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The Traveling Salesman Problem in Theory & Practice Lecture 14: More on Maximum TSP Problems 29 April 2014 David S. Johnson

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Presentation on theme: "The Traveling Salesman Problem in Theory & Practice Lecture 14: More on Maximum TSP Problems 29 April 2014 David S. Johnson"— Presentation transcript:

1 The Traveling Salesman Problem in Theory & Practice Lecture 14: More on Maximum TSP Problems 29 April 2014 David S. Johnson Seeley Mudd 523, Tuesdays and Fridays

2 Outline 1.The Asymmetric Maximum TSP 2.The “Maximum Scatter” TSP 3.The “Bottleneck” TSP 4.Outline 5.Student Presentation by Itai Feigenbaum

3 The Asymmetric Max TSP The standard maximum weight matching yields a (directed) cycle cover with cycles of length 2 possible, so the naive algorithm – Delete one edge from each cycle and patch arbitrarily into a tour only guarantees a solution that is (1/2)  OPT. An algorithm with an improved (5/8)  OPT guarantee was presented by Lewenstein & Sviridenko in a 2003 SIAM J. Discrete Math. paper. The next step was an algorithm with a (2/3)  OPT guarantee, presented by Kaplan, Lewenstein, Shafrir, & Sviridenko in a 2005 J. ACM paper. Current state-of-the-art: An algorithm with a (3/4)  OPT guarantee, claimed in an arXiv posting by Katarazyna Paluch on 15 Jan If you are willing to assume the Δ -Inequality (but no polyhedral norm), the current champ is an algorithm with a (35/44)  OPT guarantee was presented by Kowalik & Mucha in Algorithmica (2011). (Improving 3/4 to ) (They also have an algorithm with an (7/8)  OPT guarantee for the symmetric Max TSP, assuming the Δ -inequality but no polyhedral norm, that went unmentioned last week, and improves on 7/9 without the Δ -inequality.) Non-Triangular Asymmetric History: ➟ ➟ ➟

4 Getting a (2/3)  OPT Guarantee More Simply [Paluch, Elbassioni, & van Zuylen (2012)] Idea: Avoid having 2-cycles in the maximum matching via using “half- edges” and a gadget (also used in the subsequent (3/4)  OPT algorithm). out i in i tail(i,j)head(i,j) v i {i,j} head(j,i) out j in j tail(j,i) v j {i,j} w(i,j)/2 w(j,i)/2

5 Getting a (2/3)  OPT Guarantee More Simply [Paluch, Elbassioni, & van Zuylen (2012)] out i in i tail(i,j)head(i,j) v i {i,j} head(j,i) out j in j tail(j,i) v j {i,j} w(i,j)/2 w(j,i)/2 Weight w(i,j): Yields directed edge (i,j) from i to j. Weight 0: No edge between i and j.

6 Getting a (2/3)  OPT Guarantee More Simply [Paluch, Elbassioni, & van Zuylen (2012)] out i in i tail(i,j)head(i,j) v i {i,j} head(j,i) out j in j tail(j,i) v j {i,j} w(i,j)/2 w(j,i)/2 Weight w(j,i): Yields directed edge (j,i) from j to i. Weight 0: No edge between i and j.

7 Getting a (2/3)  OPT Guarantee More Simply [Paluch, Elbassioni, & van Zuylen (2012)] out i in i tail(i,j)head(i,j) v i {i,j} head(j,i) out j in j tail(j,i) v j {i,j} w(i,j)/2 w(j,i)/2 Weight (w(i,j)+w(j,i))/2: Yields undirected “double-in” edge between i and j.

8 Getting a (2/3)  OPT Guarantee More Simply [Paluch, Elbassioni, & van Zuylen (2012)] out i in i tail(i,j)head(i,j) v i {i,j} head(j,i) out j in j tail(j,i) v j {i,j} w(i,j)/2 w(j,i)/2 Weight (w(i,j)+w(j,i))/2: Yields undirected “double-out” edge between i and j.

