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Local Search Stuffs

Local Search Stuffs. In these slides, you will find: Solution Representation Local Move/Neighborhood Swap k-Nodes Swap k-Edges Distance Metric For Traveling Salesman Problem (TSP) Knapsack Problem (KP). TSP: Solution Representation.

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Local Search Stuffs

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  1. Local Search Stuffs • In these slides, you will find: • Solution Representation • Local Move/Neighborhood • Swap k-Nodes • Swap k-Edges • Distance Metric • For • Traveling Salesman Problem (TSP) • Knapsack Problem (KP)

  2. TSP: Solution Representation • The solution for Traveling Salesman Problem (TSP) is a tour of cities • Usually represented as a circular array A of vertices/nodes/cities. • The range of array indices is [0..n-1] • The starting city is irrelevant, but usually A[0] • A[i] is the city visited at step i • The edges of the tour are defined by (A[i], A[i+1]) for all i[0..n-2] plus the circular edge: (A[n-1],A[0]) • Here is an example of 5 cities (n=5) 0 1 3 2 4

  3. TSP: Local Move/Neighborhood • There are several ways to locally modify TSP tour • We list down 2 of them: • Swap k-vertices/nodes/cities • Swap k-edges

  4. Swap k-vertices/nodes/cities • k  2, but usually k=2 • Neighborhood size (respectively, the running time of the local search) grows with larger k. • There are more options for k>2 • There are k! ways to permute k cities. • Swapping k-vertices maintains the feasibility of TSP tour. • This type of local move for TSP is not good • It usually doesn’t fix crossings in the tour • But rather add more crossings… • Therefore local neighbors’ quality is usually poorer • A bad feature for local search with gradient-descent type

  5. Example: Swap(index 1,index 3) 0 0 1 1 3 3 2 4 2 4

  6. Swap k-edges • k2, but usually 2 or 3 • Again, neighborhood size (respectively, the running time of the local search) grows with larger k.. • Swapping k-edges maintains the feasibility of TSP tour. • This local move is better than swap k-vertices • Typically used in the best performing local search for TSP. • The best performing heuristic for TSP: Lin-Kernighan heuristic,uses variant of this swap k-edges… • The local neighborhood induced by this move usually contain tours/solutions with improving quality (crossings fixed). • We can fix crossings using shorter steps than swap k-vertices • In circular array and for k=2, this move is implemented by reversing the content of the sub-array/sub-tour.

  7. Example: Swap(1-3/0-2,1-2/0-3) 0 0 1 1 3 3 2 4 2 4

  8. TSP: Distance Metric • The most appropriate (natural) distance function to measure distance between two TSP tours is bond/permutation/edge distance, defined as: • The number of different edges between solution a and b. • Using appropriate data structure, this distance function can be computedin linear time O(n), where n is the number of cities. • Using the previous example (A: original tour, B: example for 2-nodes swap, C: example for 2-edges swap) • BondDistance(A,B) = 2 • BondDistance(A,C) = 2

  9. KP: Solution Representation • The solution for Knapsack Problem (KP) is a set of taken items • Usually represented as a bit string of n items • If b[i] = 0, it means item i is not taken • If b[i] = 1, it means item I is taken • Here is an example of 5 items (n=5)

  10. KP: Local Move/Neighborhood • There are several ways to locally modify KP solution • We list down 2 of them: • Toggle k-bits • Swap k-bits

  11. Toggle k-bits • k  1, but usually k=1 • Toggle bit toggles the state of b[i] • From 0 to 1 and vice versa. • Feasibility of the solution must be checked afterwards. • This local move is usually not used alone • When the knapsack is full, there is no way to insert any more item (toggle b[i] == 0 to b[i] == 1) • Toggling off will decrease the solution quality, without proper escape mechanism, the local search may get stuck easily.

  12. Example: Toggle (Bit 1)

  13. Swap k-Bits • k2, but usually 2 • For k=2, pick 2 indices i,j such that b[i] != b[j] • One item is inside but the other is outside the knapsack. • Swap them. • Feasibility of the solution must be checked afterwards. • This local move is usually used with Toggle k-Bits • This local move doesn’t degrade the solution quality as much as pure Toggle k-Bits. • But complementing this neighborhood with Toggle k-Bits neighborhood usually yield stronger local search heuristic. • Careful search strategy must be employed for dealing with boundary cases of feasibility/infeasibility.

  14. Example: Swap (Bit 1, Bit 4)

  15. KP: Distance Metric • The most appropriate (natural) distance function to measure distance between two KP solution is hamming/exact match distance, defined as: • The number of different bits between solution a and b. • We can determine the distance with one linear scan O(n). • Using the previous example (A: original knapsack, B: example for toggle 1 bit, C: example for swap 2 bits) • HammingDistance(A,B) = 1 • HammingDistance(A,C) = 2

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