Very Large Scale Neighborhood Search Neighborhood Search Combinatorial Optimization: minimize (f(S) : S F) f is typically linear, F is finite Neighborhood Function: For each S F, there is a neighborhood N(S); We say that S is a local optimum if f(S) d f(T) for all T N(S); 2 Neighborhood Search Neighborhood Search (local improvement algorithm) begin initialize with some S F; while S is not a local optimum do replace S by some T N(S) such that f(T) < f(S); end 3 TSP and 2-exchanges 1 1 2 3 4 5 6 7 The original tour 9 10 2 3 8 7 6 5 4 9 A 2-neighbor of the original tour 10 We say that a tour T’ is a 2neighbor of a tour T if it is possible to obtain T’ from T by adding two edges and deleting two edges. The operation is called a 2-exchange. T’ = T + (3,8) + (4,9) - (3,4) - (8,9). Obtained by the operation Flip[4,8] 8 When we say city i, we really mean the city that is in position i of the current tour. (Or you may assume that the current tour is 1, 2, 3, ..., n) 4 A non-optimal tour 5 6 1 4 7 20 14 3 19 18 15 17 8 16 9 10 12 13 2 11 5 Improving using a two exchange 5 4 6 3 1 4 5 7 2 20 14 3 6 19 18 15 17 8 16 9 10 12 13 2 7 11 A 2-exchange is obtained by deleting two edges of the tour and adding two edges 6 Neighborhood Search We say that a tour T’ is a 2-neighbor of a tour T if it is possible to obtain T’ from T by adding two edges and deleting two edges. This can be generalized to 3-neighbors, and k-neighbors. Neighborhood Search: begin start with some tour T. while there is a neighbor T’ of T with a lower objective value, replace T by T’ end 7 More on Neighborhood Search A solution is 2-optimum if there is no 2-neighbor that has a lower length. A solution is locally optimum if there is no neighbor with an improved objective. Neighborhood search in its “vanilla” form stops when a locally optimum solution is found. Most neighborhood search relies on neighborhoods that can be searched exhaustively. 8 Some simple extensions Extension 1. For k = 1 to 1000, select a random tour T. Find a locally optimal tour starting with T. Then choose the best of these tours. Extension 2. 1. Let T be a tour 2. Find a locally optimal tour T’ starting at T 3. perturb T’ (perhaps using a random 5 exchange) 4. replace T by T’ and return to step 1. Other extensions: simulated annealing and tabu search 9 Very Large Scale Nbhd (VLSN) search This talk: Focuses on VERY LARGE neighborhoods that can be searched very efficiently (preferably in polynomial time) or are searched heuristically. often exponentially large neighborhoods Rule of Thumb for Larger Neighborhoods: improved local optima greater search time 10 relation to other search techniques min cost flows, cycle canceling Multicommodity flows: column generation But both these problems are special case of linear programming. We will use this technique for hard problems as well. 11 I’m planning on using the 2-opt neighborhood. It has n2 neighbors. Pretty good. But please check out larger neighborhoods. 12 Two survey papers "A Survey of Very Large Scale Neighborhood Search Techniques", Ahuja, Ergun, Orlin, and Punnen [2002] http://web.mit.edu/jorlin/www/ “A study of exponential neighborhoods....” Deineko and Woeginger [2000] 13 Personal view of VLSN search I view it as liberating z it does not restrict neighborhood search to exhaustively searching a neighborhood z permits many alternatives to standard neighborhood search I view it as a very practical method for addressing problems that are large For decades researchers have argued that solving special cases of problems is potentially of use because it can help us solve more general cases. z This is one way of realizing the potential 14 An example of VLSN: Independent 2-exchanges The original tour T 1 2 3 4 5 6 7 8 9 10 Flip[7,9] and obtain T1 1 2 3 4 5 6 9 8 7 10 8 9 10 Flip[2,4] and obtain T2 1 4 3 2 5 6 7 Two exchanges, Flip[i,j] and Flip[i’,j’] are independent if i’ > j+1 (or i > j’+1). 