Balanced connected task allocations for multi-robot systems: An exact flow-based integer program and an approximate tree-based genetic algorithm

Abstract Task allocation for a multi-robot system is often imposed with various constraints to promote efficiency and avoid unwanted consequences. This study focuses on the min-max balanced connected q-partition problem (BCPq) that seeks the early completion time of a multi-robot system while satisfying mission constraints. In the problem, each environmental region is modeled as a weighted node of a graph and the graph is to be partitioned into a predefined number, q, of disjoint node sets, such that nodes in each set are connected and the largest “sum of the set” is as small as possible. The problem is NP-hard in general cases, and has a brute-force search space exponential to the number of nodes. BCPq arises in many fields besides robotics. To date, existing work has mainly been conducted on special versions of the problem, such as small q or special types of graphs. In this study, we propose two approaches for the general case: one is exact and suitable for graphs of less than 200 nodes, while the other is approximate and can handle graphs with up to 3000 nodes. Specifically, this study presents the first exact mixed integer linear program (MILP) in the literature that can partition an arbitrary graph into a given arbitrary number of components. The MILP is based on a flow model and can solve graphs with 100 more nodes than the existing MILP that can only tackle 2-partition in the q = 2 case. The study also presents an approximate genetic algorithm (GA) based on tree partition and tree evolution. It is the first GA that can handle BCPq (q ≥ 2) problems. On graphs of less than 500 nodes and 2-partition, the GA achieved all the optima in 20 runs. On q ≤ 12, this GA also can achieve results close to the optima after a few individuals were evolved, i.e., a gap smaller than 3% of the ideal average value. Moreover, the GA is scalable to q as the consumed time is nearly stable, independent of q.

[1]  J. Kruskal.  On the shortest spanning subtree of a graph and the traveling salesman problem , 1956 .

[2]  Weili Wu,et al.  Max-min weight balanced connected partition , 2013, J. Glob. Optim..

[3]  Diego Andina,et al.  Distributed Bees Algorithm for Task Allocation in Swarm of Robots , 2012, IEEE Systems Journal.

[4]  Giuliano Armano,et al.  Multiobjective clustering analysis using particle swarm optimization , 2016, Expert Syst. Appl..

[5]  Yoshiko Wakabayashi,et al.  Approximation and Inapproximability Results on Balanced Connected Partitions of Graphs , 2007, Discret. Math. Theor. Comput. Sci..

[6]  Anthony Stentz,et al.  A comprehensive taxonomy for multi-robot task allocation , 2013, Int. J. Robotics Res..

[7]  Konstantin Andreev,et al.  Balanced Graph Partitioning , 2004, SPAA '04.

[8]  Shahram Payandeh,et al.  Multi-robot, dynamic task allocation: a case study , 2013, Intelligent Service Robotics.

[9]  Nicolas Jouandeau,et al.  A Survey and Analysis of Multi-Robot Coordination , 2013 .

[10]  Pedro M. S. Carvalho,et al.  On spanning-tree recombination in evolutionary large-scale network problems - application to electrical distribution planning , 2001, IEEE Trans. Evol. Comput..

[11]  Sukhamay Kundu,et al.  A Linear Tree Partitioning Algorithm , 1977, SIAM J. Comput..

[12]  Xin Jin,et al.  Navigation of autonomous vehicles for oil spill cleaning in dynamic and uncertain environments† , 2014, Int. J. Control.

[13]  Shalabh Gupta,et al.  SLAM based shape adaptive coverage control using autonomous vehicles , 2015, 2015 10th System of Systems Engineering Conference (SoSE).

[14]  Mario Ventresca,et al.  Global search algorithms using a combinatorial unranking-based problem representation for the critical node detection problem , 2012, Comput. Oper. Res..

[15]  Jitendra Malik,et al.  Normalized Cuts and Image Segmentation , 2000, IEEE Trans. Pattern Anal. Mach. Intell..

[16]  Dragan Matic.  A mixed integer linear programming model and variable neighborhood search for Maximally Balanced Connected Partition Problem , 2014, Appl. Math. Comput..

[17]  Janka Chlebíková,et al.  Approximating the Maximally Balanced Connected Partition Problem in Graphs , 1996, Inf. Process. Lett..

[18]  Elon Rimon,et al.  Spanning-tree based coverage of continuous areas by a mobile robot , 2004, Annals of Mathematics and Artificial Intelligence.

[19]  Janka Chlebíková,et al.  Structural and Algorithmic Properties of 2-Community Structures , 2018, Algorithmica.

[20]  Ronald I. Becker,et al.  Max-min partitioning of grid graphs into connected components , 1998 .

[21]  Jozef Kratica,et al.  Solving the Maximally Balanced Connected Partition Problem in Graphs by Using Genetic Algorithm , 2008, Comput. Informatics.

[22]  Vasco Furtado,et al.  A Heuristic Method for Balanced Graph Partitioning: An Application for the Demarcation of Preventive Police Patrol Areas , 2008, IBERAMIA.

[23]  Marc Carreras,et al.  Efficient seabed coverage path planning for ASVs and AUVs , 2012, 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[24]  Dylan A. Shell,et al.  Large-scale multi-robot task allocation via dynamic partitioning and distribution , 2012, Auton. Robots.

[25]  Yong-Hyuk Kim,et al.  Genetic approaches for graph partitioning: a survey , 2011, GECCO '11.

[26]  George Karypis,et al.  Multilevel k-way Partitioning Scheme for Irregular Graphs , 1998, J. Parallel Distributed Comput..

[27]  Hakim Mitiche,et al.  A taxonomy for task allocation problems with temporal and ordering constraints , 2017, Robotics Auton. Syst..

[28]  Martin E. Dyer,et al.  On the complexity of partitioning graphs into connected subgraphs , 1985, Discret. Appl. Math..

[29]  Neng Fan,et al.  Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming , 2012, COCOA.

[30]  Yunjun Gao,et al.  Unsupervised analysis of top-k core members in poly-relational networks , 2014, Expert Syst. Appl..

[31]  Greg N. Frederickson,et al.  Optimal algorithms for tree partitioning , 1991, SODA '91.

[32]  Stefan Voß,et al.  A heuristic approach for dividing graphs into bi-connected components with a size constraint , 2016, J. Heuristics.