Consider the ratio of probabilities above: minT' L(T',G) = L* (the optimal tour). Simulated annealing The Traveling Salesman Problem, Princeton Univ. 3. repeat Question: what happens to r as T increases to infinity? s.t. Return the order of vertices visited in pre-order. and let L* be the optimal tour-length) Pick initial T to make the probability 0.95 (close cuts, branch-and-cut and various tricks to solve 2392-city problem. T = some starting tour "climb" out of local minima. Optimal solutions take a long time , New York: Plenum. 14. if degree(j) = 2 L* / n a constant Associate "energy" with "cost". Simulated annealing will allow jumps to higher-cost states. The first heuristic to produce solutions within a constant of optimal. Soc., 55, 1959, pp.299-327. Expand the node that adds the least overall cost to the (partial) objective function. Famous result: a graph has an Euler tour if and only if all 5. endfor Maintain an external array of pointers into tree, one per node. constraints for the TSP. If we plot objective function "lines": Overview The Traveling Salesman Problem (TSP) is possibly theclassic discrete optimization problem. max cTx Now consider a pre-order tree walk from the root, and low-energy states: Is the min-1-tree a good bound? 1976: Sahni-Gonzalez result Other operations that need to be supported: [WP-1] 2-OPT: Simulated annealing and Tabu search. use a gradient that points in the right direction. that no edges have a common vertex Complete graph. improvement can be found. 3. repeat 12. [Appl2006] Can be computed fast (MST) To the right, it's positive. solution overall. 10. add (i,j) segment to partial tour Key ideas in LKH-1: Find successor using tour-order (instead of numeric order). 8. noChange = false best k neighbors (e.g., k=20). Part of some neurological tests. The algorithm avoids these. The idea: some variables might change too slowly with [Vale1997]. 8. endif cutting planes local-search does). Tour segmentation: be the edges. Repeatedly apply the iteration // Start with any tour, e.g., in input order Less exploration of search space (even if you search a Analysis: 3:4, 1991, pp. Each xi is an edge in the current tour. Then define Thus, L <= W. Symp. problem. An IPT-iteration: Each node represents a city. Maintain "tabu" lists: Then define a spanning tree. Try to increase K gradually at each iteration. traveling salesman problem, 2-opt algorithm c# implementation Padberg and Hong, 1980: 318-city problem. [Lin1973] as a tree-search (the "branch" part): Three key algorithms, all major milestones in the We will show that 3. minTour = s is as high as possible in G'. High memory requirements. Branch-and-bound. Famous result: a graph has an Euler tour if and only if all Similarly, for a tour T', Estimating the Held-Karp lower bound for the geometric TSP. In practice: need to experiment with different temperature The more aligned, the lower the system "energy". local-search algorithm. L takes a shorter route than W (triangle inequality). Problem landscape: You'll get a detailed solution from a subject matter expert that helps you learn core concepts. 2. T = selectInitialTemperature() Suppose we decide to climb out of local minima. LH/L* O(log(n) / loglog(n)) Question: what happens to r as T decreases to zero? Best-tour: at all times LK records the best tour found so far. by decreasing temperature T. does this mean the last one is optimal? Scheduling of vehicles from a central depot to a number of right half-space pointed to by the vector VT(). Intuitively, this means: [Sahn1976] The general idea (an example): [Chri1976] Euclidean Traveling Salesman and other Geometric Problems. schedules for a particular problem. is as high as possible in G'. Thus, W < 2L*. Why? Next, (1,2) gets added Estimating the Held-Karp lower bound for the geometric TSP. TSP's importance in computer science: Also, some experimentation will be need for the temperature schedule. An LP (Linear Programming) problem is (in standard form): [Rose1977]. One approximation: reduce number of edges by considering only The scalability of traveling salesperson problem (TSP) algorithms for handling large-scale problem instances has been an open problem for a long time. [Clar1964] with energies E(s2) > E(s1) 13. else if expCoinFlip (s, s') (local alignments): Always re-run with several (wildly) different starting solutions. Let cij = cji = the cost of the The method searches the nearest neighbor of this Superpoint. Note: LK is actually a little more complicated than 5. 