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Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma and Sven Koenig Macao, China 08/13/2019

Outlines • Background: • Multi-Agent Path Finding. • Conflict-Based Search. • CG heuristics for Conflict-Based Search. • Two more informed heuristics: • DG heuristics. • WDG heuristics. • Experimental results. • Summary. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 2 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Multi-Agent Path Finding (MAPF) • Given: 2 • A graph; • A set of agents, each with a start 1 1 location and a goal location. • Goal: 2 • Find collision-free paths for all agents; • Minimize the sum of path costs. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 3 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Conflict-Based Search (CBS) [Sharon et al. 2015] Time 3 Time 2 Time 1 4 Sum of path costs = 8 2 3 1 C2 2 1 Agent 2 cannot be Agent 1 cannot be 1 2 at cell C2 at time 3. at cell C2 at time 3. A B C D 4 2 2 4 3 1 1 3 C2 C2 2 1 1 2 1 2 2 1 A B C D A B C D Sum of path costs = 9 Sum of path costs = 9 Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 4 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Conflict-Based Search (CBS) [Sharon et al. 2015] • CBS searches in a binary tree in a best-first manner according to the sum of path costs. g-value = sum of path costs h-value = ? Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 5 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Conflict-Based Search (CBS) [Sharon et al. 2015] • CBSH [Felner et al. 2018] adds admissible heuristics to CBS. • A conflict is cardinal iff all shortest paths of the both agents traverse the conflicting location at the conflicting time. • A cardinal conflict is an admissible h-value of 1. non-cardinal cardinal Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 6 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

CBSH [Felner et al. 2018] • Cardinal conflict graph • Minimum Vertex Cover ℎ 𝐷𝐻 = 3 We call this CG Heuristics. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 7 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Can We Get Better Heuristics? • Two agents are dependent iff every pair of their shortest paths has at least one conflict. • A pair of dependent agents is an admissible h-value of 1. • Two agents that have cardinal conflicts are dependent. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 8 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

DG Heuristics • Dependency graph • Minimum Vertex Cover ℎ 𝐸𝐻 = 4 Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 9 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Can We Get Better Heuristics? • The weight for a pair of agents is the difference between the minimum sum of the costs of their conflict-free paths and the sum of their shortest path costs. • The weight is an admissible h-value for the pair of agents. • The weight for a pair of dependent agents is at least one. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 10 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

WDG Heuristics • Edge-weighted • Edge-weighted dependency graph Minimum Vertex Cover ℎ 𝑋𝐸𝐻 = 7 Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 11 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Experiments Dense grid Empty grid Large grid 20x20 empty grid 20x20 grid with 192x192 grid with 30% randomly 51% blocked cells blocked cells Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 12 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Experiments ℎ value at the root node. ℎ ∗ ℎ ∗ ℎ ∗ ℎ value at the root node ℎ 𝑋𝐸𝐻 ℎ 𝑋𝐸𝐻 ℎ 𝑋𝐸𝐻 ℎ 𝐸𝐻 ℎ 𝐸𝐻 ℎ 𝐷𝐻 ℎ 𝐸𝐻 ℎ 𝐷𝐻 ℎ 𝐷𝐻 Agents ℎ 𝑋𝐸𝐻 ≥ ℎ 𝐸𝐻 ≥ ℎ 𝐷𝐻 Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 13 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Experiments Runtime breakdown per expanded node. Runtime (ms) Runtime for CBS Runtime for constructing graphs Runtime overhead of computing Runtime for solving MVC heuristics Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 14 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Experiments Success rate (= % solved instances) within 1 minute. WDG DG WDG Success rate WDG DG CG CG DG CG Agents Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 15 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.

Summary • Two admissible heuristics for CBS, DG and WDG, by reasoning about pairwise dependency between agents: • h-value: ℎ 𝑋𝐸𝐻 ≥ ℎ 𝐸𝐻 ≥ ℎ 𝐷𝐻 . • Runtime overhead: relatively small. • Overall performance: WDG is better than DG, which in turn is better than CG. • Future work: • Generalize these heuristics to groups larger than pairs of agents, e.g., to triples and quadruples. • Study admissible or inadmissible heuristics for sub-optimal CBS- based algorithms. Improved Heuristics for Multi-Agent Path Finding with Conflict-Based Search . 16 Jiaoyang Li, Ariel Felner, Eli Boyarski, Hang Ma, and Sven Koenig.