minT L(T,G) + Recall: greedy-local-search generates one state (tour) after for all i,j S. The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. The Traveling Salesman Problem: A Case Study in Local Optimization. [Kirk1983] Algorithmic Oper. From 1999-2009, Keld Helsgaun that is the last line to intersect the feasible region. Op.Res., 6, 1958, pp.791-812. (di - 2). Always add all neighborhood minimums. Algorithmic Oper. Padberg and Rinaldi, 1987-88: combined multiple types of starting points. // Swap a random pair of points References and further reading Grotschel and Holland, 1987: 666-city problem. with energy [Clar1964]. European J. Op. [Mobi1999]. [Rose1977]. Context 1 . Recall problem with binary trees: can go out of balance. The second one you move to the point closest to one of the two ends of your . With reversals, need to change direction for each flip (when recursing). Called a sub-tour constraint. xi R N.Christofides. Op.Res., 12 ,1964, pp.568-581. Allow some limited backtracking. LKH-1 idea: prefer 1-tree edges that go to neighbors. Springfield, Sullivan, Taylorville, Urbana. 12. until noChange of accepting a high-cost jump. Initialization { Start with a partial tour with just one city i, randomly chosen; nd the city j for which c ij (distance or cost from i to j) is minimum and build the partial tour (i;j). performing the shortest_path algorithm, by coding out a function. Pick the best K (the best tour) along the way. Problem landscape: For each pair of tours i,j, perform an IPT-iteration. algorithm). [Bear1959] The construction heuristics: Nearest-Neighbor, MST, LH/L* 2p(n) Let e1, e2, , e2k min-match-cost LO/2 L*/2. Called branch-and-cut. Gain criterion used by algorithm: L < 2L*. O(n) per trial edge-swap (local alignments): Bound is logarithmic: s.t. Apply the cutting-plane approach. local-search does). Some milestones: Suppose randomly-selected neighbor is s' with cost C(s') > C(s). Claim: the tour's length is no worse than twice the optimal C++ TSP implementation that outputs a tour given a list of points. Algorithm: expCoinFlip (s, s') Proof verification and hardness of approximation problems. Add constraints to force the LP-solutions towards integers. Climbing out of local minima: D.L.Applegate, R.E.Bixby, V.Chvatal and W.J.Cook. Intuitively, this means: Need a policy for removing items, e.g., R.Karp. construction heuristics. Res., Vol.2, 2007, pp.33--36. Then, W = 2M (each edge is traversed twice). Remove heaviest edge in cycle. Op.Res., 18, 1970, pp.1138-1162. for the following 4-point Euclidean TSP. until full-tour found prev(a): the previous node in tour order. 376-384. list of "already-visited" states and exclude these from each neighborhood. 6. Observe: if we remove any one edge from a tour, we will get Allow K to increase beyond 5. 5. History of applying IP to TSP: is negative. Can produce reasonable solutions. Implementing next(a): The Traveling Salesman Problem (TSP) is possibly the classic Step-1 - Finding Adjacent Matrix Of the Graph D.L.Applegate, R.E.Bixby, V.Chvatal and W.J.Cook. Let W() = minT L(T,G) + TSP is an NP-hard problem, which means it's computationally hard to obtain the optimal result. In contrast, a local-search heuristic works as follows: An improved bound: Reversals are noted by marking intermediate nodes, e.g. Then, the desired "best" Held-Karp bound is: Why slow-cooling works: N ear es t N ei ghbor H eur i s t i c: There are many solutions to the TSP problem and in fact - as discussed - this is a known hard problem of mathematics. Question: what happens to r as T decreases to zero? Decrease the weights for vertices with 1-min-tree degree Starts with a tour and repeatedly improves, until no Press, 2006. 2-OPT moves. Implementation: stepsizes (m) are chosen properly: landscape does NOT help point towards the global minimum. Won by a CMU mathematician (and others). Finally, add last two vertices and hub into final tour: First find the minimum spanning tree (using any MST Georgia Tech website on TSP. You try a sequence of flips (the LK-move). The traveling-salesman problem and minimum spanning - find out the shortest edge connecting current vertex and an unvisited vertex V. 3. Identifies all possible valid swap segments. 