Sequential Incremental-Value Auctions

Transcription

1 Sequential Inremental-Value Autions Xiaoming Zheng and Sven Koenig Department of Computer Siene University of Southern California Los Angeles, CA Abstrat We study the distributed alloation of tasks to ooperating robots in real time, where eah task has to be assigned to exatly one robot so that the sum of the latenies of all tasks is as small as possible. We propose a new aution-like algorithm, alled Sequential Inremental-Value (SIV) aution, whih assigns tasks to robots in multiple rounds. The idea behind SIV autions is to assign as many tasks per round to robots as possible as long as their individual osts for performing these tasks are at most a given bound, whih inreases exponentially from round to round. Our theoretial results show that the team osts of SIV autions are at most a onstant fator larger than minimal. Introdution We study the distributed alloation of tasks to ooperating robots in real time, where eah task has to be assigned to exatly one robot so that the team ost is as small as possible. We do this in the ontext of multi-robot routing, where the robots have to visit targets in the plane so that eah target is visited by some robot (Dias et al. 2005). The terrain, the loations of all robots and the loations of all targets are known. The team ost is the sum of the latenies of all targets, where the lateny of a target is the time when it gets visited. Aution-like algorithms (short: autions) promise to solve multi-robot routing problems with small ommuniation and omputation osts sine the robots ompress information into a small number of bids, whih they ompute in parallel and then exhange (Dias et al. 2005; Lagoudakis et al. 2005; Koenig et al. 2008). Thus, autions promise to be able to ontrol robots in real-time, whih is important to prevent robots from being idle eah time they alloate targets among themselves. Robotis researhers have reently studied the use of Sequential Single-Item (SSI) autions for multi-robot routing (Tovey et al. 2005). SSI autions proeed in multiple rounds, until all targets are assigned to robots. During eah round, SSI autions assign exatly one additional (previously unassigned) target to some This material is based upon researh supported by, or in part by, NSF under ontrat/grant number , ARL/ARO under ontrat/grant number W911NF and ONR in form of a MURI under ontrat/grant number N The presented views and onlusions are those of the authors alone. We thank Craig Tovey for providing his operations researh expertise to us and proofreading this paper very arefully. Copyright 2010, Assoiation for the Advanement of Artifiial Intelligene ( All rights reserved. robot so that the team ost inreases least (= hill-limbing priniple). In this paper, we propose a new type of aution, alled Sequential Inremental-Value (SIV) aution, that makes use of the hill-limbing priniple in a different way to assign targets to robots in multiple rounds. The idea behind SIV autions is to assign as many tasks per round to robots as possible as long as their individual osts for performing these tasks are at most a given bound, whih inreases exponentially from round to round. Our theoretial results show that the team osts of SIV autions are at most a onstant fator larger than minimal, whih is better than the guarantee on the team osts provided by SSI autions. Multi-Robot Routing We now formalize multi-robot routing problems. A multirobot routing problem onsists of a set of robots A = {a 1,...,a n } and a set of targets T = {t 1,...,t m }. The initial loations of all robots an be different. Any tuple (T a1,...,t an ) of pairwise disjoint bundles of targets T ai T is a partial assignment of the multi-robot routing problem. We define the path ost path a (T a ) to be the smallest possible travel distane of robot a for visiting all targets T a from its initial loation. We assume that all distanes satisfy the triangle inequality. We define the robot ost robot a (T a ) to be the smallest possible total lateny (= sum of the latenies) of all targets T a on any path that visits all targets T a from the initial loation of robot a, where the lateny of a target is the time when robot a visits it (measured in the travel distane of robot a, whih assumes that the robot moves at unit speed). Finally, we define the team ost of a partial assignment (T a1,...,t an ) to be the total lateny a A robot a (T a ). Any partial assignment with a A T a = T (that is, eah target is visited by exatly one robot) is a omplete assignment of the multi-robot routing problem. Our objetive is to determine a omplete assignment of a given multi-robot routing problem and the order in whih eah robot should visit the targets assigned to it so that the resulting team ost is small. A variety of appliations require a small total lateny. An example is finding all vitims in searh and resue ions. Most existing work is on minimizing the total lateny for single robots, alled the traveling repairman problem (Blum et al. 1994). The first onstant fator approximation algorithm (Blum et al. 1994) was later improved (Goemans and Kleinberg 1996). There is muh less work on minimizing the total lateny for multiple robots, alled the k-traveling repairman problem.