15 Independent 2-exchanges T1 1 2 3 4 5 6 9 8 7 10 T2 1 4 3 2 5 6 7 8 9 10 T3 1 4 3 2 5 6 9 8 7 10 Flips [2,4] and [7,9] are independent • costs can be calculated independently • one interval is to the left of the other (used for efficient searching) c(T3) - c(T0) = [c(T1) - c(T0)] + [c(T2) - c(T0)]. cost of [7,9] cost of [2,4] 16 Pairs that are not independent 1 2 8 7 6 5 4 3 9 10 1 4 3 2 5 6 7 8 9 10 1 2 9 8 7 6 5 4 3 10 1 2 3 4 7 6 5 8 9 10 1 2 5 4 3 6 7 8 9 10 1 2 3 4 5 8 7 6 9 10 17 Dynasearch/ Ejection Chains, and more We say that T’ is an independent 2-exchange neighbor of T if T’ can be obtained via k independent 2-exchanges for some k. Size of neighborhood = 4(n1.7548…) Dynasearch: Potts and Van de Velde [1996] O(n2) DP algorithm. Ejection chains: Glover [1992]. 18 The Improvement Graph 1 1 2 3 4 5 6 9 8 7 10 2 1 4 3 2 5 6 7 8 9 10 Improvement Graph 1 2 3 4 5 6 7 8 9 10 11 cij: cost of keeping city i fixed and city j fixed and flip cities i+1, ..., j-1 Min cost collection of independent 2-exchanges: shortest path from node 1 to node n+1. O(n2) time. Note: there are O(n2) 2-exchanges 19 Summary of the Method Create improvement graph Search improvement graph no One can incorporate tabu search by adjusting the improvement graph Is stopping rule reached? yes End 20 Insertion based neighborhoods Before 1 2 3 4 5 6 7 8 9 10 After 1 2 4 5 6 3 7 8 9 10 Improvement Graph 1 2 3 4 5 6 7 8 9 10 11 cij: cost of keeping city i fixed and city j fixed and inserting city i+1 after city j-1. Min cost collection of independent 2-exchanges: shortest path from node 1 to node n+1. O(n2) time. 21 Some advantages for VLSN search They can provide a limited form of look-ahead They can provide a limited form of parallelism They can be very effective in practice. They are appealing. 22 Cyclic Exchange for Partitioning Problems S : set of objects Partition S into subsets (S1, S2, S3, .. , Sp) Define a (possibly nonlinear) cost function f(Si) for each subset Si Find the partition S1, S2, … , Sp such that ¦p=1,K f(Sp) is minimum. 23 Partitioning Problems 15 14 Given a set S of 20 objects 12 1 7 2 3 5 18 16 10 11 6 8 9 17 13 19 20 4 Partition the set into four subsets 2 10 19 16 15 18 1 3 7 9 6 12 11 20 14 13 4 8 17 5 24 The Cost Structure 2 10 19 16 15 18 1 3 7 9 6 12 11 20 14 13 4 8 17 5 The cost for part Si is f(Si). Find the partition with minimum total cost. 25 Vehicle Routing Problems, Scheduling Problems, Clustering Problems, and more 5 4 9 6 8 1 7 26 11 10 12 0 17 14 18 15 13 19 16 22 27 25 21 24 23 20 Vehicle Routing: Assign trucks to pick up cargoes so as to minimize total travel time and meet service requirements. Note: f(Si) may be difficult to compute for a subset Si. 26 Capacitated Minimum Spanning Tree Problem (CMST) 5 4 9 6 8 1 7 26 11 10 12 0 17 14 18 15 13 19 16 22 27 25 21 No link (arc) carries more than K units of flow (at most K nodes per subtree). 24 23 20 Features: A central computer to be connected to a number of terminals Each terminals has a demand of 1 (homogeneous case) Minimize the total cost of connection. 27 Some references on CMST Exact Algorithms Chandy and Russell [1972], Chandy and Lo [1973], Elias and Fergusan [1974], Kershenbaum and Boorstyn [1983], Gavish [1982, 1983, 1985], Gouveia and Paixao [1991], Malik and Wu [1993], Gouveia [1993, 1995], Hall [1995], Gouveia and Martins [1996] Heuristic Algorithms Esau and Williams [1966], Martin [1967], Sharma and El-Bardai [1970], Whitney [1970], Frank et al. [1971], Chandy and Russell [1972], Elias and Ferguson [1974], Kershenbaum [1974], Kershenbaum and Chou [1974], Karnaugh [1976], Gavish and Altinkemer [1986, 1988], Gavish [1991], Sharaiah et al. [1995], and Amberg et al. [1996] 28 The Cyclic Exchange Neighborhood (Multi-Swap) S2 S1 19 2 10 S4 16 15 18 18 1 3 7 9 6 12 20 14 14 11 13 4 8 S3 17 5 Each subset either (i) is unchanged or (ii) has one item inserted and another item deleted The number of neighbors can grow exponentially in the number of items. 