1992: Arora et al result What vertex weight for We'll need some identify sub-tours (cycles) and add corresponding "|S|-1" constraints. TSP [Nearest Neighbour] - Given The distances Matrix min-match and MST strength of metals. Various bounds on particular heuristics (see below). The gradient descent algorithm is exactly this idea: prev(a): the previous node in tour order. Many of these edges are "good" edges for the tour. T = T' LH/L* 1+ In more detail: [Rose1977]. G' has the same vertices and edges as G. Easy to implement (since MST can be found efficiently). Roskilde University. particular neighborhood more thoroughly). 17. endfor Temporarily remove vertex 1 (and its edges) and Let's examine the partitioning idea: Salesman Problem. The standard heuristics (construction, K-OPT) give tours Unfortunately, this may not yield a feasible solution: S.Arora. What is the difference between the optimal tour and the min-1-tree Key ideas: What is the tour represented by the above tree? In a valid tour, The first thing we need to sort out is how we'll generate neighbor candidates. An Effective Heuristic Algorithm for the Traveling- Step-1 - Finding Adjacent Matrix Of the Graph 7. while |VH| > 2 for the following 4-point Euclidean TSP. LK is usually between 1-2% off. Karmarkar's algorithm: provably polynomial and practically Pick k centroids. What is the tour represented by the above tree? Today, there are several families of cutting-plane 5. endfor Tabu Search: A Tutorial, - if all the vertices in domain are visited, then terminate. Back to vertex-weight optimization: During this time, LP techniques improved greatly Observe: if we remove any one edge from a tour, we will get for the above modified graph G'? Data structures for traveling salesmen. Very efficient implementations available, both commercial m of these. Linear programming: Key idea: cool metal slowly during the forging process. During this time, LP techniques improved greatly. S.Lin and B.W.Kernighan. x 0 Always add all neighborhood minimums. 215V310, 1997. move rightwards The general idea: iT trees. there's no point in further execution. Local optima: Thermal cycling: Increase the weights for vertices with 1-min-tree degree L(T,G') = L(T,G) + iT (diT)i. Input: a collection of points (representing cities). [Chan1994] The gradient at a point x is the value of f'(x). Then, jump to it with probability At each step extend the partial tour to the nearest unvisited neighbor of the last city in the partial tour, until there are no unvisited cities remaining. xi's and yi's to repeat. Branch-and-bound. Just because it has an appealing origin, simulated annealing Let L(T',G) = cost of tour T' using graph G. Relax sequentiality allow some Here, we add a scaling factor in case f'(x) either of these edge sets. Approximate solutions: the Clarke-Wright heuristic starting points. This is an inequality constraint that could be added to the With slow-cooling, alignments are closer to optimal (global alignment): solution overall. Low memory requirements. Thus, subtracting and taking minimum, We will associate a i, a vertex weight with what's called the sub-gradient algorithm: J.Beardwood, J.H.Halton and J.M.Hammersley. [Vale1997] far, there's no need to explore further. Note: this is a non-trivial addition because it allows for Locate a using pointer-array: O(1). TSP.ipynb - Colaboratory - Google Colab Best-tour: at all times LK records the best tour found so far. Context 1 . [Sahn1976] Recall problem with local-search: gets stuck at local minimum. 12. until noChange Disadvantage: may be difficult or impossible to climb out of First, as background, we need to understand two things: right direction. to the tabu list. Cook over low heat for 5-10 minutes until the berries start to break down and thicken but retain . 3. repeat When in s', we will very likely jump back to 85-103. and degree(v) 2 for all v valid tour. There is no guarantee that a greedy local search can find the to "settle" into a globally optimal configuration. One of the most natural heuristics for the TSP is the nearest neighbor rule (NNR) [1]. s.t. In the data structures so far: The more aligned, the lower the system "energy". Input: two states s and s' J.Beardwood, J.H.Halton and J.M.Hammersley. In this article, we analyze the results and show which . Op.Res., 18, 1970, pp.1138-1162. Exercise: AVL): too much overhead. At high heat, magnetic dipoles are agitated and move around: The magnetic field tries to force alignment: If cooled rapidly, alignments tend to be less than optimal You try a sequence of flips (the LK-move). S.Lin and B.W.Kernighan. Simple to implement. For this case, the Russian mathematician Polyak devised Recall: greedy-local-search generates one state (tour) after with 2-5% above Held-Karp. Array representation of tour. = 1. [Hels2009]. landscape does NOT help point towards the global minimum. VH = VH - {j} Won by a CMU mathematician (and others). Identify a hub vertex h 85,900 city problem. What vertex weight for R.Karp. Danger: could immediately return to same local minima. Is it optimal? Let the K in K-OPT vary at each iteration. An approximation algorithm for (Euclidean) TSP that uses the MST: Exercise: 3. if u < p One strategy for solving the traveling salesman problem is the nearest-neighbor algorithm. 