3-OPT is what you can get by considering replacing 3 edges. background in linear programming. that can be generated. [Chri1976]. Polynomial Time Approximation Schemes for TA' and TB' development of LP: 6. Our presentation will follow the one in tours TA and TB trees. Implementation for other problems, e.g., BPP What can we say theoretically? Graphically, the slope of the tangent to the curve at x. In this article, we analyze the results and show which . flip takes O(1) pointer manipulations. Always re-run with several (wildly) different starting solutions. DATALOGISKE SKRIFTER (Writings on Computer Science), No. T = T' A "best" (minimal weight) subset of edges with the property Key ideas in LKH-1: An analysis of several heuristics for the traveling salesman First, a definition: [Aror1998]. Arrays: A related problem: the Knight's tour. Branch-and-bound. added a number of sophisticated optimizations to the basic LK algorithm: LKH-2: Lin-Kernighan-Helsgaun, Part 2 tours TA and TB D.J.Rosenkrantz, R.E.Stearns and P.M.Lewis. Picture a 3D surface representing the cost, A large part of the difficulty in solving combinatorial Consider the distance from the current vertex to all of its neighbors that have not already been visited. 11. minTour = s' 102:1, 1997, pp.157-175. max cTx Question: what happens to r as T increases to infinity? Problem-partitioning: One approach: re-run local-search many times with different We probabilistically jump to s', a higher-cost state. European J. Op. 102:1, 1997, pp.157-175. What better way to start experimenting with simulated annealing than with the combinatorial classic: the traveling salesman problem (TSP). This is just the high-level idea G.A.Croes. The python program plots the basic tour (result of Nearest Insertion Neighbor algorithm) and the improved tour (result of the optimization algorithm). 17. endif can be converted to an equivalent one in standard form (with Exercise: Worst-case analysis of a new heuristic for the travelling salesman problem. M.Held and R.M.Karp. 0. R.Simha. For these, try both 0 and 1 (branch-and-bound idea). iT 4- to 5-OPT is much better than 3- to 4-OPT. described above, but these are the key ideas. 376-384. Performance: [Mobi1999] Find a minimal matching of these odd-degree vertices and add G2 = c(x1) - c(y1) A method for solving traveling salesman problems. The Greedy Algorithm for the Symmetric TSP. 10. endif Heat metal bar to high temperature in magnetic field. Best-tour: at all times LK records the best tour found so far. Effective heuristics. [Karp1972] To the right, it's positive. Suppose (2,3) provides the most savings: Mathematical Programming Computation, 2009. The general idea (an example): 1,904,711-city problem solved within 0.056% of optimal (in 2009) converge to an integer solution. Mathematical Programming Computation, 2009. L(T',G') = L(T',G) + iT' 2i. lecture in algorithms course. First, consider a simple exhaustive search, organized Euclidean Traveling Salesman and other Geometric Problems. Combinatorial optimization by iterative partial transcription. 9. until noChange best. Exercise: nextState method for each new problem. Narendra Karmarkar's interior-point method (1984). "Cost" of a tour: total length of tour. 13. else if expCoinFlip (s, s') Padberg and Rinaldi, 1987-88: combined multiple types of Example of a splay-step: two mini-rotations: Consider the optimal tour on just these (even # of) Pick a better neighbor to move to (or even best neighbor). cutting planes Example: For each pair of tours i,j, perform an IPT-iteration. Find successor using tour-order (instead of numeric order). He visits the nearest unvisited city, until all cities are visited, and then returns to the starting city [65,68]. Remove heaviest edge in cycle. Each yi is NOT in the current tour. First identify a "hub" vertex: The problem is: the MST can avoid using edges that the tour In tabu-search, you maintain a list of "tabu tours". 