Held–Karp algorithm

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The Held–Karp algorithm, also called the Bellman–Held–Karp algorithm, is a dynamic programming algorithm proposed in 1962 independently by Bellman[1] and by Held and Karp[2] to solve the traveling salesman problem (TSP), in which the input is a distance matrix between a set of cities, and the goal is to find a minimum-length tour that visits each city exactly once before returning to the starting point. It finds the exact solution to this problem, and to several related problems including the Hamiltonian cycle problem, in exponential time.

Algorithm description and motivation

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Number the cities  , with   designated arbitrarily as a "starting" city (since the solution to TSP is a Hamiltonian cycle, the choice of starting city doesn't matter). The Held–Karp algorithm begins by calculating, for each set of cities   and every city   not contained in  , the shortest one-way path from   to   that passes through every city in   in some order (but not through any other cities). Denote this distance  , and write   for the length of the direct edge from   to  . We'll compute values of   starting with the smallest sets   and finishing with the largest.

When   has two or fewer elements, then calculating   requires looking at one or two possible shortest paths. For example,   is simply  , and   is just the length of  . Likewise,   is the length of either   or  , whichever is shorter.

Once   contains three or more cities, the number of paths through   rises quickly, but only a few such paths need to be examined to find the shortest. For instance, if   is shorter than  , then   must be shorter than  , and the length of   is not a possible value of  . Similarly, if the shortest path from   through   to   is  , and the shortest path from   through   to   ends with the edge  , then the whole path from   to   must be  , and not any of the other five paths created by visiting   in a different order.

More generally, suppose   is a set of   cities. For every integer  , write   for the set created by removing   from  . Then if the shortest path from   through   to   has   as its second-to-last city, then removing the final edge from this path must give the shortest path from   to   through  . This means there are only   possible shortest paths from   to   through  , one for each possible second-to-last city   with length  , and  .

This stage of the algorithm finishes when   is known for every integer  , giving the shortest distance from city   to city   that passes through every other city. The much shorter second stage adds these distances to the edge lengths   to give   possible shortest cycles, and then finds the shortest.

The shortest path itself (and not just its length), finally, may be reconstructed by storing alongside   the label of the second-to-last city on the path from   to   through  , raising space requirements by only a constant factor.

Algorithmic complexity

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The Held–Karp algorithm has exponential time complexity  , significantly better than the superexponential performance   of a brute-force algorithm. Held–Karp, however, requires   space to hold all computed values of the function  , while brute force needs only   space to store the graph itself.

Time

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Computing one value of   for a  -element subset   of   requires finding the shortest of   possible paths, each found by adding a known value of   and an edge length from the original graph; that is, it requires time proportional to  . There are    -element subsets of  ; and each subset gives   possible values of  . Computing all values of   where   thus requires time  , for a total time across all subset sizes  . The second stage of the algorithm, finding a complete cycle from   candidates, takes   time and does not affect asymptotic performance.

For undirected graphs, the algorithm can be stopped early after the   step, and finding the minimum   for every  , where   is the complement set of  . This is analogous to a bidirectional search starting at   and meeting at midpoint  . However, this is a constant factor improvement and does not affect asymptotic performance.

Space

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Storing all values of   for subsets of size   requires keeping   values. A complete table of values of   thus requires space  . This assumes that   is sufficiently small enough such that   can be stored as a bitmask of constant multiple of machine words, rather than an explicit k-tuple.

If only the length of the shortest cycle is needed, not the cycle itself, then space complexity can be improved somewhat by noting that calculating   for a   of size   requires only values of   for subsets of size  . Keeping only the   values of   where   has size either   or   reduces the algorithm's maximum space requirements, attained when  , to  .

Pseudocode

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Source:[3]

function algorithm TSP (G, n) is
    for k := 2 to n do
        g({k}, k) := d(1, k)
    end for

    for s := 2 to n−1 do
        for all S ⊆ {2, ..., n}, |S| = s do
            for all k ∈ S do
                g(S, k) := minm≠k,m∈S [g(S\{k}, m) + d(m, k)]
            end for
        end for
    end for

    opt := mink≠1 [g({2, 3, ..., n}, k) + d(k, 1)]
    return (opt)
end function
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Exact algorithms for solving the TSP

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Besides Dynamic Programming, Linear programming and Branch and bound are design patterns also used for exact solutions to the TSP. Linear programming applies the cutting plane method in integer programming, i.e. solving the LP formed by two constraints in the model and then seeking the cutting plane by adding inequality constraints to gradually converge at an optimal solution. When people apply this method to find a cutting plane, they often depend on experience, so this method is seldom used as a general method.

The term branch and bound was first used in 1963 in a paper published by Little et al. on the TSP, describing a technique of combining smaller search spaces and establishing lower bounds to enlarge the practical range of application for an exact solution. The technique is useful for expanding the number of cities able to be considered computationally, but still breaks down in large-scale data sets.

It controls the searching process through applying restrictive boundaries, allowing a search for the optimal solution branch from the space state tree to find an optimal solution as quickly as possible. The pivotal component of this algorithm is the selection of the restrictive boundary. Different restrictive boundaries may form different branch-bound algorithms.

Approximate algorithms for solving the TSP

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As the application of precise algorithm to solve problem is very limited, we often use approximate algorithm or heuristic algorithm. The result of the algorithm can be assessed by C / C* ≤ ε . C is the total travelling distance generated from approximate algorithm; C* is the optimal travelling distance; ε is the upper limit for the ratio of the total travelling distance of approximate solution to optimal solution under the worst condition. The value of ε >1.0. The more it closes to 1.0, the better the algorithm is. These algorithms include: Interpolation algorithm, Nearest neighbour algorithm, Clark & Wright algorithm, Double spanning tree algorithm, Christofides algorithm, Hybrid algorithm, Probabilistic algorithm (such as Simulated annealing).

References

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  1. ^ ‘Dynamic programming treatment of the travelling salesman problem’, Richard Bellman, Journal of Assoc. Computing Mach. 9. 1962.
  2. ^ 'A dynamic programming approach to sequencing problems’, Michael Held and Richard M. Karp, Journal for the Society for Industrial and Applied Mathematics 1:10. 1962
  3. ^ "Dynamic programming" (PDF). January 2020. Archived from the original (PDF) on 2015-02-08.