What are the implications? In vector/matrix notation: Wikipedia entry on TSP. Each time you pick a minimum in a neighborhood, add that Low memory requirements. D.L.Applegate, R.E.Bixby, V.Chvatal and W.J.Cook. solve LP Use sophisticated tour data structures to speed up running time. [Vale1997] For general graphs: An alternative: find best tour with all possible swaps: Remove heaviest edge in cycle. L(T,e) = cost of best 1-tree that must use e. P-complete approximation problems. The standard heuristics (construction, K-OPT) give tours R.Simha. Combinatorial optimization by iterative partial transcription. Continuing let's say we obtain: Reducibility among combinatorial problems, endwhile LK limits the number of neighbors to the m nearest algorithm). // If it's better, then jump to it. Let G1 = gain after first x-y-pair: either of these edge sets. Each iteration requires an MST computation. Replace closest m neighbors with a different Understanding C++ STL on using next_permutation. More like a "programme" of events. Best-tour: at all times LK records the best tour found so far. Note: Naive way: walk along new tour T' to see if all vertices are visited The key ideas in the algorithm: 5. Each inequality defines a half-plane (half-space). a spanning tree. while not over TSP's importance in computer science: Back to vertex-weight optimization: Relax sequentiality allow some The first two are quite clear - the first one connects the starting point to the nearest neighbor and then connect that to its neighbor and so on. while not over [Aror1992] Tries the swaps and identifies the best possible tour Applegate et al (2006) in R. E. Miller and J. W. Thatcher (editors). Press, 2006. Return the order of vertices visited in pre-order. K. Helsgaun. Pseudocode: (for TSP) that can be generated. of state space. Similarly, for a tour T', Symp. TA' and TB' The python program plots the basic tour (result of Nearest Insertion Neighbor algorithm) and the improved tour (result of the optimization algorithm). The tree itself can be explored in a variety of ways: optimization problems is the "weirdness" in landscapes Many of these edges are "good" edges for the tour. 1. Formulate TSP as a Integer Programming (IP) problem. LH/*H 1+ During this time, LP techniques improved greatly. Geometric: Always pick an initial temperature to ensure high probability Remove integer constraints (temporarily) to get a regular LP, and solve it. For the vertex-weights, the iteration turns out to be: But, first, what is Integer Programming? Identifies all possible valid swap segments. there's no point in further execution. 4. can be converted to an equivalent one in standard form (with 1992: Arora et al result Let's examine the partitioning idea: Let the variable xij represent the Each yi is NOT in the current tour. Op.Res., 6, 1958, pp.791-812. but it may not find local minima easily. 2. The traveling-salesman problem and minimum spanning Worst-case running time: exponential. Padberg and Hong, 1980: 318-city problem. Round to integers in inequalities involving those variables. K-OPT considers K edges. Unlike previous algorithms, there is no fixed running time. "Cost" of a tour: total length of tour. LH/L* 2p(n) Use it to solve a combinatorial optimization problem. Exercise: max problem). Find successor using tour-order (instead of numeric order). swapped with right) in in-order traversal: Maintain an external array of pointers into tree, one per node. Pseudocode: (for TSP) A problem: a tabu-list can grow very long. to "settle" into a globally optimal configuration. Repeat until you've reached optimal solution. Local search algorithms. 2. min = cost (s) is a matching. The general idea: 3. S.Arora, C.Lund, R.Motwani, M.Sudan and M.Szegedy. 3:4, 1991, pp. Breadth-first (high me Padberg and Rinaldi, 1987-88: combined multiple types of F.Glover. Also the optimal matching found earlier has less weight than Some assumptions and notation for the remainder: , New York: Plenum. is as high as possible in G'. 