In the multi-agent path finding problem (MAPF) we are given a set of agents each with respective start and goal positions. The task is to find paths for all agents while avoiding collisions aiming to minimize an objective function. Two such common objective functions is the sum-of-costs and the makespan. Many optimal solvers were introduced in the past decade - two prominent categories of solvers can be distinguished: search-based solvers and compilation-based solvers. Search-based solvers were developed and tested for the sum-of-costs objective while the most prominent compilation-based solvers that are built around Boolean satisfiability (SAT) were designed for the makespan objective. Very little was known on the performance and relevance of the compilation-based approach on the sum-of-costs objective. In this paper we show how to close the gap between these cost functions in the compilation-based approach. Moreover we study applicability of various techniques developed for search-based solvers in the compilation-based approach. A part of this paper introduces a SAT-solver that is directly aimed to solve the sum-of-costs objective function. Using both a lower bound on the sum-of-costs and an upper bound on the makespan, we are able to have a reasonable number of variables in our SAT encoding. We then further improve the encoding by borrowing ideas from ICTS, a search-based solver. Experimental evaluation on several domains show that there are many scenarios where our new SAT-based methods outperforms the best variants of previous sum-of-costs search solvers - the ICTS, CBS algorithms, and ICBS algorithms.
More A*-based Algorithms. Enhanced Partial Expansion () @cite_49 avoids the generation of surplus nodes (i.e. nodes @math with @math where @math is the optimal cost; we assume standard A* notation with @math ) by using a priori domain knowledge. When expanding a node @math generates only the children @math with @math and the smallest @math -value among those children with @math ( @math stands for @math in the context of MAPF). The other children of @math are discarded. This is done with the help of a domain-dependent operator selection function (OSF). The OSF returns the exact list of operators which will generate nodes @math with the desired @math . Node @math is then re-inserted into OPEN setting @math to the @math -value of the next best child of @math . In this way, EPEA* avoids the generation of surplus nodes and dramatically reduces the number of generated nodes. An OSF for MAPF can be efficiently built as the effect on the @math -value of moving a single agent in a given direction can be easily computed. For more details see @cite_49 .
@cite_49
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When solving instances of problem domains that feature a large branching factor, A* may generate a large number of nodes whose cost is greater than the cost of the optimal solution. We designate such nodes as surplus. Generating surplus nodes and adding them to the OPEN list may dominate both time and memory of the search. A recently introduced variant of A* called Partial Expansion A* (PEA*) deals with the memory aspect of this problem. When expanding a node n, PEA* generates all of its children and puts into OPEN only the children with f = f(n). n is reinserted in the OPEN list with the f-cost of the best discarded child. This guarantees that surplus nodes are not inserted into OPEN. In this paper, we present a novel variant of A* called Enhanced Partial Expansion A* (EPEA*) that advances the idea of PEA* to address the time aspect. Given a priori domain-and heuristic-specific knowledge, EPEA* generates only the nodes with f = f(n). Although EPEA* is not always applicable or practical, we study several variants of EPEA*, which make it applicable to a large number of domains and heuristics. In particular, the ideas of EPEA* are applicable to IDA* and to the domains where pattern databases are traditionally used. Experimental studies show significant improvements in run-time and memory performance for several standard benchmark applications. We provide several theoretical studies to facilitate an understanding of the new algorithm. An @math maximum distance separable (MDS) array code of length @math , dimension @math , and sub-packetization @math is formed of @math matrices over a finite field @math , with every column of the matrix stored on a separate node in the distributed storage system and viewed as a coordinate of the codeword. Repair of a failed node (recovery of one erased column) can be performed by accessing a set of @math surviving (helper) nodes. The code is said to have the optimal access property if the amount of data accessed at each of the helper nodes meets a lower bound on this quantity. For optimal-access MDS codes with @math , the sub-packetization @math satisfies the bound @math . In our previous work (IEEE Trans. Inf. Theory, vol. 63, no. 4, 2017), for any @math and @math , we presented an explicit construction of optimal-access MDS codes with sub-packetization @math . In this paper, we take up the question of reducing the sub-packetization value @math to make it to approach the lower bound. We construct an explicit family of optimal-access codes with @math , which differs from the optimal value by at most a factor of @math . These codes can be constructed over any finite field @math as long as @math , and afford low-complexity encoding and decoding procedures. We also define a version of the repair problem that bridges the context of regenerating codes and codes with locality constraints (LRC codes), which we call group repair with optimal access . In this variation, we assume that the set of @math nodes is partitioned into @math repair groups of size @math , and require that the amount of accessed data for repair is the smallest possible whenever the @math helper nodes include all the other @math nodes from the same group as the failed node. For this problem, we construct a family of codes with the group optimal access property. These codes can be constructed over any field @math of size @math , and also afford low-complexity encoding and decoding procedures. We consider a system, comprising a library of @math files (e.g., movies) and a wireless network with a @math transmitters, each equipped with a local cache of size of @math files and a @math receivers, each equipped with a local cache of size of @math files. Each receiver will ask for one of the @math files in the library, which needs to be delivered. The objective is to design the cache placement (without prior knowledge of receivers’ future requests) and the communication scheme to maximize the throughput of the delivery. In this setting, we show that the sum degrees-of-freedom (sum-DoF) of @math is achievable, and this is within a factor of 2 of the optimum, under uncoded prefetching and one-shot linear delivery schemes. This result shows that (i) the one-shot sum-DoF scales linearly with the aggregate cache size in the network (i.e., the cumulative memory available at all nodes ), (ii) the transmitters’ caches and receivers’ caches contribute equally in the one-shot sum-DoF, and (iii) caching can offer a throughput gain that scales linearly with the size of the network. To prove the result, we propose an achievable scheme that exploits the redundancy of the content at transmitter’s caches to cooperatively zero-force some outgoing interference, and availability of the unintended content at the receiver’s caches to cancel (subtract) some of the incoming interference. We develop a particular pattern for cache placement that maximizes the overall gains of cache-aided transmit and receive interference cancellations. For the converse, we present an integer optimization problem which minimizes the number of communication blocks needed to deliver any set of requested files to the receivers. We then provide a lower bound on the value of this optimization problem, hence leading to an upper bound on the linear one-shot sum-DoF of the network, which is within a factor of 2 of the achievable sum-DoF. We consider a system of @math servers inter-connected by some underlying graph topology @math . Tasks arrive at the various servers as independent Poisson processes of rate @math . Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in @math . Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case @math is a clique. Since the servers are exchangeable in that case, the queue length process is quite tractable, and it has been proved that for any @math , the fraction of servers with two or more tasks vanishes in the limit as @math . For an arbitrary graph @math , the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph @math is said to be @math -optimal or @math -optimal when the occupancy process on @math is equivalent to that on a clique on an @math -scale or @math -scale, respectively. We prove that if @math is an Erd o s-R 'enyi random graph with average degree @math , then it is with high probability @math -optimal and @math -optimal if @math and @math as @math , respectively. This demonstrates that optimality can be maintained at @math -scale and @math -scale while reducing the number of connections by nearly a factor @math and @math compared to a clique, provided the topology is suitably random. It is further shown that if @math contains @math bounded-degree nodes, then it cannot be @math -optimal. In addition, we establish that an arbitrary graph @math is @math -optimal when its minimum degree is @math , and may not be @math -optimal even when its minimum degree is @math for any @math .