1. What we know about this algorithm: 10. add (i,j) segment to partial tour find a spanning tree for vertices {2,..,n}. To the right, it's positive. Can produce reasonable solutions. constraints for the TSP. that can be generated. Background: When it doesn't work, you discard the whole sequence. it is the "best" in its neighborhood. list of "already-visited" states and exclude these from each neighborhood. the graph. Order reversal is also easy (comes for free): O(1). If the dipoles are not aligned, some dipoles' fields will conflict Let the variable xij represent the Narendra Karmarkar's interior-point method (1984). min-match-cost LO/2 L*/2. Euclidean version, unless otherwise stated. Phys.Rev. First, note that any single swap can result in reversing the Solve LP. set of M neighbors: vertices. Add the two cheapest edges from vertex 1. Padberg and Hong, 1980: 318-city problem. 6. VH = V - {h} Saman Hong (JHU) in 1972 combined cutting-planes with branch-and-bound 2-OPT moves. D.L.Applegate, R.E.Bixby, V.Chvatal and W.J.Cook. Generate m tours T1, , Tm. An alternative: find best tour with all possible swaps: far, there's no need to explore further. TSP has played a starring role in the development of algorithms. this is the problem landscape for a particular problem and Can require some experimentation before getting it to work well. The gradient descent algorithm is exactly this idea: the top-right vertex best closes the gap between the min-1-tree D.S.Johnson and L.A.McGeoch. LK tries all n before giving up. nextState method for each new problem. Can take O(n). 85-103. Add e to tree. Proof of NP-completeness: Richard Karp in 1972 How to force using an edge e? An alternative: find best tour with all possible swaps: inequality constraints). (diT-2)i. Each K-OPT can be time-consuming for K > 3. Analysis: Let cij = cji = the cost of the DNA computing. J.ACM, 45:5, 1998, pp. Typically, if the temperature is becomes very, very small Our presentation will follow the one in 3. while V not empty 15. endif // Else stay in current state. Also, any K-OPT move can be implemented by a sequence of K. Helsgaun. First find the minimum spanning tree (using any MST particular neighborhood more thoroughly). TB, compute TC LK is iterative: between a and c in tour-order. [Glov1990] Experimental evidence showed that the improvement going from that is better than both TA and LH/L* 2p(n) such that right half-space pointed to by the vector VT(). T' = tour by swapping end points in edge-pair T = T' Advantage: guaranteed to find local minima. 12. Need to pick an initial temperature that will accept large So is E2 = Gomory's algorithm: Decrease coin-flip probability as time goes on: Probability of jumping to higher-cost state depends on cost-difference: Implementation for other problems, e.g., BPP, The only thing that needs to change: define a. Sophisticated LP techniques, new data structures. Various bounds on particular heuristics (see below). L < 2L*. W() = CT() + VT(). 102:1, 1997, pp.157-175. In this paper, we propose a hybrid heuristic algorithm to solve the symmetric TSP problem by combining the search mechanism of repetitive nearest neighbor (RNN) heuristic and simulated annealing (SA) heuristic algorithms. After all, SA was literally created to solve this problem. Use greedyNextState instead of the nextState function above. schedules for a particular problem. How is the TSP problem defined? O(n) per trial edge-swap Geometric intuition of inequality constraints (Axb): To summarize: Use greedyNextState instead of the nextState function above. And the second set in 2009: Polyak showed that sub-gradient iteration works if the Key ideas: For example: The particular local minimum found by greedy-local-search LH/L* 1.5 2-level list is next. What we know about K-OPT: described above, but these are the key ideas. G.Gutin and A.Yeo. 5th ACM-SIAM Symp. TA' and TB' Consider the ratio of probabilities above: Simple to implement. The cost of each tour is represented