9 Getting to a Tour The maximum weight matching in our constructed half-weigh-edge graph has total edge weight at least that of a maximum weight tour in the original graph. It also yields a set of edges in our original graph (ignoring directions) that is a cover by cycles of length at least 3. We will – take two copies of each edge in this cover (with each undirected edge represented by two directed edges in opposite directions*, and with the total weight being at least 2  OPT), – construct three sets P 1, P 2, and P 3 of vertex-disjoint directed paths from these copies, and – extend each arbitrarily to a tour. The largest of these three tours will thus have to be at least (2/3)  OPT. Here are the details: *Note: 2  weight{i,j} = 2(weight(i,j)+weight(j,i))/2 = weight(i,j) + weight(j,i)

10 Case 1: Directed Cycle Consider a cycle H in which all the edges are directed (and hence must be consistently directed). Pick two adjacent edges e i, e j from H. – Put {e i,e j } in P 1. – Put H – {e i } in P 2. – Put H – {e j } in P 3. Note that each edge goes into exactly two of the sets. Note also that this adds no cycles to any of the P i since in each case at least one edge is deleted from H. eiei ejej

11 Case 2: Cycle containing undirected edges, not all directed edges with same orientation Consider a cycle H containing at least one undirected edge, and with no consistent direction for all the directed edges it contains. Note that there must be equal numbers of “double-in” and “double-out” undirected edges, and these must alternate in the cycle, perhaps separated by paths of directed edges (consistently directed). – Put all clockwise directed edges and all undirected edges (converted to clockwise directed edges) into P 1. – Put all counter-clockwise directed edges and undirected edges (converted to counter- clockwise directed edges) into P 2. – Put all directed edges into P 3. inoutinout inoutin

12 Case 3: Cycle containing undirected edges, all directed edges with the same orientation Consider a cycle H containing at least one undirected edge, and with all the directed edges it contains having the same orientation, say clockwise. Note that there must be equal numbers of “double-in” and “double-out” undirected edges, which now must come as adjacent pairs. Pick one undirected out edge e out and one undirected in edge e in. Direct all the undirected edges in the same orientation (that of the directed edges, if any are present). – Put all edges except e in in P 1. – Put all edges except e out in P 2. – Put e out and e in into P 3, both with their directions reversed. outinoutine out e in

13 What about non-symmetric unit balls? These yield “quasi-norms” (triangle inequality but no symmetry) and our results carry over, although now the faces do not pair up and we need a tunnel for each face, with a directed edge from i to j having to enter a tunnel from the front and exit from the rear. So we have Theorem [BFJTWW, 2003]: Suppose we are given a polyhedral quasi-norm for R d, d ≥ 2, whose unit ball has f faces. Then the Maximum TSP for a set of N points in R d can be solved in time O(N 2f- 2 logN) on a real number RAM. (A polynomial-time algorithm for the special case of R 2 with a triangle for the unit ball was presented in [Serdyukov, 1995].) A modified version of Serdukov’s “asymptotically optimal” algorithm seems still to apply, although now only with a guarantee of (1 – 1/N) 2f  OPT(C) This is because we cannot merge cycles for free unless they contain edges that use the same tunnel in the same direction, and you need three cycles sharing a tunnel to guarantee this.

14 The Maximum Scatter TSP [Arkin, Chiang, Mitchell, Skiena, Yang, SIAM J. Comput. (1990)] Find the largest D such that there exists a Hamilton cycle with all edges of length D or more. Applications: – Mobile Bankrobber Problem. – Rivet Sequencing Problem. – X-ray Imaging Sequencing Problem Solvable by logN calls to a Hamilton Circuit code… Without the Δ -inequality, no constant-factor approximation algorithm can exist unless P = NP. With the Δ -inequality, one can find a tour with all edge lengths at least as long as (1/2)  OPT(C) in O(N 2 ) time. Generalization: Find the largest D for which there exists a Hamilton cycle such that all cities that are within k steps of each other are at least distance D apart (the “Min-Max k-Neighbor” problem).

15 The Bottleneck TSP Find the smallest D for which there exists a Hamilton cycle with all edges of length D or less. Applications: Transportation of perishable goods Inter-asteroid space delivery service Etc. Optimization Solvable by logN calls to a Hamilton Circuit code… Or by one call to a Min TSP code that can handle very large edge lengths: Only the ordering, not the values, of the edge lengths matter. Let the distinct edge lengths in instance I be d 0 < d 1 < d 2 < … < d k. Let I’ be the modified instance in which we replace these lengths by g 0 = 1, g 1 = N, g 2 = N 2, and, in general, g i = N i. Then we have OPT Bottleneck (I) = d i for i = min {h:OPT(I’) ≤ N h+1 }.

