Toward Optimizing Static Target Search Path Planning Nassirou Lo T-OptLogic Ltd. Quebec City, Canada nassirou.lo@t-optlogic.com Jean Berger DRDC Valcartier Quebec City, Canada jean.berger@drdc-rddc.gc.ca Martin Noel UQAM, TELUQ Quebec City, Canada noel.martin@teluq.uqam.ca Abstract—Discrete static open-loop target search path planning is known to be a NP (non-deterministic polynomial) -Hard problem, and problem-solving methods proposed so far rely on heuristics with no way to properly assess solution quality for practical size problems. Departing from traditional nonlinear model frameworks, a new integer linear programming (ILP) exact formulation and an approximate problem-solving method are proposed to near-optimally solve the discrete static search path planning problem involving a team of homogeneous agents. Applied to a search and rescue setting, the approach takes advantage of objective function separability to efficiently maximize probability of success. A network representation is exploited to simplify modeling, reduce constraint specification and speed-up problem-solving. The proposed ILP approach rapidly yields near-optimal solutions for realistic problems using parallel processing CPLEX technology, while providing for the first time a robust upper bound on solution quality through Lagrangean programming relaxation. Problems with large time horizons may be efficiently solved through multiple fast subproblem optimizations over receding horizons. Computational results clearly show the value of the approach over various problem instances while comparing performance to a myopic heuristic. Keywords: search path planning, search and rescue, linear programming, static, open-loop I. INTRODUCTION Target search path planning is a pervasive problem occurring over a variety of civilian and military domains such as homeland security, emergency management and search and rescue/respond. Target search and in particular search and rescue (SAR) problems may be characterized through multiple dimensions and attributes including: one-sided search in which targets are non-responsive toward searcher’s actions, two- sided, describing target behavior diversity (cooperative, non- cooperative or anti-cooperative), stationary Vs. moving target search, discrete Vs. continuous time and space search (efforts indivisibility/divisibility), observation model, static/dynamic as well as open and closed -loop decision models, pursued objectives, target and searcher multiplicity and diversity. Early work on related search problems emerges from search theory [1], [2]. Search-theoretic approaches mostly relate to the effort (time spent per visit) allocation decision problem rather than path construction. Based upon a mathematical framework, efforts have increasingly been devoted to algorithmic contributions to handle more complex dynamic problem settings and variants [3], [4]-[6]. In counterpart, many contributions on search path planning may be found in the robotics literature in the area of robot motion planning [7] and, namely, terrain acquisition [8], [9] and coverage path planning [10],[11], [12]. Robot motion planning explored search path planning, primarily providing constrained shortest path type solutions for coverage problem instances [13], [14]. These studies typically examine uncertain search environment problems with limited prior domain knowledge, involving unknown sparsely distributed static targets and obstacles. Recent taxonomies and comprehensive surveys on target search problems from search theory and artificial intelligence/distributed robotic control perspectives may be found in [15], [4], [16]-[18] respectively. However, despite a large body of work published on various problem models, the SAR problem even in its simplest form remains computationally hard [4]. The open-loop static (offline planning) problem in particular, still presents strong interest and challenges in a variety of situations such as major disaster management or urban/military combat search-and-rescue operations where any potential gain in life saving and efficiency is worth the investment. Such circumstances include cases for which gathered information during search cannot be instantly exploited, or high-level organizational resource allocation decision-making processes aimed at exploring and assessing anticipated offline solution plan prior to costly resource deployment. Those situations may typically result from unavailable expertise/knowledge or insufficient/limited information processing technology capability at searcher’s disposal, or the prevalence of current organizational structure, process, policy and constraints or security conditions. In spite of the development of many heuristics and approximate problem-solving techniques to face the curse of dimensionality [19]-[21], [4], [18] for the static SAR, published problem- solving heuristics mostly fail to provably estimate real performance optimality gap for practical size problems, questioning their real expected relative efficiency, ignoring potential feasible gains to be possibly further considered. In this paper, we propose a new exact integer linear programming formulation along with an approximate technique to near-optimally solve the discrete static search path planning problem involving a team of homogeneous agents. In that setting, a team of centrally controlled