5. for all possible edge-pairs in T This is an idea from TSP's importance in computer science: Temporarily remove vertex 1 (and its edges) and 10. endif [Appl2006], Results: million city problem with 0.058% of Held-Karp. between(a,b,c): determine whether b is 1. repeat Recall: there are n choices for t1, is negative. Next, (1,2) gets added [Sahn1976] Guess what the large cost increase might be. J.Algorithms, Vol.18, 1995, pp.432-479. (GR) and the nearest neighbor algorithm (NN), popular choices for tour construction heuristics, work at acceptable level for the Euclidean TSP, but produce . The general form of the TSP appears to have been first studied by mathematicians during the 1930s in Vienna and at Harvard, notably by , who defines the problem, considers the Karl Menger obvious brute-force algorithm, and observes the non-optimality of the nearest neighbour heuristic. Thus, L <= W. Example of a splay-step: two mini-rotations: Initially, for first tour: in-order traversal is the tour: Reversals are noted by marking intermediate nodes, e.g. A simple algorithm for the min-1-tree: But consider what needs to be done to reverse a segment: Order reversal is also easy (comes for free): Recall problem with binary trees: can go out of balance. < 2. S.Arora. Thus, L <= W. 7. while |VH| > 2 i,j xi,j |S|-1 because probability of escaping a local minimum is miniscule. Also the optimal matching found earlier has less weight than A simplex-move is a move to a neighboring corner. // Note: this is a different K than in K-OPT. The traveling-salesman problem and minimum spanning The Shortest Path Through Many Points. S.Sahni and T.Gonzalez. You'll get a detailed solution from a subject matter expert that helps you learn core concepts. e-[C(s') - C(s)] / kT Optimization by Simulated Annealing. What's an example of an instance that's metric but not Euclidean? Simply running greedy-local-search on multiple starting The first set were added in 1999: tour's length. The idea: First, as background, we need to understand two things: General k-opt submoves for the Lin-Kernighan TSP heuristic. Thus O(n) per trial edge-swap Key ideas: and open-source. The feasible region is sometimes called the. 6. if T' < T If a flip is performed correctly, it will result in a Suppose we use a graph where the vertices represent states. A preview : [Chri1976]. and a "new-putative-edge". Consider the ratio of probabilities above: that is the last line to intersect the feasible region. does this mean the last one is optimal? 4. LH/L* 2p(n) Cambridge Phil. Tour: on Discrete Algorithms, 1994, pp.150-159. T = some starting tour // Perhaps by using Christofides. Run LKH-2 once to find a tour. consistently produces the worst tour Find the following 14 cities in Illinois/Indiana Doubly-linked lists: 1. p = exp ( -(cost(s') - cost(s)) / T) Each time you pick a minimum in a neighborhood, add that Compute starting cost as cost of going through hub: Identify "savings" for each pair of vertices: Take shortcuts and add them to final tour, as long as no For a differentiable function, the gradient "points" in the Note: E1 Neighbor limitation: L* / n a constant A single IPT trial-swap between 1992: Arora et al result Interfaces, 20:1, 1990, pp.74-94. Is it optimal? Segment-tree is usually best. The word program has different meaning than we are but it may not find local minima easily. [Aror1998]. Saman Hong (JHU) in 1972 combined cutting-planes with branch-and-bound Unlike previous algorithms, there is no fixed running time. - mark V as visited. // Randomly select a neighboring state. 7. noChange = false [Sahn1976]. Simulated annealing will allow jumps to higher-cost states. its vertices have even degree. Note: &alpha(e)=0 for any edge in min-1-tree. The problem is: the MST can avoid using edges that the tour Simply running greedy-local-search on multiple starting 4. Observe the following: Each inequality defines a half-plane (half-space). 6. Applegate et al (2006) Let L(T',G) = cost of tour T' using graph G. flip a coin to decide whether to jump to higher-cost state [Rose1977]. with energies E(s2) > E(s1) This will be a lower bound for the optimal tour. (dT1, , dTn). Let T be a 1-tree and T' be a tour. [Fred1995] For this case, the Russian mathematician Polyak devised solution overall. Then pick add two cheapest edges from vertex 1. Intuitively, this means: Typically, if the temperature is becomes very, very small Use a tabu-list to create freshness in exploration. Karmarkar's algorithm: provably polynomial and practically Observe the following: Low-energy states are more probable at low temperatures: 12. Algorithm: 9. if cost(s') < min consistently produces the. and open-source. Using a splay-tree for a tour: 5. else that can be generated. 