2 1 funtion SSI-Aution (T, A) 2 inputs: T : the set of targets T 3 A: the set of robots A 4 outputs: {T a} a A: a omplete assignment 5 for eah robot a A do 6 T a ; 7 while (T ) do 8 /* Annuniation Stage */ 9 the autioneer announes T to eah robot a A; 10 /* Bidding Stage */ 11 for eah robot a A do 12 for eah target t T do 13 b t a robot a (T a {t}) robot a (T a); 14 robot a submits b t a to the autioneer; 15 /* Winner-Determination Stage */ 16 (a, t) arg min (a A,t T) b t a ; 17 T a T a {t}; 18 T T \ {t}; Figure 1: Sequential Single-Item Autions A onstant fator approximation algorithm for the speial ase where the initial loations of all robots are idential (Fakharoenphol, Harrelson, and Rao 2003) was later extended to the ase where eah target has a given repair-time (Jothi and Raghavahari 2007). Our SIV autions generalize the idea behind the former algorithm to the ase where the initial loations of all robots an be different. Sequential Single-Item (SSI) Autions Sequential Single-Item (SSI) autions (Tovey et al. 2005) assign targets to robots in multiple rounds (whih explains the term sequential ) and, during eah round, assign exatly one additional (previously unassigned) target to some robot (whih explains the term single-item ) so that the team ost inreases least, see Figure 1. All targets are initially unassigned (Lines 5-6). The autioneer starts a new round as long as there are still unassigned targets (Line 7). Eah round onsists of three stages (Line 8-18): First, the autioneer announes the unassigned targets to eah robot in the annuniation stage (Line 9). Seond, eah robot bids on eah unassigned target in the bidding stage the inrease in its robot ost in ase it has to visit the target it bids on in addition to all targets already assigned to it in previous rounds (Line 13), whih is similar to previous work on marginalost bidding in ContratNet (Sandholm 1996). The robot an determine its bids in parallel with the other robots. In the proess, it needs to alulate the total latenies of given sets of targets, whih is alled the traveling repairman problem and is NP-hard (Blum et al. 1994). Thus, the robot needs to approximate these alulations even though it an determine its bids in parallel with the other robots, whih is often done with the heapest-insertion heuristi (Lagoudakis et al. 2005). Third, the autioneer hooses a bid with the smallest bid ost as the winning bid and assigns the winning target to the winning robot in the winner-determination stage (Line 16-17), whih terminates the round. Ties an be broken in an arbitrary way. The following theorem provides the best known bounds on the team osts of SSI autions. Theorem 1 ((Lagoudakis et al. 2005)) The team osts of SSI autions an be at least a fator of Ω 1/3 ) larger than minimal, even if eah robot alulates its robot osts exatly. They are at most a fator of O 2 ) larger than 1 funtion SIV-Aution (T, A, b) 2 inputs: T : the set of targets T 3 A: the set of robots A 4 b: a onstant in (1, 2) 5 outputs: {T a} a A: a omplete assignment 6 j 0; 7 for eah robot a A do 8 T a ; 9 while (T ) do 10 j j + 1; 11 B b j+1 ; 12 A A; 13 while (A ) do 14 /* Annuniation Stage */ 15 the autioneer announes T and B to eah robot a A ; 16 /* Bidding Stage */ 17 for eah robot a A do 18 T a arg max T T: path a (T T ; ) B 19 robot a submits T a to the autioneer; 20 /* Winner-Determination Stage */ 21 a arg max a A T a ; 22 A A \ {a}; 23 T a T a T a ; 24 T T \ T a ; Figure 2: Sequential Inremental-Value Autions minimal, whether eah robot alulates its robot osts exatly or uses the heapest-insertion heuristi to determine them approximately in polynomial time. Sequential Inremental-Value (SIV) Autions Sequential Inremental-Value (SIV) autions assign targets to robots in multiple rounds (whih explains the term sequential ) and, during eah round, assign the largest number of additional (previously unassigned) targets to eah robot with the onstraint that the path ost of the set of targets assigned to a robot in this round is at most a given bound, whih inreases exponentially from round to round (whih explains the term inremental-value ), see Figure 2. All targets are initially unassigned (Lines 7-8). The autioneer starts a new round as long as there are still unassigned targets (Line 9). In the beginning of eah round, the autioneer multiplies the bound with a given onstant b (Line 10-11). Different from SSI autions, eah round of SIV autions onsists of A iterations (Line 14-24). Eah iteration onsists of three stages and assigns a number of unassigned targets to some robot as follows: First, the autioneer announes the unassigned targets and the bound to eah eligible robot in the annuniation stage. A robot is eligible in a round iff it has not been a winning robot in that round. Seond, eah eligible robot bids as many unassigned targets as possible in the bidding stage with the onstraint that the path ost of the set of these targets is at most the given bound (Line 18-19). Note that the bids onsist of sets of targets without bid osts, that robots ignore the targets already assigned to them when determining their bids, and that eligible robots an always submit a bid (whih ould be the empty set). Third, the autioneer hooses a bid with the largest number of targets as the winning bid and assigns the winning targets to the winning robot in the winner-determination stage (Line 21-23), whih makes the robot ineligible in the urrent round and terminates the iteration. The last robot beoming ineligible terminates the urrent round. Ties an be broken in an arbitrary way.

