Symmetry Breaking for k-Robust Multi-Agent Path Finding
SSymmetry Breaking for k -Robust Multi-Agent Path Finding Zhe Chen, Daniel Harabor, Jiaoyang Li, Peter J. Stuckey Monash University University of Southern California { zhe.chen,daniel.harabor,peter.stuckey } @monash.edu, [email protected] Abstract
During Multi-Agent Path Finding (MAPF) problems, agentscan be delayed by unexpected events. To address such situa-tions recent work describes k -Robust Conflict-Based Search( k -CBS): an algorithm that produces a coordinated andcollision-free plan that is robust for up to k delays for anyagent. In this work we introduce a variety of pairwise sym-metry breaking constraints, specific to k -robust planning, thatcan efficiently find compatible and optimal paths for pairs ofcolliding agents. We give a thorough description of the newconstraints and report large improvements to success rate ina range of domains including: (i) classic MAPF benchmarks,(ii) automated warehouse domains, and (iii) on maps fromthe 2019 Flatland Challenge, a recently introduced railwaydomain where k -robust planning can be fruitfully applied toschedule trains. Multi-Agent Path Finding (MAPF) is a coordination prob-lem where we need to find collision-free paths for a teamof cooperating agents and is known to be NP-hard ongraphs and grids (Yu and LaValle 2013; Banfi, Basilico, andAmigoni 2017). When MAPF problems are solved in prac-tice, agents can sometimes be unexpectedly delayed duringplan execution; e.g. due to exogenous events or mechanicalproblems. Currently, there exist two principal approaches tohandle such delays. The first approach involves robust exe-cution policies (Ma, Kumar, and Koenig 2017; H¨onig et al.2019). Here dependencies are introduced to guarantee thatagents execute their plans in a specific and compatible or-der. Another approach is to reason about potential delays atthe planning stage which involves computing robust plans .Following (Atzmon et al. 2018) we say that a plan is k -robust if the individual path of each agent remains valid forup to k unexpected delays of that agent. In other words, pro-vided each agent waits for no more than k timesteps on theway to its target, its plan is guaranteed to be collision-free.In addition to their execution benefits, k -robust plans arevaluable in application areas where agents must maintainminimum safety distances. Such constraints appear in rail * scheduling, quay crane scheduling, planning for warehouserobots and others. In these settings a k -robust plan naturallyprovides k timesteps of distance between agents that are aremoving, while also allowing agents to stay close if they arenot moving. Furthermore, k -robust plans can be used in con-junction with robust execution policies to benefit from bothmethods (Atzmon et al. 2020).To compute k -robust plans, Atzmon et al. (2018) propose k -robust Conflict-Based Search ( k -CBS), a robust variant ofthe popular and well known branch-and-replan strategy usedfor MAPF (where k = 0) (Sharon et al. 2015), and a SAT-based solution, which cannot solve problems on large gridslike brc202d DAO map(Stern et al. 2019). A main problemwith CBS is that the algorithm is extremely inefficient whenreasoning equivalent permutations of conflicts that can oc-cur between pairs of agents, such as rectangle symmetry (Liet al. 2019), corridor symmetry, and target symmetry (Liet al. 2020). Further complicating the situation is that thereasoning techniques proposed to handle these situations, donot always extend straightforwardly to the k -robust case.To address this gap in the literature we introduce a varietyof specialised k -robust symmetry breaking constraints thatdramatically improve performance for the k -CBS algorithm.Experimental results show very large gains in success ratefor k -CBS; not only on classic MAPF benchmarks but alsoin two application specific domains: in warehouse logistics ,where k -robust plans are desirable and in railway scheduling where k -robust plans are mandatory. We consider a multi-agent coordination problem where theoperating environment is an undirected (e.g. gridmap) or di-rected (e.g. rail network) graph G = ( V, E ) . We restrictgraph G to be a 4-neighbour grid and we place upon it m agents { a ...a m } . Every agent a i is assigned an initial ver-tex s i and a goal vertex g i . Time is discretised into unit-sizesteps. In each timestep, agents can move to an adjacent ver-tex or wait at the current location. Each move or wait actionhas an associated unit cost.We say that a path