29 A Path Exchange S2 S1 19 2 10 S4 16 15 18 1 3 7 9 6 12 20 14 14 11 13 4 8 S3 17 5 After a path exchanges one subset increases by 1 and another decreases by 1 30 A Simplification for this talk We consider only cyclic exchanges Improvement graph: used to identify profitable cyclic exchanges Suggested originally by Thompson and Orlin [1989] and Thompson and Psaraftis [1993] Thompson [1989] Other References Ahuja, Orlin, and Sharma [1998, and 2003] Gendereau, Guertin, Potvin, and Sequin [1998] Maria G. Scutella, Antonio Frangioni, Emiliano Necciari [2000] 31 The Effect of a Cyclic Exchange, Multi-Swap S2 S1 19 2 10 S4 16 15 18 18 1 11 3 9 7 6 12 20 14 14 Let S2 be the 2nd subset prior to the exchange. Let S’2 be the subset after. Then S’2 = S2 + {18} – {2}. Let c18,2 = f(S2) – f(S2+ {18} – {2}) = increase in cost of 2nd part after the exchange. Let c2,14 = f(S4) – f(S4+ {2} – {14}) = increase in cost of 4th part after the exchange. 32 The improvement Graph G S[j] S[i] 19 2i 10 7 16 6 12 20 14 j 15 Create a node in G for each element in the partition i cij j Let ci,j = f(S[j]) – f(S[j] + {i} – {j}) = increase in cost of part S[j] after an exchange that moves i to S[j] and moves j out. 33 About the improvement graph S2 S1 19 2 10 S4 16 15 18 18 1 3 7 9 6 12 20 14 14 11 13 4 8 S3 17 5 Let “increase in the cost” from the exchange is the cost of the corresponding cycle in the improvement graph. A negative cost cycle leads to an improving exchange. “Increase” is c4,18 + c18,2 + c2,14 + c14,4 34 The Improvement Graph There is a one-to-one correspondence between cyclic exchanges and “subset-disjoint” directed cycles in G. (At most one node enters or leaves per subset). A negative cost subset-disjoint cycle in G defines an improving cyclic exchange. Identifying a negative cost subset-disjoint cycle is NP-hard. We can find negative cost subset-disjoint cycles effectively in practice using either heuristics or implicit enumeration. 35 Data for the CMST Problem. Ahuja, Orlin, Sharma [1998] Percentage deviation 10 5 0 -5 -10 Nbhd 1 -15 -20 Nbhd 2 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 Problem number Comparison on heterogeneous demand problems. R. K. Ahuja, J. B. Orlin, and D. Sharma. 1998. New neighborhood search structures for the capacitated minimum spanning tree problem. Working Paper, Sloan School of Management, MIT, Cambridge, MA, submitted to Mathematical Programming. 36 Airline Scheduling Schedule Generation Fleet Assignment Through Assignment Maintenance Routing Crew Scheduling 37 Airline Fleet Assignment Model Assign planes of different types to different flight legs so as to minimize the cost of assignment K fleets of aircraft Flight legs to be assigned Fleet Assignment Model Assignment of a plane to each leg Cost of assigning a plane type to a flight leg Flight coverage and aircraft integrality Aircraft balance Fleet size for each of K different fleets Possibly: connections, maintenance, and more 38 On Neighborhood Search Usual neighborhood search does not seem appropriate One cannot swap a fleet type on one leg for a fleet type on another leg. But we can swap a fleet type on a sequence of legs for a fleet type on other legs. 39 Single A-B Swaps (before) Orlando Atlanta Cincinnati Boston New York Wash. D.C. Raleigh 8a 8a 8a 8a 8a 8a 8a 9a 9a 9a 9a 9a 9a 9a 10a 10a 10a 10a 10a 10a 10a 11a 11a 11a 11a 11a 11a 11a 12a 12a 12a 12a 12a 12a 12a 1p 1p 1p 1p Type A 1p Plane 1p 1p 2p 2p 2p 2p 2p 2p 2p 3p 3p 3p 3p 3p 3p 3p 4p 4p 4p 4p 4p 4p 4p 5p 5p 5p 5p 5p 5p 5p 6p 6p 6p 6p 6p 6p 6p 7p 7p 7p 7p 