3 Analysis of SIV Autions Eah eligible robot a bids as many unassigned targets as possible in the bidding stage of an SIV aution with the onstraint that the path ost path a (T ) of the set of these targets T is at most the given bound B. The problem of alulating its bids is NP-hard (whih an be shown by reduing Hamiltonian path problems to it). Thus, the robot needs to approximate the alulations of its bids even though it an determine its bids in parallel with the other robots. We assume that the robot approximates the alulations of its bids by hanging Line 18 of SIV autions to 18 T a Bid(T, a, B); The funtion Bid(T,a,B) determines, for k = 0,..., T, a rooted k-mst (whose root is the initial loation of robot a) for the unassigned targets T and returns the targets T in the rooted k-mst with the largest k with the onstraint that the path ost path a (T ) is at most the given bound B. 1 Determining rooted k-msts and determining the path osts of given sets of targets are both NP-hard (Ravi et al. 1994). We therefore assume that the funtion Bid(T,a,B) uses, for k = 0,..., T, an (1/α)-approximation algorithm for determining a rooted k-tree (whose root is the initial loation of robot a) for the unassigned targets T and returns the targets T in the rooted k-tree with the largest k with the onstraint that the travel distane for irumnavigating the rooted k-tree (whih is twie the ost of the rooted k-tree) is at most the given bound B. The robot visits the targets assigned to it at the end of the SIV aution by moving with minimal travel distane from eah target to the next. It visits the targets it was assigned in earlier rounds before targets it was assigned in later rounds and the targets it was assigned in the same round in the order given by irumnavigating the orresponding tree. We now analyze the team osts of SIV autions using the following notation: {T a } a A : any omplete assignment of the multi-robot routing problem and the order in whih eah robot should visit the targets assigned to it so that the resulting team ost is minimal (short: the optimal assignment) - if there is more than one, hoose one arbitrarily; = a A robot a (Ta): the team ost of the optimal assignment (short: minimal team ost); n j : the number of targets whose latenies are larger than 0.5αb j+1 in the optimal assignment {Ta } a A ; n j : the number of unassigned targets in the beginning of the jth round of an SIV aution; and T j : the set of unassigned targets in the beginning of the jth round of the SIV aution whose latenies are at most 0.5αb j+1 in the optimal assignment. The onstant b influenes the runtime and the resulting team ost and an be hosen arbitrarily from the interval 1 The ost of a tree is the sum of the osts of its edges. An unrooted k-tree is a tree that ontains exatly k targets, while a rooted k-tree is a tree that ontains k targets plus the root, whih is the loation of the robot. A rooted or unrooted k-minimum Spanning Tree (k-mst) is a rooted or unrooted (respetively) k-tree of minimal ost. (1,2). We assume that the distane from the robot to any target is larger than b, whih implies that n 0 and n 1 are equal to the number of targets. This relationship an be enfored by eliminating all targets at the loations of the robots and then suffiiently dereasing the units in whih distanes are measured. Lemma 1 Bid(T,a,B) returns at least T targets if there exists a set of targets T T with path a (T ) 0.5αB. Proof: Consider 0 k = T T. There exists a rooted k-tree whose ost is at most 0.5αB sine path a (T ) 0.5αB. Thus, the ost of the rooted k-mst is at most 0.5 αb. The (1/α)-approximation algorithm returns a rooted k-tree whose ost is at most 0.5B and the travel distane for irumnavigating the rooted k-tree (whih is twie the ost of the rooted k-tree) is at most the given bound B. Thus, Bid(T,a,B) returns at least k targets. The number of targets assigned to robots during the jth round of the SIV aution is n j n j+1. The following theorem shows that this number is at least half of the number of the unassigned targets in the beginning of the jth round whose latenies are at most 0.5αb j+1 in the optimal assignment. Theorem 2 For all j 1, 0.5 T j n j n j+1. Proof: Let T a,j be