753-782. Famous Beardwood-Halton-Hammersley result landscapes often have very little structure to exploit. What to do? (dT1, , dTn). B.Chandra, H.Karloff and C.Tovey. Simply running greedy-local-search on multiple starting Naive way: walk along new tour T' to see if all vertices are visited [Rose1977]. An approximation algorithm for (Euclidean) TSP that uses the MST: Identify a hub vertex h Let cij = cji = the cost of the Tries the swaps and identifies the best possible tour 2-level list is next. Nearest-neighbor heuristic: In Aarts, E. H. L.; Lenstra, J. K., Local Search in Combinatorial Optimisation, John Wiley and Sons Ltd, pp. Repeat. Can require some experimentation before getting it to work well. K-means clustering: D.J.Rosenkrantz, R.E.Stearns and P.M.Lewis. In a valid tour, D.J.Rosenkrantz, R.E.Stearns and P.M.Lewis. Pick any vertex to be the root of the tree. and degree(v) 2 for all v Start at bottom-left corner, and visit all squares exactly once and return to the start. Find a minimal matching of these odd-degree vertices and add valid tour. A problem: a tabu-list can grow very long. Polyak showed that sub-gradient iteration works if the E, 59:4, 1999, pp.4667-74. With slow-cooling, alignments are closer to optimal (global alignment): [Held1970] When it doesn't work, you discard the whole sequence. An approximation algorithm for (Euclidean) TSP that uses the MST: Res., Can be computed fast (MST) Combinatorial optimization by iterative partial transcription. max W(). An alternative: find best tour with all possible swaps: 3-OPT is what you can get by considering replacing 3 edges. // Start with any tour, e.g., in input order Let T be a 1-tree and T' be a tour. To summarize: Each xi is an edge in the current tour. The idea: some variables might change too slowly with Suppose the states s1, s2, Find best tour for each cluster. All four heuristics above were constructive Exact solution techniques: TSP as an IP problem . Judicious choice of cutting-plane heuristics. Three key algorithms, all major milestones in the Applegate et al (2006) Local Optima and Problem Landscape Identifies all possible valid swap segments. Advantages of simulated annealing: LH/L* = O(log n) [CS153] Summary: slow-cooling helps because it gives molecules more time is not guaranteed to work LH/L* 1+ pp. Original cutting plane idea due to Dantzig, Fulkerson and Johnson in 1954. Soc., 55, 1959, pp.299-327. General k-opt submoves for the Lin-Kernighan TSP heuristic. Input: two states s and s' Need to pick an initial temperature that will accept large 2-level list is next. 10. endif To help with tentative flips. However, it can take a long time. space than a greedy-local-search. Note: LK is actually a little more complicated than Find best tour for each cluster. next(a): the next node in tour order. next(a): the next node in tour order. Polynomial Time Approximation Schemes for 1. 2. u = uniformRandom (0, 1) To help with tentative flips. Combinatorial optimization by iterative partial transcription. x2, y2, , the top-right vertex best closes the gap between the min-1-tree First identify a "hub" vertex: What we know about Nearest-Neighbor: 12. endif The more aligned, the lower the system "energy". LH/L* 0.25 n1/2k. Analysis: Let L(T,G) = cost of 1-tree T using graph G. This is both an approximation strategy and a greedy strategy. i(m+1) Local search algorithms. An Effective Implementation of the Lin-Kernighan Traveling Salesman Heuristic, depends on the start state: Consider TSP using a particular local-search algorithm: Suppose we use a graph where the vertices represent states. T' = tour by swapping end points in edge-pair Use T = T - a, where a is a constant like 0.0001. Initially, for first tour: in-order traversal is the tour: [Glov1990]. In practice: need to experiment with different temperature Note: an LP problem with equality constraints Then we would use different searching algorithms to obtain a sub-optimal solution in reasonable time. undirected edge (i,j). IP problem. - set current vertex to V. 4. Next, (1,2) gets added every vertex i. First, note that any single swap can result in reversing the while not over Cambridge Phil. Based on key observation about LK: An alternative: find best tour with all possible swaps: Notice that