7p 7p 7p Type B Plane 40 Single A-B Swaps (after) Orlando Atlanta Cincinnati Boston New York Wash. D.C. Raleigh 8a 8a 8a 8a 8a 8a 8a 9a 9a 9a 9a 9a 9a 9a 10a 10a 10a 10a 10a 10a 10a 11a 11a 11a 11a 11a 11a 11a 12a 12a 12a 12a 12a 12a 12a 1p Type A B Type Plane 1p Plane 1p 1p Type A B Type 1p Plane Plane 1p 1p 2p 2p 2p 2p 2p 2p 2p 3p 3p 3p 3p 3p 3p 3p 4p 4p 4p 4p 4p 4p 4p 5p 5p 5p 5p 5p 5p 5p 6p 6p 6p 6p 6p 6p 6p 7p 7p 7p 7p 7p 7p 7p 41 Multi A-B Swaps B A B A A B After Before B A B A B A 1 red path directed out and 1 black path directed out 1 red path directed in and 1 black path directed in reversing B arcs leads to a directed cycle. 42 A-B Graph (Talluri [1996]) Construct the A-B Graph, consisting of legs flown by planes of types A and B. Reverse directions for arcs of type B. Cost of (i,j) is the cost of moving i from one type to another. For the diagram, please see: K.T. Talluri. Swapping Applications in a Daily Airline Fleet Assignment. Transportation Science, 30, 1996. 43 Computational Results: Research with United Airlines Ahuja, Orlin, Sharma [2000] 1609 Flights Legs, 13 Fleet types. Initial solution: guaranteed to be within .1% of optimal. Local Search Tabu Search Increase in Fleeting Profits per year -$0.7 million $3.2 million if sole objective Increase in Through Profits per year $27.8 million Increase in Total Profits per year Running Time $27.1 million 20-30 secs -$1.7 million $3.2 million if sole objective $30.3 million $28.6 million 15-20 min Through flight: direct connecting flight with the same flight number Based on: R.K. Ahuja, J.B. Orlin and D. Sharma. Very Large-Scale Neighborhood Search. International Transactions in Operations Research 7, 2000, pp. 301-317. 44 Neighborhoods Based on Polynomial Time Algorithms for Special Cases Vast literature based on polynomially solvable special cases. Methodology: turn a special case into a neighborhood. Illustrations: Pyramidal Tours Halin Graphs 45 Pyramidal Tours and Neighbors 1 2 4 3 Tour is monotone increasing from node 1 to node n both clockwise and counterclockwise 7 Best pyramidal tour : O(n2) time. Pyramidal neighbor: a tour T’ that is 5 pyramidal if the initial tour is 1, 2, ..., n 6 9 Best pyramidal neighbor : O(n2) time. 8 10 11 12 13 Pyramidal neighborhood and extensions: Sarvanov and Doroshko [1981] Carlier and Villon [1990], and Burkard, Deineko, and Woeginger [1998]. 46 Halin Graphs A Halin Graph is a tree whose nodes do not have degree 2, and the leaves are connected in the plane to form a cycle. 47 Halin Graphs One can find min cost tours in Halin Graph in O(n) time. Cornuejols, Naddef, and Pulleyblank [1983] 48 Halin Neighbors Approach: embed a Tour inside a Halin graph, and optimize over the neighborhood. Suggested by Punnen Requires: a polynomial time algorithm for embedding T in a Halin Graph 49 A Generic Approach Let X be an NP-hard problem. Let X’ be a restriction of X, where X P. Develop a subroutine CreateNeighborhood(S) INPUT: feasible subset S, for (F, f) X OUTPUT: Instance (F’, f) X’, with F’ F, and S F’ 50 A Generic Approach INPUT: feasible subset S, for (F, f) X OUTPUT: Instance (F’, f) X’, with F’ F, and S F’ S is feasible for the special case F’ that is solvable in polynomial time. We refer to F’ as the X’-induced neighborhood of S. The X’-induced neighborhood can be searched in polynomial time. 51 More on the generic approach Essential aspect: for a given solution x*, solve a restriction of the original problem for which x* is feasible. Any solution in the restriction is a neighbor. 52 Conclusions VLSN Search is another tool in our heuristic toolkit Key is defining a neighborhood that can be searched efficiently and that leads to good local optima Search techniques z Network Flow approaches and DP z Polynomial algorithms for subproblems z IP and DP and … approaches 53