the subset of targets in T j that are visited by robot a in the optimal assignment. Let Ta,j be the subset of targets in T a,j that are not assigned to any robot during the jth round of the SIV aution. The set of targets assigned to robot a during the jth round an be partitioned into the following sets: T self a,j is the subset of targets in T a,j that are assigned to robot a during the jth round; Ta,j in is the subset of targets in a at a,j that are assigned to robot a during the jth round; and Ta,j out is the subset of targets not in T j that are assigned to robot a during the jth round. There are three important relationships among these sets: a,j + Ta,j in out + Ta,j ) = n j n j+1 sine the sets T self a,j, Ta,j in out and Ta,j for all a A partition the set of targets assigned to robots during the jth round. T self a,j + Ta,j in a,j, Ta,j in Ta,j + Ta,j ) = T j sine the sets and T a,j for all a A partition the set T j. self a,j + Ta,j in out + Ta,j ) a A a,j + ). The left-hand side of the inequality is the number of targets assigned to robots during the jth round. In the beginning of the iteration of the jth round where robot a is the winning robot, the targets in T self a,j and Ta,j are unassigned with path a (T self a,j Ta,j ) path a (T a,j ) 0.5αb j+1. Thus, robot a wins at least T self a,j + Ta,j targets by Lemma 1. Summing over all agents A yields the relationship. a A in Solving Relationships (2) and (3) for Ta,j, adding the resulting inequalities and dividing by two yields in a A Ta,j 0.5 j self a A a,j + Ta,j out )). Substituting this inequality into Relationship (1) yields n j

4 a 1 a 2 1 t 1 t 3 1 t t 4 Figure 3: Example 1 n j j + self a A a,j + Ta,j out )) 0.5 T j. Corollary 1 For all j 1, n j+1 0.5(n j + n j). Proof: Let n j be the number of assigned targets in the beginning of the jth round of the SIV aution whose latenies are at most 0.5αb j+1 in the optimal assignment. First, n j T n j sine the right-hand side of the inequality is the number of assigned targets in the beginning of the jth round. Seond, T = T j + n j + n j by definition. Third, 0.5 T j n j n j+1 by Theorem 2. Putting it all together, 0.5(n j n j ) = 0.5 n j n j)) 0.5 n j n j ) = 0.5 T j n j n j+1. The inequality of Corollary 1 an be tight, as shown in Figure 3. Assume that α = 1, b = 2 ǫ (where ǫ is a small positive onstant so that b 3 6). Then, the optimal assignment is Ta 1 = {t 1,t 2 }, Ta 2 = {t 3 } and Ta 3 = {t 4 }. Thus, n 0 = n 1 = 4 (sine b < 2), n 1 = n 2 = 4 (sine b 2 < 4), and n 2 = 0 (sine path a 1 ({t 1,t 2 }) = 3 0.5b 3 and path a 2 ({t 3 }) = path a 3 ({t 4 }) = 2 0.5b 3 ). If all three robots bid {t 3,t 4 } during the seond round of the SIV aution and robot a 1 is the winning robot, then no targets are assigned to robots a 2 and a 3 in the seond round. Thus, n 3 = 2 and n 3 = 0.5(n 2 + n 2 ). Corollary 1 allows us to follow (Fakharoenphol, Harrelson, and Rao 2003) to show that the team osts of SIV autions are at most a fator of O(1/α) larger than minimal. Lemma 2 The minimal team ost satisfies 0.5α(b 1) j 0 2 a 3 b j n j Proof: Consider any target t whose lateny is in the range (0.5αb j+1,0.5αb j+2 ] in the optimal assignment. First, target t ontributes to n i for all 0 i j sine n i is the number of targets whose latenies are larger than 0.5αb i+1 in the optimal assignment. Seond, 0.5α(b 1) j i=0 bi = 0.5α(b j+1 1) is at most the lateny of target t in the optimal assignment. Summing over all targets T yields the lemma. Lemma 3 The team ost of an SIV aution satisfies 2 j 0 Proof: Consider any target t that is assigned during the (j + 1)st round of the SIV aution. First, target t ontributes to n i+1 for all 0 i j sine n i+1 is the number of unassigned targets in the beginning of the (i + 1)st round of the SIV aution. Seond, the lateny of target t in the assignment produed by the SIV aution is at most 2(b j+3 b 2 )/(b 1) beause the path ost of the set of targets assigned a robot during the ith round is at most b i+1 and the robot ould return to its initial loation before visiting the targets assigned to it in future rounds (but atually moves with minimal travel distane from eah target to the next), resulting in target t having lateny at most