is k -robust if for each location v vis-ited at time t by agent a no other agent visits the location inthe time interval [ t, t + k ] . We call a k -delay vertex conflict the situation where agent a i at timestep t and a j at timestep t (cid:48) = t + ∆ , ∆ ∈ [0 , k ] , visit the same location v . We de- a r X i v : . [ c s . A I] F e b a) Rectangle (b) Corridor (c) Target Figure 1: Examples of rectangle, corridor and target conflictswhen k =0, reproduced from (Li et al. 2020).note such a conflicts by (cid:104) a i , a j , v, t, ∆ (cid:105) . When k = 0 it isfurther possible for two agents to cross the same edge in op-posite directions at the same time, resulting in a so-called edge conflict . Notice however that with k > such a cross-ing will always result in a vertex conflict. Thus for k > wedo not need to model edge conflicts.A solution to the problem (equiv. a k -robust plan ) is a setof paths, one for each agent a i , which moves each agent a i from its start location s i to its goal location g i such that thereare no k -delay conflicts with the plans of any other agent.Our objective is to find a k -robust plan which minimises thesum of all individual path costs (SIC). k -Robust Conflict-Based Search k -Robust Conflict-Based Search ( k -CBS) (Atzmon et al.2018) is a two-level search algorithm specialised from clas-sical MAPF ( k =0) (Sharon et al. 2015). At the high-level, k -CBS searches (in a best-first way) a binary constraint tree( CT ) where each node is a complete assignment of paths toagents (i.e. a plan). The process of finding such single agentpaths constitutes the low level of k -CBS. Here individualagents find paths (via A*) from start to target while subjectto a set of collision-avoiding constraints .The search process of k -CBS proceeds as follows: At eachiteration k -CBS expands the CT node with lowest f -cost. Ifthe current node is conflict-free then that node is a goal andthe search terminates having found a least-cost feasible plan.Otherwise, the current node must contain at least one pair ofagents that are in collision. Suppose for example that theconflict is (cid:104) a i , a j , v, t, ∆ (cid:105) . This situation occurs when agent a i at timestep t and a j at timestep t (cid:48) = t + ∆ , ∆ ∈ [0 , k ] ,visit the same location v . To resolve the conflict k -CBS re-plans each of the two affected agents and thus generates twonew candidate plans. To each child node is added a timerange constraint which resolves the situation now and inall future descendant nodes derived from each respectivechild. The constraint (cid:104) a i , v, [ t, t + k ] (cid:105) says that agent a i isnot allowed to occupy vertex v at any timestep in the range [ t, t + k ] . The second child node, where a j is replanned, re-ceives a similar constraint: (cid:104) a j , v, [ t, t + k ] (cid:105) . k -CBS continues in this way, splitting and searching,while the current CT node contains any conflict. This ap-proach is solution complete and optimal. It guarantees tofind a k -robust plans (Atzmon et al. 2018), if any such planexists, since the union of valid plans permitted by the twochild nodes is the same as at the parent node (i.e. addingconstraints does not eliminate valid solutions). (a) (b) Figure 2: (a)A collision-free solution, orange path, is elimi-nated, if we simply put ”thick” barrier constraints to resolvea k -delay rectangle conflict when k = 4 . (b) How number ofhigh-level nodes expanded by k -CBS increases with k andthe size of the rectangle area. Conflict selection strategies:
Deciding which conflictto resolve next is critical to the success of ( k -)CBS. In thiswork we follow Boyarski et al. (2015) where authors classifyconflicts as cardinal , semi-cardinal and non-cardinal :• A conflict C is cardinal if replanning for any agent in-volved in the conflict increases the SIC.• A conflict C is semi-cardinal if replanning for one agentinvolved in the conflict always increases the SIC whilereplanning for the other agent does not.• A conflict C is non-cardinal otherwise.In Boyarski et al. (2015) it is shown that resolving cardi-nal conflicts first can dramatically reduce the size of theresulting CBS tree. After all cardinal conflicts are resolvedwe choose semi-cardinal conflicts and finally non-cardinal.Similar to that work we use a Multi-valued Decision Dia-gram (MDD) to classify conflicts. Each MDD records allnodes (i.e., vertex-time pairs) that can appear on an opti-mal path for each agent. (Semi-)cardinal conflicts require theconflict node to be a singleton for (resp. one) both agents, i.e.all optimal paths must pass through the node. CBS Heuristics (CBSH):
