happy after the teraton. Denote the ndex set of those tass as t K. Snce these tass dd not receve any bd from other robots snce r s prevous teraton, ther prces (and hence ther values) to r do not change. Meanwhle any other tass prce ether reman the same or ncrease after recevng bds, so ther values to r reduce. So tass n t K must stll be n the new assgnment to mae r almost happy. Snce the bddng process to get newly assgned tass s the same, the newly assgned tass must also be n the new assgnment to mae r almost happy (due to smlar proof for the frst teraton). So the concluson s true for each teraton τ of r,.e., after each teraton τ of r, r s newly assgned tass together wth the tas prces p j (τ + 1) eep r almost happy. Snce Theorem 2 holds true for all robots, we get the corollary below. Corollary 1. When Algorthm 1 for all robots termnates, the acheved assgnment and prce are almost at equlbrum. Theorem 3 below answers the second queston (at the begnnng of Secton IV-C3), and gves performance guarantee for Algorthm 1. Theorem 3. When Algorthm 1 for all robots termnates, the acheved assgnment {(,(l 1,...,l N )) = 1,..., } must be wthn N ε of an optmal soluton. Proof: Denote ({(,(l 1,...,l N )) = 1,..., }) as any feasble assgnment,.e., N ( t l ) =1 N ( t l ) =1 Tm 1,,m : = 1,..., ;m = 1,...,n s N j ( =1 t l j ) = /0 f j (12) Denote {p j j = 1,...,n t } as the set of tas prces when Algorthm 1 termnates for all robots and {p j j = 1,...,n t } as any set of tas prces. Frst, we want to gve an upper bound for the optmal soluton. N (a l p l ) ( (N ) max) =1,...,ns (max(a j p j )) j T =1 N =1 N =1 (a l p l ) (a l ) n t j=1 p j + ( (N ) max) =1,...,ns (max(a j p j )) j T ( (N ) max) =1,...,ns (max(a j p j )) j T Snce t holds true for any set of prce and any feasble assgnment, we have A B, where A s the optmal total payoffs of any feasble assgnment. A = B = mn p j : j=1,...,n t B n t ( p j : j=1,...,n t j=1 = mn max N l satsfy (12) =1 p j + (a l ) ( (N ) max) =1,...,ns (max(a j p j ))) j T On the other hand, accordng to Corollary 1, we have N =1 N a l =1 N (a l p l ) n t j=1 B (N ) ( max )(max =1,...,n s (N ) ( max )(max =1,...,n s p j + N ε A j T (a j p j )) j T (a j p j )) N ε N ε =1 a l s the total payoffs of the acheved assgnment by Algorthm 1, and A N =1 a l A N ε So t s wthn N ε of an optmal soluton. Please note, f all the payoffs are ntegers, and we set ε < 1 nr N, the acheved assgnment wll be optmal. V. DISTRIBUTED AUCTION ALGORITHM N ε In ths Secton, we brefly dscuss how to combne our algorthm wth consensus technques to mae the algorthm totally dstrbuted. In Algorthm 1, each robot r can bd on ts own, however, t needs to access global p j (τ) nformaton ether from a shared memory or from communcatng wth a centralzed auctoneer. Recently, consensus algorthms have been ntroduced to combne wth the aucton algorthm, so that the shared memory/centralzed auctoneer can be removed [8], [12]. Next we brefly tal about the basc dea. Consder a connected networ G of all robots, each robot can fnally get some global nformaton, based oepeated local nteracton wth ts neghbors. For example, n maxmumconsensus [27], each robot r R has an ntal value of tas j as p j, and wants to get the maxmum ntal value among all robots, p j = max r R p j (denote r the robot whch gets the

9 ntal value p j ). The maxmum ntal value p j can propagate to the whole connected networ, f every robot eeps updatng ts value usng the local maxmum value among ts neghbors as follows. Suppose that at teraton τ, each robot r has the value of tas j as p j (τ). Startng from ntal value p j (0), the robot needs to update ts value: p j(τ + 1) = max p j(τ) (13) N + where N + = {} N, and N s the set of r s neghbors n networ G. Eventually, each robot can get the true maxmum value of tas t j, and the number of teratons that each robot r gets the true value p j would be the length of the shortest path from r to r, whch s at most the number of robots. Smlar dea apples to the aucton algorthm as shown below. Modfcaton of Algorthm 1 to form a dstrbuted algorthm: Suppose at teraton τ, the prce of tas t j that r mantans s p j (τ), then the vector of prces that r mantans s that [p 1 (τ), p 2 (τ),..., p n t (τ)], where n t s the number of tass. At the begnnng of Algorthm 1, we can add a part where r updates ts prce nformaton of each tas t j, p j (τ), usng maxmum-consensus approach as shown n Equaton 13. r may use underestmated prce for bddng durng some teratons due to two factors: (a) r mantans the prce of all tass usng local maxmum nstead of global maxmum; (b) the prce of each tas at each teraton may ncrease (due to new bds). However, the current