Developing a Deterministic Patrolling Strategy for Security Agents Nicola Basilico, Nicola Gatti, Francesco Amigoni

N. Basilico, N. Gatti, F. Amigoni Scenario 2

N. Basilico, N. Gatti, F. Amigoni Strategic Patrolling Determination of an efficient strategy for the patroller agent A patrolling strategy determines how the patroller moves in the environment An efficient patrolling strategy minimizes the intrusion probability Usually, patrolling strategies use randomized movements [Agmon et al., ICRA2008] [Paruchuri et al., AAMAS2008] [Amigoni et al., IAT2008] [Basilico et al., AAMAS2009] Sometimes, a deterministic patrolling strategy (a fixed path) can prevent intrusions Problem: determine an efficient deterministic patrolling strategy for a security agent 3

N. Basilico, N. Gatti, F. Amigoni Problem Formulation: Environment The environment is represented as a weighted directed graph G = (V,A,w,d) V: vertexes are locations to patrol A: arcs are direct connections between adjacent locations w:A ➛ R: weights are temporal costs to traverse an arc d:V ➛ R: penetration times for vertexes time the intruder needs to penetrate the vertex time interval between two successive visits of the patroller to the vertex 4

N. Basilico, N. Gatti, F. Amigoni Problem Formulation: Agents The patroller agent moves between adjacent vertexes Movement between v and v’ requires a time w(v,v’) The patroller agent senses the presence of the intruder only in its current vertex The intruder can appear at any vertex at any time Once intruder appears at vertex v, it stays there for d(v) time 5

N. Basilico, N. Gatti, F. Amigoni Problem Formulation: Solution Sequence of vertexes: S(1), S(2), …, S(s) Time interval between visits to two vertexes S(j) and S(j’): Σ i=j, j+1…,j’-1 w(S(i),S(i+1)) A solution is a sequence of vertexes such that: 1. It is cyclical S(1) = S(s) 2. Every vertex is visited at least one time 3. When indefinitely repeating the sequence, the time interval between two successive visits to a vertex v is not larger than d(v) If a solution is found, no intrusion can occur A rational intruder will never attempt to enter When an intruder attempts to enter, it is always detected by the patroller The length s of the solution is part of the problem 6

N. Basilico, N. Gatti, F. Amigoni Related Works Visiting vertexes of graphs under temporal deadlines has been extensively studied Deadline travel salesman problems [Tsitsiklis, Networks 1992] Vehicle routing problems with time windows [Kolen et al., Operations Research 1987] Period routing problems [Christofides and Beasley, Networks 1984] … However, our problem has relative deadlines (“two successive visits to a vertex v”) and is a feasibility problem 7

N. Basilico, N. Gatti, F. Amigoni The Proposed Solving Algorithm (1) Idea: formulate the problem of finding a solution as a Constraint Satisfaction Problem (CSP) Variables: S(j) (their number is a priori unknown!) Domains: D(S(j)) = V (they will be restricted!) Constraints: 1. Cycle: S(1) = S(s) 2. Every vertex is visited at least one time 3. When indefinitely repeating the sequence, the time interval between two successive visits to a vertex v is not larger than d(v) Solution of the CSP: assignment of values to variables such that all the constraints are satisfied 8

N. Basilico, N. Gatti, F. Amigoni The Proposed Solving Algorithm (2) Our CSP is solved by a brute-force backtracking search Forward checking is used to reduce search complexity Forward checking eliminates from the domains the values that are not compatible with the current partial assignment 9 S(1)=v S(1)=v S(2)=v’ S(1)=v S(2)=v’ S(3)=? S(1)=v S(2)=v’’

N. Basilico, N. Gatti, F. Amigoni The Proposed Solving Algorithm: Example (1) 10 S(1)=01 S(1)=01 S(2)=02 S(1)=01 S(2)=02 S(3)=03 D(S(2)) = {02,05} D(S(3)) = {01,03,05} 05 is eliminated because: w(01,05)+w e (05,03)+w e (03,01) > d(03) > 14

N. Basilico, N. Gatti, F. Amigoni The Proposed Solving Algorithm: Example (2) 11 stopping criterion!backtracking

N. Basilico, N. Gatti, F. Amigoni Some Properties of the Proposed Solving Algorithm The algorithm is sound and complete If a solution exists, it is independent of the first vertex S(1) At least a solution (if one exists) has a temporal length not larger than max i ∊ V {d(v i )} Definition of a new stopping criterion With linear environments, if a solution exists, then the linear sequence is a solution Computational complexity is hard to characterize 12

N. Basilico, N. Gatti, F. Amigoni Experimental Results Randomly-generated graphs with n vertexes and m ∊ [n, (n-1)n] arcs 500 runs for each value of n Metrics: % of termination within 10 minutes Computational time (for terminated runs) 13

N. Basilico, N. Gatti, F. Amigoni Conclusions Method for finding deterministic patrolling strategies for security agents Method for finding a cyclic sequence of vertexes on a graph under temporal constraints Future works: More sophisticated stopping criteria for blocking search earlier Extending patroller’s sensing abilities beyond the current vertex Extension to multiagent settings 14

N. Basilico, N. Gatti, F. Amigoni Thank you! 15