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Open image in new window Fig. 1 Conflict between departments' objective.

Open image in new window Fig. 1 Conflict points at signalized intersections Open image in new window Fig. 2 Conflict points at a single-lane Roundabout.

L has a BTP if and only if there exists a set of q − 1 conflict - free triangles in L. Suppose L = { a 1, …, a 2 q − 1 }.

1 Conflict has been defined as a multidimensional construct, involving a process whereby two or more people perceive opposition of another, 2 and a process whereby one's interests, norms and values are opposed by another.

The motives were categorized as: (1) conflict with prison officer (threatening), (2) minor assault without physical harm, (3) starting a fight following a verbal conflict (reactive violence), (4) deliberate use of violence without prior verbal conflict (instrumental violence).

L has a BTP if and only if there exists a set of q − 1 conflict - free triangles in L. The proof of NP-completeness of the BTP problem (Theorem 3) is by a reduction from the Numerical Matching with Target Sums (NMTS) problem, (Garey and Johnson, 1979).

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Two kinds of conflicts might occur as a result of crop change in an agricultural landscape: (1) conflicts among multiple ecosystem services i.e., internal conflicts and (2) conflicts among multiple stakeholders i.e., external conflicts.

Because we cannot have more than q − 1 conflict-free triangles in a solution, OPT ≤ q − 1.

Then, I has a solution if and only if a particular multiset L I has a BTP (i.e. a set of 2m − 1 conflict-free triangles).

In the previous section, we showed that for a given multiset L of 2 q − 1 elements, each BTP for L corresponds to a collection of q − 1 conflict-free triangles.

Because L can have at most q − 1 conflict-free triangles, we define the max ⁡ -BTP problem to be the problem of finding the maximum sized set of conflict-free triangles.

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