true prce nformaton wll eventually propagate to r n at most teratons (gven the networ s connected). So after combnng wth consensus technques, the performance of Algorthm 1 does not change except that the convergence tme may be delayed by at most tmes, where s the dameter of the robot networ. After the modfcaton, the only nowledge each robot needs to now s ts own budget, as well as the payoffs between tself and each tas. The prce update and bddng procedure can be mplemented ether n synchronous or asynchronous way. Durng each bddng teraton, each robot needs to communcate wth ts drect neghbors to update the local maxmum tas prce. The average number of messages each robot needs to communcate s the average node degree n the networ. The sze of each message s the number of tass. Almost-optmalty of the modfed algorthm: Smlar proof as for Theorem 1 can be used to prove that the new algorthm wth consensus technque would also termnate n fnte number of teratons at a feasble soluton f there exst at least one such soluton. Theorem 2 also holds true f we change the prce n the theorem from true values to robots estmate from local maxmum,.e., all robots are almost happy wth respect to ts mantaned tas prce each tme after ts bddng teratons; snce we assume the robot connecton networ s connected, the accurate tas prce nformaton at teraton τ (.e., the global hghest bd prce of the tass at that tme), would eventually propagate to the whole networ wthn at most teratons. When the algorthm termnates, the prce nformaton stored by all robots does not change and must reach the true values due to propagaton, so Theorem 2 holds true for the true prce values. Thus Theorem 3 also holds true. So each robot n a connected networ can mae decsons based on updated local prce nformaton from ts own neghbors. Therefore the aucton algorthm becomes totally dstrbuted for both decson process and the nformaton collectng process. VI. EXTENSIONS In ths secton, we dscuss a few extensons to the basc problem formulaton n Problem 1, ncludng the relaxaton of budget constrant (7) and tas group constrant (8). A. Relaxaton of budget constrant In Problem 1, each robot has budget constrant,.e., the number of tass robot r can perform s exactly N. In ths subsecton, we relax ths constrant so that each robot mght not exhaust all ts budget, n other words, the number of tass robot r s assgned to s bounded by N (but can be any nonnegatve number smaller than N ): n t f j N, = 1,..., j=1 To solve the extended problem n a centralzed or decentralzed way, we just need to modfy the nput nstances n the followng way: snce the total budgets of robots must be no less than the number of tass,.e., N n t, we add N n t vrtual tass (denote the set of vrtual tass as T V ) to the orgnal tass. Every sngle vrtual tas s formng a separate tas group. The payoffs between any vrtual tas and any robot s set to be dentcal,.e., a 1 j = a 2 j, two robots 1, 2, and tas t j T V. Then we can apply the same algorthms descrbed n Secton IV-B and IV-C. The vrtual tass are auxlary and only exst n the nput to the algorthm, and get removed n the output assgnment soluton,.e., f a robot s assgned to z vrtual tass after the algorthms termnate, the robot would have z remanng unused budgets. The soundness and completeness of the method above drectly come from the soundness and completeness of the algorthms n Secton IV. The optmalty of the method can be proved as follows. Accordng to Theorem 3, for the new nput nstance wth vrtual tass, we have A = j J a j A N ε, where J s the set of tass assgned to robot r, ncludng the possbly assgned vrtual tass. Snce the vrtual tass have the same payoffs for any robot, we can cancel ther payoffs n our assgnment soluton A and the optmal soluton A, whch leads to A = a j A j J N ε, where J s the set of tass assgned to robot r, excludng the possbly assgned vrtual tass. To solve the extended problem n a dstrbuted way, we cannot drectly use the method above. The reason s that each