Different from (Atzmon et al.2018) but following (Felner et al. 2018), we also exploitknown cardinal conflicts to derive a minimal cost increase heuristic for our high-level A* search. This strategy isknown to improve the performance of CBS and has becomea common approach in leading MAPF ( k = 0) solvers. k -Symmetries Symmetries in MAPF occur when two agents repeatedly runinto one another along equivalent individually optimal paths.Figure 2a shows an example for k = 4 . Notice that eachagent has available a number of optimal-cost paths. How-ever every optimal path of agent a is in collision with ev-ery optimal path of agent a and vice versa. Without detect-ing such situations k -CBS will repeatedly split, node afternode, growing the CT tree in order to enumerate all pos-sible collisions in the highlighted rectangle area. Figure 2bshows the result: each time k -CBS splits, it potentially dou-bles the amount of search yet remaining to reach the goalnode. The size of the CT tree grows exponentially as thesize of the rectangle area increases, causing k -CBS to returnimeout failure. Note that as k increases, the size of the CTtree also grows exponentially, which makes the problem ex-tremely difficulty to solve. All this difficulty can be avoidedby recognising there exists a simple optimal strategy: one ofthe two agents has to wait. For k = 0 one agent must waitfor one timestep. For k > one agent may need to waitfor more than one timestep. However, as k grows large theproblem can permit optimal-cost bypass routes that allowone agent to avoid the rectangle area entirely. For examplein Figure 2a, when k ≥ the solid-orange-line path becomesoptimal and agent a can reach its target without waiting.In this work we consider three distinct symmetric situa-tions which can appear in k -robust MAPF:• Rectangle Symmetries, as illustrated in Figure 1a.• Corridor Symmetries, as illustrated in Figure 1b.• Target Symmetries, as illustrated in Figure 1c.Rectangle symmetries have previously been studied in thecontext of MAPF ( k = 0) (Li et al. 2019) while Corridor andTarget Symmetries have only recently been introduced (Liet al. 2020), again in the context of MAPF ( k = 0). Eachtime authors show that symmetries are common in a rangeof standard benchmarks and they report dramatic gains inperformance when these symmetric situations are resolvedvia specialised reasoning techniques. We adapt each of theseideas to k -robust planning and we report similarly strong re-sults. Generalising these constraints is not simply academic.As we show in the experimental section, k -robust plans areneeded for important practical problems. Without the de-velopment of suitable algorithmic techniques such problemswill remain out of reach to MAPF planners. k -Robust Rectangle Reasoning Rectangle symmetries arise in k -CBS when the paths of twoagents cross topologically. The agents are heading in thesame directions (e.g. down and right in Figure 2a) and thereexists for each many different but equally shortest pathswhich arise from re-ordering their individual moves. In suchcases the standard replanning strategy of k -CBS does nothelp to immediately resolve the problem. Definition 1 ( k -delay rectangle conflict) . A k -delay rect-angle conflict between two agents occurs if all paths withcost between optimal and optimal+ k (inclusive) for the twoagents that enter a given rectangular area have a k -delayvertex conflict in the rectangular area. Rectangle conflicts pose substantial challenges for k -robustplanning in general and for k -CBS in particular because:• the agents do not have to reach positions at exactly thesame time to have a conflict;• the delay caused by the conflict can be up to k + 1 ; and• for k ≥ an agent can leave and enter the conflicting areawithout adding more than k steps to its path.To address these challenges we follow Li et al. (2019) whereauthors develop barrier constraints : a pruning strategy thatcan efficiently resolve rectangle conflicts for CBS with k =0 in a single branching step. With respect to Figure 2a, one barrier constraint prohibits agent a from occupying cells(5, 2), (5, 3), and (5,4) at timesteps 3, 4, and 5, respec-tively. Similarly, the other barrier constraint prohibits agent a from occupying cells (3, 4), (4, 4), and (5, 4) at timesteps3, 4, and 5, respectively. Notice that each barrier blocks allequivalent shortest paths and forces one agent or the other towait, thus resolving