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NP , and 2. Π0 ≤p Π for every Π0 ∈ NP where Π0 ≤p Π means that problem Π can be reduced in polynomial-time to problem Π and therefore by solving Π0 we can retrieve the solution to Π in polynomial time. NP-complete problems are the hardest problems in NP and consequently, if any NPcomplete problem could be solved in polynomial time, then all NP-complete problems could be solved in polynomial time and P=NP. Conversely, if any problem in NP is not solvable in polynomial time, then all NP-complete problems cannot be solved in polynomial time.
These dynamic programming problems are characterized by their hierarchical structure. Similarly to divide-and-conquer algorithms [34], dynamic programming subdivides (sub)problems into manageable pieces, which are solved and then used to retrieve the final solution to the overall problem. Contrary to divide-andconquer, the “manageable” pieces can be used by multiple parent (sub)problems. Dynamic programming relies on the principle of optimality [16]: 42 CHAPTER 3. SHORTEST PATH ALGORITHMS An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.
2: Illustration of curved equilength lines. 2, curved equilength lines match the constraint boundaries much better. 2) lq (P ) = L i i=1 is well-known as Holder’s q-vector norm [60] and is fundamental in the theory of classical Banach spaces (see Royden [142, Chapter 6]). 3). 3), has length larger than 1 and, hence, violates at least one of the constraints, no other path will satisfy the constraints. 3) of path length solves the multiconstrained path (MCP) problem. However, the shortest path is not guaranteed to be found with Dijkstra’s algorithm, which relies on property 21 of a linear path length definition that subsections of shortest paths are also shortest paths.