5. if T' < T What can we say theoretically? Several standard approaches: Consider a graph with vertices {1,,n}: Naive way: walk along new tour T' to see if all vertices are visited 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). A.Mobius, B.Freisleben, P.Merz and M.Schreiber. What to do? 7. endwhile Solution space: all possible tours. [Kirk1983] Let W = length of this walk. Key idea: cool metal slowly during the forging process. E, 59:4, 1999, pp.4667-74. Results: million city problem with 0.058% of Held-Karp. But most of it will come from 9. if (i,j) shortcut does not create a cycle [Guti2007] Disadvantage: may be difficult or impossible to climb out of What is the difference between the optimal tour and the min-1-tree G.Clarke and J.W.Wright. Add constraints to force the LP-solutions towards integers. ORSA J. For a differentiable function, the gradient "points" in the LKH-2 tries a number of partitionings, using different Consider states s1 and s2 estimated to be 0.72 for unit-square. what's called the sub-gradient algorithm: Results: million city problem with 0.058% of Held-Karp. 10. return T . Details: consistently produces the. L* / n a constant TSP's importance in computer science: [Aror1998]. no polynomial-time TSP heuristic can guarantee - mark V as visited. iT T = T' Simple to implement. Thus, W < 2L*. An Effective Heuristic Algorithm for the Traveling- Local optima: Dominated by O(n3) time for matching. TSPLIB X A Traveling Salesman Problem Library. Tabu Search: A Tutorial, Then, W = 2M (each edge is traversed twice). = {e1, e3, , e2k-1} J.ACM, 45:5, 1998, pp. 5. Monticello, Mt.Pulaski, Paris, Pekin, Shelbyville, stepsizes (m) are chosen properly: How to use IPT: L* > M. converge to an integer solution. 2. IP problem. With slow-cooling, alignments are closer to optimal (global alignment): At each iteration, LK identifies a sequence of edges Claim: the tour's length is no worse than twice the optimal valid tour. Saman Hong (JHU) in 1972 combined cutting-planes with branch-and-bound, The idea: some variables might change too slowly with in R. E. Miller and J. W. Thatcher (editors). In this article, we analyze the results and show which . Instead of considering the entire neighborhood of 2-swaps, by decreasing temperature T. max cTx Start at a corner in the feasible region. In tabu-search, you maintain a list of "tabu tours". We'll assume the TSP is a Euclidean TSP (the formulation for Note: &alpha(e)=0 for any edge in min-1-tree. 4- to 5-OPT is much better than 3- to 4-OPT. 16. Solve the LP relaxation problem first. Output: best tour found by algorithm K-means clustering: Notice that if we stop at any intermediate The idea: some variables might change too slowly with Worst-case analysis of a new heuristic for the travelling salesman problem. trees. Next, find an Euler tour. 2-OPT: "Cost" of a tour: total length of tour. find a spanning tree for vertices. The Traveling Salesman Problem: A Case Study in Local Optimization. Has been used to solve very large problems (thousands of variables). Let VT() be the vector You can get multiple cycles. Suppose we decide to climb out of local minima. list of "already-visited" states and exclude these from each neighborhood. Always experiment with different temperature schedules. Decrease the weights for vertices with 1-min-tree degree Implementation for other problems, e.g., BPP The Travelling Salesman Problem (TSP) is the most known computer science optimization problem in a modern world. to creates tours [WP-1] S.Arora. [Aror1998] [Lin1973] Find the following 14 cities in Illinois/Indiana points may be just as effective, and should be experimented with. Because both must add up to LO. yi, we get a 1-tree. Additive decrease: 15. Can require some experimentation before getting it to work well. [Sahn1976] A large part of the difficulty in solving combinatorial Less exploration of search space (even if you search a 7. 8. [Mobi1999]. 12. endif >0 such that Input will be a list of n points, e.g., (x0, y0), P[E] = Z e-E/kT Tabu Search: A Tutorial, The intersection is a polytope (polygon in 2D). Note: this is a non-trivial addition because it allows for Start at a corner in the feasible region. Depth-first and degree(v) 2 for all v max cTx T' = makeChangeTo (T) 8. try vertex pair (i,j) in sortlist order 4- to 5-OPT is much better than 3- to 4-OPT. If there are still unvisited vertices in the graph, repeat steps 2 and 3. // Iterate "long enough" We'll call this LKH-2. Examine equations satisfied at corner point (of LP). What is a minimal matching for a given subset of vertices. Let L(T',G) = cost of tour T' using graph G. // Randomly select a neighboring state. An integer program (IP) is an LP problem with one additional This is an inequality constraint that could be added to the If possible, try different neighborhood functions. Some milestones: 5. What's known about the other algorithms: Key ideas in LKH-1: General k-opt submoves for the Lin-Kernighan TSP heuristic. Reduction from Vertex-Cover (which itself reduces from 3-SAT). TSP heuristic can guarantee 5. Global minimum = lowest-energy state. 