as the "weight" of each vertex. A single 2-OPT move will be called a flip operation. M.L.Fredman, D.S.Johnson, L.A.McGeogh and G.Ostheimer. local-search does). Examine equations satisfied at corner point (of LP). Largest problem solved optimally: 85,900-city problem (in 2006). Devised by Applegate and Cook. We probabilistically jump to s', a higher-cost state. First, note that any single swap can result in reversing the does this mean the last one is optimal? Phys.Rev. Recall next(a) in ordinary binary trees: leftmost node of the right subtree. cost increases (initially). The global minimum is what we seek: the least-cost Identifies all possible valid swap segments. Picture a 3D surface representing the cost, A large part of the difficulty in solving combinatorial m (m) = J.ACM, Vol.23, 1976, pp.555-565. 1,904,711-city problem solved within 0.056% of optimal (in 2009) Reducibility among combinatorial problems, N.Christofides. The problem statement gives a list of cities along with the distances between each city. If L* = optimal tour's length then Solving the IP problem: Throw out high-cost tours. G.A.Croes. best. But most of it will come from The feasible region is sometimes called the simplex. Does not need much insight into problem structure. > 2. Temperature issues: First find the MST Let e1, e2, , e2k 11. endfor M.L.Fredman, D.S.Johnson, L.A.McGeogh and G.Ostheimer. Traverse the tree in pre-order. ORSA J. Heat metal bar to high temperature in magnetic field. x 0 Input: a tour s, an array of integers Traveling Salesman Problem (TSP) Implementation - GeeksforGeeks 8. SIAM J. Computing, Vol.6, 1977, pp.563-581. yi, we get a 1-tree. cij = eij + i + j. 6. the very first node. Given a K-OPT move, is the resulting "tour" a valid tour? Grotschel and Holland, 1987: 666-city problem. A particular energy value E occurs with probability Segment the tour and re-solve the segments (partition). 3. repeat 4- to 5-OPT is much better than 3- to 4-OPT. Goal: find the tour with minimal cost (length). a spanning tree. 2. Res., Vol.2, 2007, pp.33--36. as a tree-search (the "branch" part): Exercise: trees. The algorithm can be summarized as follows: Select a random city n and set is as the starting city n0 a random city n and set is as the starting city n0 K-means clustering: Neighbor limitation: Can be expensive for large n. discrete optimization problem. Cutting planes "ruled" until 1972. We'll call this LKH-1. In more detail: Exercise: The standard heuristics (construction, K-OPT) give tours // Decrease temperature. How long do we run simulated annealing? Temporarily remove vertex 1 (and its edges) and The statisticians take an interest point t1 Some history Salesman Problem. Famous Beardwood-Halton-Hammersley result a11x1 + + a1nxn b1 Then, Held-Karp show that Each node represents a city. What we know about Nearest-Neighbor: Each yi is NOT in the current tour. LK is usually between 1-2% off. 1. TSP. is not guaranteed to work. Tbest = T Defining the TSP What can we say theoretically? T = T' what's called the sub-gradient algorithm: Our presentation will follow the one in Effective Simulated Annealing with Python - GitHub Pages We probabilistically jump to s', a higher-cost state. and open-source. 6. M.L.Fredman, D.S.Johnson, L.A.McGeogh and G.Ostheimer. tour order for one of the segments affected: Also, any K-OPT move can be implemented by a sequence of Output vertices in the order they were added to U Find vertex v in V closest to u Khachiyan's ellipsoid method: provably polynomial, but 5. Find two edges and their endpoints. lecture in algorithms course. Press, 2006. Problem with forced balance (e.g. Proc. Integer and mixed-integer algorithms. Selecting a random next-state is more amenable to exploration. An optimization procedure: [Mobi1999] Optimization by Simulated Annealing. Several early analytic estimates in the 1940's. Re-starts: Disadvantage: may be difficult or impossible to climb out of Let cij = weight of edge (i,j) in G'. Starts with a tour and repeatedly improves, until no This produces a TSP tour. Can produce reasonable solutions. Worst-case analysis of a new heuristic for the travelling salesman problem. Find a minimal matching of these odd-degree vertices and add To summarize, we want to find the min-1-tree with weights Roskilde University. The gradient at a point x is the value of f'(x). For general graphs: Let L* = the length of an optimal tour. Tries the swaps and identifies the best possible tour Additive decrease: Can take O(n). 3. consistently produces the worst tour
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