the conflict.A straightforward idea for extending this strategy to k > would be to increase the “thickness” (number of timesteps)of the barrier constraints. We therefore introduce temporalbarrier constraints , which unify the (temporal) range con-straints of (Atzmon et al. 2018) and the (spatial) barrier con-straints of (Li et al. 2019). Definition 2 (Temporal barrier constraint) . Given a vertex-time pair p = ( u, t ) we denote with ot ( p, v ) the opti-mal time to reach vertex v = ( v x , v y ) from vertex u =( u x , u y ) starting at timestep t . We compute the optimaltime by adding to t the Manhattan distance from u to v : ot ( p, v ) = t + | v x − u x | + | v y − u y | . A w temporal barrierconstraint , denoted B ( a x , V, p, w ) , forbids an agent a x , cur-rently at vertex-time pair p , from visiting any vertex v ∈ V at its optimal time or up to w timesteps later. That is, the bar-rier constraint B ( a x , V, p, w ) is the set of time-range vertexconstraints (cid:104) a x , v, [ ot ( p, v ) , ot ( p, v ) + w ] (cid:105) , v ∈ V . To resolve the k -robust rectangle conflict in Figure 2a, onemight think that we can replace the two barrier constraintsused by (Li et al. 2019) with two k temporal barrier con-straints at the exit of the rectangle. However, this approachis not complete! For example, when k = 4 , the orange line isa collision-free path for agent a , no matter what path agent a takes. But this solution is eliminated by the w = 4 tem-poral barrier constraints. Therefore, we propose a novel ap-proach that enlarges the rectangle area being reasoned aboutand adding temporal barrier constraints based on it with ad-justed thickness so that we can preserve the completenessand optimality by taking into consideration such bypasses. Given a 4-neighbor grid map, we define a rooted rect-angle R as the set of vertices occurring in the rectan-gle defined by the two corner points D = ( D x , D y ) theroot corner, and E = ( E x , E y ) the opposite corner. Asshown in Figure 3, the illustrated rectangle is defined by D = (4 , and E = (5 , . Given D and E , we de-fine four sides: S R ) = { ( D x , D y ) .. ( E x , D y ) } , S R ) = { ( D x , D y ) .. ( D x , E y ) } , S R ) = { ( E x , D y ) .. ( E x , E y ) } ,and S R ) = { ( D x , E y ) .. ( E x , E y ) } . We define the shiftedside Sj ( R ) l , j ∈ [1 , , l ∈ [0 , (cid:98) k (cid:99) ] , as the side Sj ( R ) shifted away from the center of R by l grid locations, notethe Sj ( R ) cannot be shifted beyond the start or goal loca-tions of agents involved in the conflict. Define the y -shiftedstart location D yl as the root corner D shifted l locationsalong y -axis away from R , and the x -shifted start location D xl as the start location shifted l locations along y -axis awayfrom R . Similarly define E yl and E xl .Given these definitions, the shifted barrier constraintswith some particular thickness can define a k -delay rectan-gle conflict , that is all paths for two agents a and a thatigure 3: (1) Example of temporal barrier constraints. In thiscase k = 2 , S R ) l and S R ) l are shifted for l = 1 grid location away from S R ) and S R ) , and k = 4 so S R ) l and S R ) l are shifted for l = 2 timestepsfrom S R ) and S R ) . (2) The yellow area is the rectangle R . Adding the light and dark orange areas gives rectangle R (cid:48) . (3) The solid red line is a path traversing rectangle R and must have k -delay conflicts. The dashed red line is apath bypassing the rectangle R , but traversing rectangle R (cid:48) and must have k -delay conflicts as well. The green dashedline is a collision-free path bypassing the rectangle R (cid:48) . (4)Extended constraints (Step Temporal Barrier Constraint) de-fined in Section 5.2 fill the gaps in the barriers for the darkorange area.cross these barriers must inevitably result in a k -delay con-flict. Theorem 1.