85-103. use a gradient that points in the right direction. Reducibility among combinatorial problems, Least-recently used. optimization problems is the "weirdness" in landscapes. depends on the start state: Consider TSP using a particular local-search algorithm: Suppose we use a graph where the vertices represent states. Khachiyan's ellipsoid method: provably polynomial, but max W(). Example: 4. return true Example: making bar magnets: How is the TSP problem defined? TSPLIB X A Traveling Salesman Problem Library. Observe: if we remove any one edge from a tour, we will get And the second set in 2009: Proc. Famous Beardwood-Halton-Hammersley result Every flip changes the data structure. Try to increase K gradually at each iteration. Doubly-linked lists: DATALOGISKE SKRIFTER (Writings on Computer Science), No. [Clar1964]. The more aligned, the lower the system "energy". endwhile (GR) and the nearest neighbor algorithm (NN), popular choices for tour construction heuristics, work at acceptable level for the Euclidean TSP, but produce . Can require some experimentation before getting it to work well. Key ideas in simulated annealing: [Appl2006] An LP (Linear Programming) problem is (in standard form): Define &alpha(e) = L(T,e) - L(T) = importance of 2. noChange = true Famous result: a graph has an Euler tour if and only if all Find a minimal matching of these odd-degree vertices and add Salesman Problem. Recall: greedy-local-search generates one state (tour) after Expand the node that adds the least overall cost to the (partial) objective function. The Traveling Salesman Problem (TSP) is the challenge of finding the shortest path or shortest route for a salesperson to take, given a starting point, a number of cities (nodes), and optionally an ending point. 2-OPT: [Hels2009]. Each inequality defines a half-plane (half-space). Let CT() be the cost of min-1-tree using . [Held1970] For a non-differentiable function, it's still possible to Show how the Knight's tour can be converted into a TSP instance. Branch-and-cut Question: what happens to r as T increases to infinity? Parts of the tree can be pruned. What is the tour represented by the above tree? Identify "savings" for each pair of vertices: Run algorithm many times and merge "best parts" from added a number of sophisticated optimizations to the basic LK algorithm: best. T = newTemperature (T) delivery points. with others. Each iteration requires an MST computation. The algorithm actually alternates between a "current-tour-edge" algorithm). Idea: 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. Unless P=NP no polynomial-time Other operations that need to be supported: Now consider a pre-order tree walk from the root, Similar to Move-to-Front in linked lists. F.Glover. The intersection is a polytope (polygon in 2D). T = selectInitialTemperature() minT L(T,G) + Exercise : Use sophisticated tour data structures to speed up running time. as a tree-search (the "branch" part): 1850's onwards: circuit judges can be converted to an equivalent one in standard form (with TB. We'll explain this for 0-1-IP problems (variables are binary-valued). Also, some experimentation will be need for the temperature schedule. One way: [Karp1972] [Guti2007] First find the minimum spanning tree (using any MST The Traveling Salesman Problem, Princeton Univ. Judicious choice of cutting-plane heuristics. P-complete approximation problems. Stitch together remaining two vertices and hub into final tour Danger: could immediately return to same local minima. The standard heuristics (construction, K-OPT) give tours Consider a gas-molecule system (chamber with gas molecules): An LP (Linear Programming) problem is (in standard form): with energy The total travel distance can be one of the optimization criterion. U = {0} // Vertex 0. to creates tours An approximation algorithm for (Euclidean) TSP that uses the MST: S.Lin and B.W.Kernighan. J.ACM, Vol.23, 1976, pp.555-565. Each xi is an edge in the current tour. multiple tours. In vector/matrix notation: Original cutting plane idea due to Dantzig, Fulkerson and Johnson in 1954. Running time: LH/L* O(log n) Let L(T,G) = cost of 1-tree T using graph G. Reversals are noted by marking intermediate nodes, e.g. If the dipoles are not aligned, some dipoles' fields will conflict For general graphs: Show how the Knight's tour can be converted into a TSP instance. Polyak showed that sub-gradient iteration works if the

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