# A Simple, but NP-Hard, Motion Planning Problem

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2 even when all blocks are 1x2 dominoes (?). However, there exist situations in which the pianomover s problem is easy. If the robot is navigating in the plane and the obstacle regions are polygonal sets, then a straightforward cell decomposition will efficiently reveal the existence or non-existence of a path from the start to the goal (?;?). In this paper, we demonstrate that the corresponding minimum constraint removal problem is NP-hard, even when the obstacles are restricted to being convex polygons. This is done via a reduction from the maximum satisfiability problem for quadratic Horn clauses, a special case of MAXHORNSAT that remains NP-hard (?). This augments Hauser s discrete NP-hardness result. While it is possible to transform the convex polygon minimum constraint removal problem into a discrete, graph-based constraint removal problem by associating each connected region intersected by a particular set of obstacles with a single graph vertex, the graphs and obstacles obtained from this transformation are very restricted (see Figure 2). The graphs must be planar, and the subgraph induced by the set of vertices intersected by a particular obstacle must be connected (though other restrictions also apply). The graphs used by Hauser in the discrete NP-hardness proof are generally non-planar, and the graphs induced by the set of vertices intersected by an obstacle are not generally connected. Section 2 formally defines the minimum constraint removal problem. Section 3 describes the maximum satisfiability problem for quadratic Horn clauses. Section 4 describes types of obstacles that are used in the reduction. Section 5 describes the gadgets that encode the variables and clauses of the satisfiability problem. Section 6 explains how to obtain the solution to the maximum satisfiability problem from the result of the minimum constraint removal problem. Section 7 discusses the implications of this NP-hardness result and lists open problems. 2 Problem Formulation A point robot navigates in R 2. Let {O 1, O 2,..., O m } be a set of m obstacles. Each obstacle is a convex polygonal subset of R 2. It is acceptable for obstacles to be degenerate 1-gons (points) or 2-gons (line segments). It should be noted that either type of degenerate polygon can be simulated by a non-degenerate polygon that is sufficiently small (in the case of 1-gons) or sufficiently thin (in the case of 2-gons). Let q s R 2 be the starting point. Let q g R 2 be the goal point. Let y be a continuous path from q s to q g. We will treat y as a set of points, as the parameterization of the path is irrelevant. The cover of y is the set of obstacles that intersect y. A minimum constraint removal path y is a path whose cover has a minimal size among all paths from q s to q g. A minimum constraint removal S is the cover of a minimum constraint removal path. 3, 4 O 3 3 O4 O 2 2, 3 1, O 1 2, 6 O 5 O 6 O 7 Figure 2: [top] The environment from Figure 1 with each connected region intersected by a particular set of obstacles assigned a graph vertex. Since movement within a single region does not intersect any new obstacles, the minimum constraint removal problem for this obstacle set can be transformed into a discrete problem. [bottom] The graph for the discrete minimum constraint removal problem. The numbers next to a vertex denote the obstacles intersecting that vertex. 3 Maximum Quadratic Horn Clause Satisfiability A clause is a disjunction of literals. A Horn clause is a clause that contains at most one positive literal. A quadratic clause is a clause that contains at most two literals. The decision problem about the satisfiability of a conjunction of Horn clauses (HORNSAT) is solvable in linear time (?). However, the corresponding maximization problem Given a set of n Horn clauses, what is the maximum number that can be simultaneously satisfied? (MAXHORNSAT) is NPhard. Similarly, the decision problem about the satisfiability of a conjunction of quadratic clauses (QSAT) is in P, but the corresponding maximization problem (MAXQSAT, equivalent to the more well-known MAX2SAT problem) is also NP-hard , , , 7

5 q s Positive/Negative Clauses q g x j +1 x j Figure 5: A clause gadget for x j. A single low-weight obstacle blocks the right path of the variable loop and the left path of the clause loop. Another low-weight obstacle blocks the right path of the x j variable loop and the right path of the clause loop. If is true, then the left path of the clause loop can be taken without intersecting an additional obstacle. If x j is false, then the right path of the clause loop can be taken without intersecting an additional obstacle. by grey squares, and can be seen in Figure 7. 6 Reduction Since a path from q s to q g exists that does not intersect any high-weight obstacles, a minimum constraint removal path will not intersect any high-weight obstacles. Additionally, the minimum number of medium-weight obstacles that can be crossed on a path from q s to q g is c, the number of clauses. Each literal within a clause adds an obstacle to the left and right paths of the associated variable gadget loop (one obstacle used to establish the relationship between the clause and the variable, and one balancing obstacle), one of which needs to be crossed to reach the goal. Let l be the total number of literals in all clauses. A minimum constraint removal path will cross at least l low-weight obstacles while setting the truth values of the variables. Finally, each clause that is not satisfied requires the crossing of an additional lowweight obstacle. Therefore, if no clauses can be satisfied, the size of the minimum constraint removal is (a + 1)v + l + c. (1) in which a + 1 is the weight of a medium-weight obstacle. For each clause that can be satisfied, the size of the minimum constraint removal is decreased by 1. Therefore, given a set of c clauses with l total literals over v variables that produces a minimum constraint removal of size z, the maximum number of clauses that can be simultaneously satisfied is Figure 6: A clause gadget for. A single low-weight obstacle blocks the left path of the variable loop and path between the upper loops of and +1. (a + 1)v + l + c z. (2) Therefore, MAXQHORNSAT is reducible to the minimum constraint removal problem in the plane even when all obstacles are convex, indicating that the latter problem is NP-hard. A complete representation of a MAXQHORN- SAT problem as a minimum constraint removal problem is shown in Figure 7. 7 Discussion and Open Problems Navigation in R 2 with convex polygonal obstacles is a highly restricted form of path planning. The NP-hardness of the minimum constraint removal problem under these restrictions immediately implies the NP-hardness of the problem for higher dimensions and for less restricted types of obstacles. What restrictions on the environment and obstacles make the problem tractable? In the plane, the minimum constraint removal problem is trivial when the obstacles are forced to be infinite lines or half-planes. It is not known whether forcing all obstacles to be unit circles, circles, or axis-aligned rectangles makes the problem tractable, though the authors conjecture that the first is tractable and the latter two are not. One could also place restrictions on the ways in which the obstacles are allowed to intersect. The problem becomes trivial when the obstacles are allowed to intersect only pairwise. It is not known at what point restrictions of the form There is no point at which more than k obstacles intersect or Each obstacle intersects no more than k other obstacles render the problem tractable. Acknowledgements This work is supported in part by NSF grant (IIS Robotics), NSF grant (CNS Cyberphysical

6 q s x 1 x 2 x 3 x 4 A B q g E D C x 4 x 3 A = x 2 x 4 B = x 3 x 1 C = x 1 x 4 D = x 2 x 3 E = x 3 x 4 x 2 x 1 Figure 7: A MAXQHORNSAT problem with four variables and five clauses represented as a minimum constraint removal problem. High-weight obstacles surround the solid black path between q s and q g. Medium-weight obstacles are represented by dotted black lines. Low-weight obstacles are represented by grey lines and grey squares. Systems), DARPA SToMP grant HR , and MURI/ONR grant N

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