Consider a k -robust MAPF problem with anarbitrary rooted rectangle R defined by corners D and E and two arbitrary integers ≤ k , k ≤ k . Let root time rt be the minimum of the timestep when agent a or a reachesthe root corner D , l = (cid:98) k (cid:99) , l = (cid:98) k (cid:99) , p = ( D yl , rt − l ) , and p = ( D xl , rt − l ) . If the paths for agents a and a violate all of the four constraints: • a Entrance: B ( a , S R ) l , p , k ) , • a Exit: B ( a , S R ) l , p , k ) , • a Entrance: B ( a , S R ) l , p , k ) , • a Exit: B ( a , S R ) l , p , k ) ,the paths of the two agents have a k -delay vertex conflict.Proof. Assume that S R ) is the top of the rooted rectan-gle R , S R ) the bottom, S R ) the left, and S R ) theright. The other cases follow similarly. S R ) l , S R ) l , S R ) l , S R ) l are corresponding sides shifted awayfrom R . These shifted sides define a new larger rectangle R (cid:48) with two corner points D (cid:48) and E (cid:48) , as shown in Figure 3.In order to violate the first two constraints, agent a has toenter the top of rectangle R (cid:48) through a vertex v on S R ) l at timestep ot ( p , v ) or up to k timesteps later, and leavefrom the bottom through a vertex v on S R ) l at timestep ot ( p , v ) or up to k timesteps later, so it can wait for at most k timesteps in R (cid:48) but cannot leave R (cid:48) , since this would re-quire at least extra l + 2 timesteps comparing with a short-est path across R (cid:48) but l +2 > k . Every vertex v it visits in R (cid:48) is visited within the time range [ ot ( p , v ) , ot ( p , v )+ k ] . Similarly agent a enters the left of rectangle R (cid:48) througha vertex v on S R ) l at timestep ot ( p , v ) or up to k timesteps later, and leaves the right through a vertex v on S R ) l at timestep ot ( p , v ) or up to k timesteps later.Again it cannot leave R (cid:48) , since this would require at leastextra l + 2 timesteps and it can take at most k wait in R (cid:48) .Every vertex v it visits in R (cid:48) is visited within the time range [ ot ( p , v ) , ot ( p , v ) + k ] .Now since agent a crosses from top to bottom andagent a from left to right, their paths must cross, say atvertex v ∈ R (cid:48) . Assume that agent a visits vertex v at t ∈ [ ot ( p , v ) , ot ( p , v ) + k ] while agent a visits ver-tex v at t ∈ [ ot ( p , v ) , ot ( p , v ) + k ] . Since ot ( p , v ) and ot ( p , v ) share the same root time rt , ot ( p , v ) = ot ( p , v ) ,and k and k are both less than or equal to k , which make | t − t | ≤ k , there is a k -delay conflict (cid:104) a , a , v, t , t − t (cid:105) (if t ≤ t ) or (cid:104) a , a , v, t , t − t (cid:105) (if t > t ). The temporal barrier constraints proposed in Theorem 1 donot cover all situations where two agents must have a k -delay vertex conflict in a rectangle area because they do notcover the entire circumference of rectangle R (cid:48) . As the reddashed line path shown in Figure 3 illustrates, if agent a enters the top of rectangle R (cid:48) through corner D (cid:48) at the opti-mal time, and leaves from left-bottom vertex at the optimaltime. This path (red dashed line) traverses through S R ) l and is (cid:98) k (cid:99) timesteps longer than paths entering through S R ) l optimally and leaving through S R ) l optimally.Now agent a enters R (cid:48) via S R ) l at the optimal time orup to k timesteps later, so they still have a k -delay conflict.We cannot simply extend temporal barrier constraints tothe entire circumference of R (cid:48) to eliminate such conflicts, asit would also eliminate the green dashed collision-free pathin Figure 3. We thus consider a stronger reasoning method. Definition 3 (Step temporal barrier constraint) . Denotedby B step ( a x , S, p, k (cid:48) ) this constraint reduces the temporalwidth of the barrier B ( a x , S, p, k (cid:48) ) by a value of 2 for eachstep away from the original barrier. Let l (cid:48) = (cid:98) k (cid:48) (cid:99) , we addrange constraints (cid:104) a x , v, [ ot ( p, v ) , ot ( p, v ) + k (cid:48) − d ] (cid:105) foreach vertex v in the same line as S at distance d ∈ [1 , l (cid:48) ] from the original barrier. For the rectangle R (cid:48) in Figure 3, the step tempo-ral barrier constraint B step ( a , S R ) l , p , k ) consistsof B ( a , S R ) l , p , k ) and the additional range con-straints (cid:104) a , (2 , , [ rt + 1 , rt + 1] (cid:105) , (cid:104) a , (3 , , [ rt, rt + 2] (cid:105) , (cid:104) a , (6 , , [ rt + 1 , rt + 3] (cid:105) , and (cid:104) a , (7 , , [ rt + 2 , rt + 2] (cid:105) .Notice how the time ranges shrink further from the originalbarrier. This prevents the red dashed path in Figure 3. Wecan now extend Theorem 1: if the paths of agents a and a violate all of the four following constraints:• a Entrance: B step ( a , S R ) l , p , k ) • a Exit: B step ( a , S R ) l , p , k ) • a Entrance: B step ( a , S R ) l , p , k ) • a Exit: B step ( a , S R ) l , p , k ) the paths of the two agents have a k -delay vertex conflict.The proof is similar. .3 Resolution of k -Delay Rectangle Conflicts We can always resolve a rectangle conflict by four-waybranching adding to each branch one of the constraints: a Entrance, a Exit, a Entrance, and a Exit; since we knowthat one of them must be violated in any solution. But inmany cases, we can correctly resolve the rectangle conflictby two-way branching on the constraints a Exit and a Exit,if both agents satisfy the following condition:
Condition 1
All possible paths that traverse the exit bar-rier must also traverse the entrance barrier.We can use a k -MDD to check that the condition is sat-isfied. A k -MDD for agent a i is a modified Multi-ValuedDecision Diagram (MDD) (Boyarski et al. 2015) that storesall paths of agent a i from start to goal with path length nomore than k above the optimal. MDDs are widely used inCBS algorithms to store all optimal paths, the k -MDD is adirect extension.If either agent’s k -MDD shows that paths that bypass theentrance barrier and traverse the exit barrier exist, the givenconflict cannot be resolved by two-way branching on thegiven barriers, as it may eliminate conflict free paths thatbypass the entrance barrier. If any combination of k and k lead to Condition 1 being satisfied, the given conflict can re-solved by two-way branching on the exit barriers. If none ofthe combinations satisfy
Condition 1 , the given conflict willbe resolved as a normal conflict (i.e. we never do four-waybranching).We can classify k -delay rectangle conflicts as cardinal,semi-cardinal and non-cardinal using the k -MDD:• A k -delay rectangle conflict is cardinal , if all paths in the k -MDDs of both agents traverse the exit barrier, whichmeans that replanning for any agent involved in the con-flict increases the SIC.• A k -delay rectangle conflict is semi-cardinal , if only oneagent has all paths in its k -MDD traverse the exit barrier,which again means that replanning this agent involvedalways increases the SIC while replanning for the otheragent does not.• A k -delay rectangle conflict is non-cardinal if both agentshave paths in their k -MDD bypass their exit barriers.We can then prioritize selecting conflict based on cardinality. When we detect a vertex conflict during CBS we need torecognise that it’s actually a rectangle conflict, in order toperform rectangle symmetry breaking.Assume that agents a and a have a k -delay vertex con-flict (cid:104) a , a , v, t, ∆ (cid:105) . Let d and d be the moving directionswhen agents a and a enter vertex v , respectively. If theyare the same directions, then there is an earlier vertex con-flict (where they both came from). If they are opposite direc-tions, then there is no rectangle conflict.So assume d and d are orthogonal directions. Let ( B x , B y ) be the earliest vertex in the path of agent a where all moves from here to v are in direction d or d , Figure 4: Detecting a rectangle conflict when k = 2 .similarly define ( B x , B y ) for agent a . Let t b be the ear-liest timestep that a visits ( B x , B y ) and t b be the earli-est timestep that a visits ( B x , B y ) . Let ( A x , A y ) bethe latest vertex in the path of agent a where all movesfrom v to here are in directions d or d , similarly define ( A x , A y ) for agent a .Define D x to be the closer of B x and B x to v x , simi-larly for D y . Define E x to be the closer of A x and A x to v x , similarly for E y . Figure 4 shows an example.Let rt = t b + | B x − D x | + | B y − D y | and rt = t b + | B x − D x | + | B y − D y | . We define the root time rt = min( rt , rt ) .We then for each value k ∈ { , , . . . , k } and k ∈{ , , . . . , k } check if the agents satisfy Condition 1 us-ing the Step Temporal Barriers defined by these values. Wedo so by examining the k -MDD for each agent, temporar-ily blocking its entrance barrier and seeing if its exit barrieris still reachable. If not then Condition 1 holds. We try thevalues for k and k in decreasing order to find the strongestblocking conditions possible. If k = a, k = b satisfiesthe conditions we don’t investigate any pairs ( a (cid:48) , b (cid:48) ) where a (cid:48) ≤ a and b (cid:48) ≤ b .Figure 4 shows an example of the rectangle detection.In this example, agents a and a have a 2-delay conflict (cid:104) a , a , (3 , , , (cid:105) with d pointing right and d pointingdown. Then B is located at (1 , with t b = 2 and rt = 3 ,and B located at (2 , with t b = 2 and rt = 5 . Roottime rt = min ( t b , t b ) = 3 . We detect that for the case k = k = 2 the Step Temporal Barriers defined by thesevalues satisfy Condition 1 .We call the detection of rectangles and the associatedbranching K-CBSH-RM (it is a generalization of the RMtechnique defined by Li et al. (2019)).
Theorem 2.
K-CBSH-RM is correct.Proof.