if we stop at any intermediate S.Kirkpatrick, C.D.Gelatt, and M.P.Vecchi. Some assumptions and notation for the remainder: Early history: LKH-1: Lin-Kernighan-Helsgaun What can we say theoretically? [Karp1972] Alternate way of viewing this: TB, compute TC max cTx Now, walk along in Euler tour, but skip visited nodes Experimental evidence: 70-80% of these edges are in optimal tour. Unfortunately, we don't have a differentiable function. [Chri1976] NN and NND algorithms are applied to different instances starting with . Phys.Rev. Best known matching algorithm: O(n2.376) local-search algorithm. construction heuristics. Possibly the simplest to implement. 4. For the vertex-weights, the iteration turns out to be: Increase the weights for vertices with 1-min-tree degree Then, Held-Karp show that Today, there are several families of cutting-plane Add the two cheapest edges from vertex 1. For now im using nearest neighbour to find the path but this method is far from perfect, and after some research i found 2-opt algorithm that would correct that path to the acceptable level. i(m) + (m) Thermal cycling: max W(). An auxiliary segment-list: Note: we are performing a matching on an even number of vertices. Continuing let's say we obtain: can be converted to an equivalent one in standard form (with 9. until noChange In practice, it's quite efficient, approximately O(n3). optimization problems is the "weirdness" in landscapes have energies E(s1), E(s2), A tour that traverses all edges exactly once (but may repeat vertices) Devised in 1973 by Shen Lin (co-author on BB(N) numbers) and Has been used to solve very large problems (thousands of variables). 5. for i=1 to large-enough-number delivery points. 10. endif Judicious choice of cutting-plane heuristics. Exercise : V = {1, , n-1} // Vertices except for 0. Is the min-1-tree a good bound? Important result: min-matching can be found in poly-time. The last yk returns to the starting E, 59:4, 1999, pp.4667-74. The feasible region is sometimes called the simplex. Various bounds on particular heuristics (see below). With a sequence of such constraints, such a process can Goal: given two tours TA and 18. endwhile = For a differentiable function, the gradient "points" in the The algorithm actually alternates between a "current-tour-edge" Many involve some type of reversal. Overview Functions Version History Reviews (2) Discussions (4) The 2-column vector contains cities' coordinates. (diT-2)i. DATALOGISKE SKRIFTER (Writings on Computer Science), No. Periodically raise temperature and perform "re-starts". Consider states s1 and s2 A minimum-spanning tree: m of these. 4. u = most recently added vertex to U in R. E. Miller and J. W. Thatcher (editors). and the optimal tour? Expand the node that adds the least overall cost to the (partial) objective function. Pick vertex A as the root: the very first node. If L* = optimal tour's length then Consider a graph with vertices {1,,n}: Advantage: guaranteed to find local minima. right half-space pointed to by the vector VT(). 753-782. Best-tour: at all times LK records the best tour found so far. Goal: find a tour of minimal length. 5. if T' < T Problem with LK: 3. for each i,j != h Algorithm: LH/L* O(log n). 6. Graphically, the slope of the tangent to the curve at x. Repeat until you've reached optimal solution. some lines will pass through the feasible region. I would suggest solving the tsp and then solve the visual stuff. The Euclidean (points on the plane). J.ACM, Vol.23, 1976, pp.555-565. In tabu-search, you maintain a list of "tabu tours". DATALOGISKE SKRIFTER (Writings on Computer Science), No. Finally, we will show that L <= W and therefore, vertices. Worst-case analysis of a new heuristic for the travelling salesman problem. Suppose current state is s with cost C(s). A minimum-spanning tree: Note: LK is actually a little more complicated than G' has the same vertices and edges as G. Consider a subset of vertices S. 9. if cost(s') < min Branch-and-bound. Can be expensive for large n. Best known matching algorithm: O(n2.376). Note: Any K-OPT move can be reduced to a sequence of 2-OPT moves. The intersection is a polytope (polygon in 2D). for any fixed polynomial p(n). Allow K to increase beyond 5. T = some starting tour First, consider a simple exhaustive search, organized The state of the system is the particular snapshot (positions Keep increasing k as long as Gk > 0. Best known optimal algorithm: Held-Karp algorithm in 1962, O(n22n). improvement can be found. What is the difference between the optimal tour and the min-1-tree Gomory's algorithm: Geometric intuition of inequality constraints (Axb): [Hels1998] Claim: the tour's length is no worse than twice the optimal Put the fraughans or blueberries in a medium saucepan. Original cutting plane idea due to Dantzig, Fulkerson and Johnson in 1954. 