j+1 i=1 2bi+1 = 2b 2 j i=0 bi = 2(b j+3 b 2 )/(b 1). Third, 2 j i=0 bi+3 = 2(b j+4 b 3 )/(b 1) is at least 2(b j+3 b 2 )/(b 1) for all b > 1 and j 0. Summing over all targets T yields the lemma. Theorem 3 The team osts of SIV autions are at most a fator of O(1/α) larger than minimal if eah robot alulates its bids with a (1/α)-approximation algorithm for determining rooted k-msts. Proof: Continuing to follow (Fakharoenphol, Harrelson, and Rao 2003), let C = 2 j 0 bj+3 n j+1 be the upper bound on the team ost of an SIV aution from Lemma 3. Then, C = 2 j 0 = 2b 3 n j 1 (Corollary 1) 2b 3 n j 1 b j+3 0.5(n j + n j ) (n 0 j 0 = 2b 3 n 0 + j 1 = b 3 n 0 + b 3 j 0 b j n j) 2b 3 j 0 (Lemma 2) Solving for C yields b j+3 n j + j 1 b j+3 n j b j n j + b j 0 b j n j + b j 0 4b 3 1 (b 1) α + 0.5bC 8b 3 1 C (b 1)(2 b) for the given onstant b (1,2). α There exist onstant-fator approximation algorithms for determining unrooted k-msts for given sets of targets T, suh as (Garg 1996; Arora and Karakostas 2000), whih an be transformed into onstant-fator approximation algorithms for determining rooted k-msts for given sets of targets T as follows (Awerbuh et al. 1999): For x = k,..., T, determine an unrooted k-tree for the x targets in T losest to the root and then onnet it to the root. Return the resulting rooted k-tree with the smallest ost. Thus, one an implement SIV autions so that their team osts are at most a onstant fator larger than minimal.

5 t 3 t 2 t 4 t 1 a +e t 5 t 7 t t8 6 e e e e t 9 t 10 Figure 4: Example 2 Corollary 2 The team osts of SIV autions are at most a fator of O(1) larger than minimal if eah robot alulates its bids with a onstant fator approximation algorithm for determining rooted k-msts. However, onstant-fator approximation algorithms for determining rooted k-msts are slow sine they rely on primal-dual algorithms with Lagrangean relaxation (Chudak, Roughgarden, and Williamson 2004). We are not neessarily interested providing the best possible approximations but a good trade-off between runtime and the resulting team ost to prevent robots from being idle eah time they alloate targets among themselves. We now show that there exist fast approximation algorithms for determining rooted k-msts for given sets of targets T that still result in a better guarantee on the team osts than the one urrently known for SSI autions. Consider, for example, a simple nearestneighbor algorithm that determines a minimum spanning tree for the root and the k targets in T losest to the root. Theorem 4 The nearest-neighbor algorithm produes rooted k-trees whose osts are at most a fator of k larger than minimal. Proof: Let T be the targets in the rooted k-tree produed by the nearest-neighbor algorithm and T be the targets in the rooted k-mst. Let D be the largest distane from the root to any target in T and D be the largest distane from the root to any target in T. First, D D per onstrution of T. Seond, the ost of the rooted k-mst is at least D by the triangle inequality. Finally, the ost of the rooted k- tree produed by the nearest-neighbor algorithm is at most kd kd and thus at most a fator of k larger than minimal. The upper bound of Theorem 4 is tight, as shown in Figure 4. The rooted k-tree produed by the nearest-neighbor algorithm ontains the targets t 1,...,t 5 and has ost 5, while the rooted k-mst for k = 5 ontains the targets t 6,...,t 10 and has ost + 5e. The ost ratio approahes k as e approahes 0. Corollary 3 The team osts of SIV autions are at most a fator of O ) larger than minimal if eah robot alulates its bids with the nearest-neighbor algorithm for determining rooted k-msts (sine k T ). Conlusions We proposed a new aution-like algorithm, alled sequential inremental-value (SIV) aution, whih assigns as many tasks per round to robots as possible as long as their individual osts for performing these tasks are at most a given bound, whih inreases exponentially from round to round. Our theoretial results showed that the resulting sum of the latenies of all tasks is at most a onstant fator larger than minimal. Referenes Arora, S., and Karakostas, G A 2+ǫ approximation algorithm for the k-mst problem. 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