10 robot does not now other robots budget, and thus does not now how many vrtual tass there are n the modfed nput nstance. The way to resolve ths ssue s to change the bddng procedure: each tme a robot detects that t s bddng for a tas wth non-postve value, t should stop bddng for that tas and meanwhle reduce ts budget by one. The reason s that f we set the payoffs of vrtual tass to be zero n the above method, a robot would bd for vrtual tas f and only f the values of other tass are negatve; and robots would not compete for the same vrtual tass. So the modfed bddng procedure above can lead to the same soluton n a dstrbuted way wthout assumng that a robot nows other robots budgets. B. Relaxaton of tas group constrant In Problem 1, all tass are formng dsjont groups, and tas group constrant means that each robot can be assgned to at most one tas from each group. In ths subsecton, we relax ths constrant so that each robot r can be assgned to multple tass n each group T, but the number of tass t can be assgned to n each group s bounded by N, : t j T f j N,,, : = 1,...,, = 1,...,n s (14) To address ths extenson, we need modfy how the canddate bd tass are selected (lne 11 and 12) n the bddng procedure of Algorthm 1. Frst, nstead of selectng the best canddate tas from each subset T, we select the best N, tass from T to form a set J ; second, nstead of storng the ndex of the second best canddate tas from each group T, we store the ndex of the (N, + 1)-th best canddate tas, j, for future bd prce update. The modfed bddng procedure s shown below n Algorthm 2: The proof of soundness, completeness, and optmalty of Algorthm 2 s smlar to the proof for Algorthm 1. The dfference s that n the optmalty proof, nstead of showng that the best N canddate tass are selected from dfferent tas group to satsfy the basc tas group constrant (8), we need to show that the selected N tass are the best canddate tass satsfyng the extended tas group constrant (14). VII. SIMULATION RESULTS In Secton IV, we desgned Algorthm 1 for the MAP-GT problem, and proved the performance guarantee the desgned algorthm. Accordng to Theorem 3, we now that ε s a control parameter whch drectly nfluences the performance of our algorthm. In ths secton, we run smulatons n a synthetc example to chec how the control parameter ε nfluences the aucton algorthm s soluton qualty and convergence tme. Consder = 20 robots, each robot N needs to perform N = 3 tass from a set of n t = 60 tass. The tas set T can be dvded nto n s = 20 dsjont subsets, wth 3 tass n each subset. We randomly generate payoffs a j from a unform dstrbuton n (0, 20). ε n Algorthm 1 s a control parameter, related to the convergence tme and performance guarantee of the algorthm. In our smulatons, we tested dfferent values of ε. For each ε, we generated 100 samples wth dfferent payoffs drawn from the unform dstrbuton, and we compared Algorthm 2 Bddng Procedure For Robot r 1: Input: a j, p j (τ), T for all j,, < I t,i T,P > // I t : ndces of tass assgned to r durng // r s prevous teraton; I T : ther correspondng subset // ndces; P: ther correspondng bddng prces from r 2: // Update the assgnment nformaton: 3: m {1,..., I t } // m-th prevously assgned tas 4: f P(m) < p I t (m)(τ) then 5: // another robot has bd hgher tha s prevous bd 6: remove I t (m), correspondng I T (m), P(m) from I t, I T, and P, respectvely 7: end f 8: Denote N = It // number of tass stll assgned to r 9: // Collect nformaton for new bds 10: Denote v j (τ) = a j p j (τ) // value of t j to r 11: Select the best N, canddate tass from each subset T : J = arg(max(n,) ) j T v j (τ) // arg(max (N,) ) s the //operator to get ndces of the N, bggest values 12: Store the ndex of the (N, + 1)-th best canddate from each T : j = argmax j T \J v j(τ) 13: Select the N best canddate tass from J = J : 14: K = arg(max (N ) ) J v (τ) // arg(max (N ) ) s the //operator to get ndces of the N bggest values 15: Store the ndex of (N, + 1)-th best canddate tas from J : = argmax (J \K ) v (τ) 16: // Start new bds 17: Bd for t K = {t K } wth prce: 18: b = p (τ)+v (τ) max{v (τ),v j g() (τ)}+ε // Suppose //tas belongs to tas group g 19: // Update assgnment nformaton and prce nformaton: 20: Add { K } to I t, {g() K } to I T, and {b K } to P 21: Set p (τ + 1) = b for K and set p j (τ + 1) = p j (τ) for j K the mean and standard devaton of performance rato of our soluton to the optmal soluton, as well as the convergence tme of the algorthm. Fgure 2 shows how the soluton of assgnment payoffs changes wth the control parameter ε. When ε s as small as 0.1, the assgnment payoffs acheved by our algorthm almost equal the optmal soluton. When ε ncreases, the dfference between our soluton and the optmal soluton s ncreased. Fgure 3 shows how the convergence tme of our algorthm changes wth ε. The number of teratons decreases wth ε, whch means wth hgher ε, Algorthm 1 converges faster. From Fgure 2 and 3, we can see that there s a tradeoff between the soluton qualty and the convergence tme, whch can be adjusted by ε. Wth bgger ε, the algorthm converges faster at sacrfce of soluton qualty; whle wth smaller ε, the algorthm soluton s better at the cost of slower convergence tme. In ths example, ε = 1 can acheve a good balance between the above two performance ndcators. To test the effect of maxa j mna j, we fxed ε, and