8. noChange = false Note: every tour (including the optimal one) is a 1-tree. This means that larger values of W(') are in the Repeat. So is E2 = 4. return true 13. return T < 2. Always re-run with several (wildly) different starting solutions. What's known about the simplex method: 7. if cost(s') < cost(s) Note: we are performing a matching on an even number of vertices. The statisticians take an interest Unless P=NP no polynomial-time Locate a using pointer-array: O(1). = exp ([E(s2) - E(s1)] / kT) a graph-TSP is similar). Temperature issues: [Guti2007] G.A.Croes. valid tour. The global minimum is what we seek: the least-cost [Croe1958] G.Reinelt. 1 Answer. Exercise: of molecules) at any time. Karmarkar's algorithm: provably polynomial and practically Thus, the iteration tries to force the 1-min-tree to be "tour-like". The Boltzmann Distribution: Disadvantages: Science, 220 1983, pp.671-680. Just like the splay-tree, there are several different cases. F.Glover. // Record initial tour as best so far. Geometric: Data structures for traveling salesmen. Simple to implement. We seek an iterative algorithm of the form Use K=5 (prefer this value of K over smaller ones). Part of some neurological tests. Op.Res., 12 ,1964, pp.568-581. An optimization procedure: as a tree-search (the "branch" part): and the optimal tour? Cool rapidly (quench): if gradient < 0 for the above modified graph G'? The TSP describes a scenario where a salesman is required to travel between n cities. // Record best so far: Always pick an initial temperature to ensure high probability For more details on TSP please take a look here. Largest problem solved optimally: 85,900-city problem (in 2006). Fewer iterations Implementing next(a): The LK algorithm in more detail: Re-starts: If possible, try different neighborhood functions. A local-search algorithm gets "stuck" in a local minimum. In tabu-search, you maintain a list of "tabu tours". ORSA J. A problem: a tabu-list can grow very long. + c(x2) - c(y2). Narendra Karmarkar's interior-point method (1984). This is quite encouraging; altered_nn_tsp gives shorter tours and is three times faster than repeating nearest neighbors from 50 starting cities. But use of rotations is useful. D.S.Johnson and L.A.McGeoch. DNA computing. Solve the LP relaxation problem first. Tour segmentation: In contrast, a local-search heuristic works as follows: Relation between "energy" and "optimality" some lines will pass through the feasible region. LH L* + L*/2 This problem has been solved! Press, 2006. AVL): too much overhead. Then identify the odd-degree vertices The word program has different meaning than we are between a and c in tour-order. LKH-1 sorts neighbors by and uses best Goal: find lowest-energy state. In practice, it's quite efficient, approximately O(n3). and the optimal tour? 4- to 5-OPT is much better than 3- to 4-OPT. LKH-1 idea: prefer 1-tree edges that go to neighbors. 8. s = s' Each time you pick a minimum in a neighborhood, add that s.t. 215V310, 1997. U = {0} // Vertex 0. Output: a tour 388, GSIA, Carnegie-Mellon University, Pittsburgh, PA, 1976. 1. starting points. Then define Travelling Salesman Problem (TSP) : Given a set of cities and distances between every pair of cities, the problem is to find the shortest possible route that visits every city exactly once and returns to the starting point. R.Karp. Probability of jumping to higher-cost state depends on cost-difference: We seek an iterative algorithm of the form a11x1 + + a1nxn b1 4. u = most recently added vertex to U Held-Karp's idea: = {e1, e3, , e2k-1} One way: Called a sub-tour constraint. Wrong way to make a magnet: 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.
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