Since
Condition 1 holds for the chosen Step Tem-poral Barriers, and using (extended) Theorem 1, each pairof paths that violate the exit barriers must conflict. Hencethe two-way branching removes no solutions, and advancesthe search since it is violated by the current paths. k -Robust Corridor Reasoning A corridor from B to E is a chain of nodes C wherenodes in C except B and E each have exactly two neigh-bours, and B and E have exactly one neighbour in C .Figure 1b shows a corridor where B = (3 , , C = (1 , , (2 , , (3 , , (4 , , (5 , } and E = (5 , . A cor-ridor conflict (Li et al. 2020) occurs when two agents have avertex or edge conflict occurring in a corridor. In the exam-ple agents a and a conflict at vertex (3, 3). Simply addinga vertex conflict constraint will not resolve the conflict, theywill continue to conflict in the corridor. Li et al. (2020) in-troduce corridor symmetry breaking constraints which weextend here for k -delay conflicts.The difference between corridor conflicts for k -robustCBS and normal corridor conflicts is that agents need to oc-cupy vertexes for extra timesteps to avoid k -delay conflicts.Assuming there is a corridor with length of l between vertex B and vertex E , and a k -delay conflict in the corridor withagent a ( a ) moving from B to E (resp. E to B ). Let t (resp. t ) be the earliest timestep when agent a ( a ) is ableto reach vertex E (resp. B ).Clearly when planning a k -robust solution, any path of a using the corridor that reaches vertex E at or before timestep t + l + k must conflict with any paths of a using the corri-dor that reach vertex B at or before timestep t + l + k . Butthere may be alternate paths the agents can take to reach B or E . Assuming agent a can reach vertex E at timestep t (cid:48) without using the corridor and agent a can reach vertex B at timestep t (cid:48) without traversing the corridor. Hence in plan-ning a k -robust solution, any path of a that reaches vertex E at or before timestep min ( t (cid:48) − k, t + l + k ) must con-flict with any path of a that reaches vertex B at or beforetimestep min ( t (cid:48) − k, t + l + k ) .Hence the constraint (cid:104) a , E, [0 , min ( t (cid:48) − k, t + l + k )] (cid:105) ∨ (cid:104) a , B, [0 , min ( t (cid:48) − k, t + l + k )] (cid:105) must hold inall solutions. To handle the corridor constraint we branch onthis disjunction. Clearly Theorem 3. k -robust Corridor Reasoning is correct. k -Robust Target Reasoning A target conflict (Li et al. 2020) occurs when one agent a reaches its goal vertex g at timestep l , and another agent a conflicts with agent a at vertex g at some later timestep t, t ≥ l . Consider Figure 1c where agent a reaches itsgoal cell (4,2) at timestep 1, and then agent a tries to tra-verse cell (4,2) at timestep 3. Simply adding the constraint (cid:104) a , (4 , , [3 , (cid:105) causes a to wait before entering cell (4,3)at timestep 4 and then the conflict reoccurs.To avoid this Li et al. (2020) resolve the conflict bybranching on et ≤ t ∨ et > t where et is the end timefor agent a . In the first case, since agent a finishs before orat timestep t , agent a can never use location g at timestep t or after. In the second case agent a cannot finish beforetimestep t + 1 freeing up the location for agent a .Planning a k -robust solution requires the avoiding of k -delay conflict. Therefore, we branch on et ≤ t + k ∨ et >t + k . The first case forces agent a to finish before timestep t + k preventing agent a (or any other agent) from usingvertex g at timestep t , the second case forces agent a notto finish earlier so vertex g at timestep t is freed up for agent a . Again clearly Theorem 4. k -Robust Target Reasoning is correct. Note that to handle target symmetries we have to updatethe low-level path finder for agents to take into account newkinds of constraints where we restrict the end time of anagent, and where we prevent any agent from using a locationfrom some time point onwards. Both are straightforward ad-ditions. See Li et al. (2020) for details.
The implementation is based on CBS with rectangle, corri-dor and target reasoning of Li et al. (2020) and support for k -robust planning is added on top of it. It is programmedin C++ and experiments were performed on a server withAMD Opteron 63xx class CPU and 32 GB RAM. For eachmap, we keep increasing the number of agents, and for eachnumber of agent we solve 25 different instances. The timelimit is set to 90s for each instance. In result plots, K-CBSis the current state of art algorithm to plan k -robust planproposed by Atzmon et al. (2018),which is selected as thebaseline, K-CBSH is our extension of K-CBS with heuris-tics (Section 3), RM adds k -robust Rectangle reasoning, Cadds k -robust Corridor Reasoning, and T adds k -robust Tar-get Reasoning. Experiment 1: Game Maps
The MAPF research com-munity have developed a series of benchmark maps fromgames (Stern et al. 2019). They are available from movin-gai.com. We use 25 even scenarios from movingai.com toevaluate our algorithms. We run experiments on followingrepresentative maps:
Brc202d , Den520d , Random-32-32-10 ,and maze-128-128-1 .Figure 5 shows the experiment results on grid game maps.K-CBSH-RM based algorithms shows significant highersuccess rate compared with K-CBS on
Brc202d , Den520d ,and
Random-32-32-10 . Although k -robust corridor reason-ing does not help on these three maps and k -robust tar-get reasoning slightly helps, they effectively improves thesuccess rate on maze-128-128-1 . As k increases, the prob-lem becomes harder, and the success rate drops, but thesymmetry-breaking algorithms still show significant advan-tages over K-CBS. Experiment 2: Warehouse Map
We use a ×
79 Ware-house map (Li et al. 2020) with randomly generated prob-lems to evaluate the performance of robots in warehouse sys-tem. Figure 5e shows that k-CBSH significantly improvessuccess rate, k -robust target reasoning and corridor reason-ing helps to further improve the success rate of K-CBSH-RM. Here target reasoning is clearly very important. Experiment 3: Simplified Railway System
The Flatlandchallenge is a railway scheduling challenge (Swiss FederalRailways 2019) provides a simplified railway simulator us-ing a directed grid map, where trains cannot move back-wards. Railway systems have headway control, one traincannot start to enter a railway block if another train cur-rently occupies the block. Hence the railway domain re-quires k = 1 robust plans. Our experiments use flatland-rl v2.0.0 to generate experiment problems. Note, no targetconflicts can occur in the Flatland challenge scenarios sincetrains “disappear” when reaching their destination. a) Random 32x32 (b) Den520d (c) Brc202d (d) Maze 128x128 (e) Warehouse map (f) Flatland Railways Figure 5: Success rate versus number of agents on different problems.We have two settings for evaluation: (1) a ×
100 fixeddense map contains fixed 140 start and goal locations, andeach pair of start and goal are connected by 5 railways; and(2) a ×
100 incremental sparse map has one start andgoal location per agent, and each pair of start and goal areonly connected by 1 railway.The experiment on railway maps, Figure 5f shows that K-CBSH and K-CBSH-RM performs substantially better thanK-CBS. k -robust corridor reasoning helps further improveperformance on sparse railways map. Clearly the rectanglemethods are more important on the denser map, where moresymmetric conflicts are possible, and k -robust corridor rea-soning is more important on sparse maps. Ratio of Rectangle Conflicts
The Figure 6 shows the per-centage of k -delay rectangle conflicts among all resolvedconflicts using K-CBSH-RM-C-T as k increases. The statis-tics on conflicts are derived from Experiment 1 and Exper-iment 2. Clearly, as k increases, the percentage of k -delayrectangle conflicts rises on the maps where rectangle con-flicts can occur, hence the importance of k -robust rectanglereasoning is demonstrated. Figure 6: Percentage of k -delay rectangle conflicts amongall conflicts as k increases when using K-CBSH-RM-C-T. This research introduces symmetry resolution methods forgenerating k -robust plans, which are vital for robust (e.g.warehouse robotics) and safe (e.g. railway scheduling)multi-agent plans. Symmetry reasoning methods improvedramatically on K-CBS, with k -robust rectangle reasoningbeing the most important, while k -robust corridor and targetreasoning can further improve the performance. eferences Atzmon, D.; Stern, R.; Felner, A.; Wagner, G.; Bart´ak, R.;and Zhou, N.-F. 2018. Robust multi-agent path finding. In
Eleventh Annual Symposium on Combinatorial Search .Atzmon, D.; Stern, R.; Felner, A.; Wagner, G.; Bart´ak, R.;and Zhou, N.-F. 2020. Robust multi-agent path finding andexecuting.
Journal of Artificial Intelligence Research
IEEE Robotics and Automation Letters
Twenty-Fourth International Joint Conference on Artifi-cial Intelligence .Felner, A.; Li, J.; Boyarski, E.; Ma, H.; Cohen, L.; Kumar, T.K. S.; and Koenig, S. 2018. Adding Heuristics to Conflict-Based Search for Multi-Agent Path Finding. In
Proceedingsof the 28th International Conference on Automated Planningand Scheduling (ICAPS) , 83–87.H¨onig, W.; Kiesel, S.; Tinka, A.; Durham, J. W.; and Aya-nian, N. 2019. Persistent and robust execution of mapfschedules in warehouses.
IEEE Robotics and AutomationLetters
Proceedings ofthe International Conference on Automated Planning andScheduling .Li, J.; Harabor, D.; Stuckey, P. J.; Ma, H.; and Koenig,S. 2019. Symmetry Breaking Constraints for Grid-basedMulti-Agent Path Finding. In
Proceedings of the NationalConference on Artificial Intelligence, Honolulu, HI, USA ,volume 27.Ma, H.; Kumar, T. S.; and Koenig, S. 2017. Multi-agentpath finding with delay probabilities. In
Proceedings ofthe Thirty-First AAAI Conference on Artificial Intelligence ,3605–3612.Sharon, G.; Stern, R.; Felner, A.; and Sturtevant, N. R. 2015.Conflict-based search for optimal multi-agent pathfinding.
Artificial Intelligence arXiv preprint arXiv:1906.08291