16 The Bottleneck TSP Approximation Without the Δ -inequality, no constant-factor approximation algorithm can exist unless P = NP. With the Δ -inequality, one can find a tour with no edge length longer than 2  OPT(C) in O(N 2 ) time. [Doroshko & Sarvanov, 1981 (in Russian)], [Parker & Rardin, 1984 (independently, in English)]. Proof: Observation 1 -- The maximum edge length in a “bottleneck biconnected spanning subgraph” is a lower bound on OPT Bottleneck (I): – This is a bi-connected subgraph with the smallest possible longest edge (“biconnected” ⇔ “connected with no articulation points”). – Computable in time O(N 2 ) since biconnectivity can be tested in linear time using depth first search, and we can do this repeatedly for graphs omitting all edges longer than one of the smallest k edge lengths, k = 1, 2, …, N. Observation 2 –- The square of a biconnected graph has a Hamilton cycle [Fleischner, 1974], which can be produced in time O(N 2 ) [Lau, PhD Thesis, 1980]. – The “square” G 2 of graph G = (V,E) is G, augmented by edges joining the endpoints of each length-2 path in G (if the edge is not already present).

17 The Bottleneck TSP Biconnected Graph G without a Hamilton circuit. Graph G 2, now with a Hamilton circuit.

18 The Bottleneck TSP By the Δ -inequality, each added edge in G 2 has length no more than the length of the two-edge path it short-circuits, and hence no more than twice the maximum edge length in the original graph. So the maximum edge length in the Hamilton cycle we construct for G 2 is at most 2 times the maximum edge length in G. If G is a bottleneck biconnected subgraph, this is thus at most 2  OPT Bottleneck (I). QED Theorem: Assuming P ≠ NP, no polynomial-time algorithm can guarantee a solution A(I) ≤ (2-ε)  OPT Bottleneck (I) for any constant ε > 0. [Doroshko & Sarvanov, 1981 (in Russian)], [Parker & Rardin, 1984]. Biconnected Graph G without a Hamilton circuit. Graph G 2, now with a Hamilton circuit.

19 The Asymmetric Bottleneck TSP Theorem [An, Kleinberg, Shmoys, Proc. APPROX & RANDOM, 2010]: Given an instance of the asymmetric TSP that obeys the triangle inequality, there is a polynomial-time algorithm that guarantees a tour whose maximum edge length is no more than (logN/loglogN)  OPT Bottleneck (I). Proof uses technology from the analogous result for the standard asymmetric TSP of [Asadpour, Goemans, Madry, Gharan, & Saberi, Proc. SODA, 2010]

20 Course Review

21 References, Web Resources, Etc.Definitions Applications Lecture 1: Introduction

22 Computational Equivalence of Cycle, Path, Symmetric, and Asymmetric Versions World Record TSP solutions and Concorde

23 Lecture 1: Introduction Metrics and Rounding Conventions Exploiting the Δ- Inequality in Theory and PracticeThe Held-Karp Bound

24 Lectures 2-3: NP-Hardness NP-completeness and the Hamilton Circuit Problem for Grid Graphs Hardness of Approximation, the PCP Theorem, MaxSNP, and the TSP

25 Lectures 3-4: Poly-Time Special Cases Polynomial Time Algorithm for the Gilmore-Gomory Scheduling Problem Polynomial Time Algorithm for Bounded- Height Rectangular Grid Graphs 2 O(√N) Time Algorithm for Planar Graphs

26 Lectures 4-5: Tour Construction Heuristics Upper Bounds, Lower Bounds, Performance in Practice

27 Lecture 6: Exploiting Geometry K-d Trees Sometimes with Geometric Lemmas Speeding Up Algorithms with K-d Trees Including Empirical Comparisons

28 Lectures 7-8: Local Optimization DIMACS Challenge Website Tour Data Structures 2-Opt Neighbor Lists

29 Lectures 7-8: Local Optimization Starting Tours 4-Opt and the Double Bridge Move 3-Opt and the Partial Sum Theorem Lin-Kernighan

30 Lectures 7-8: Local Optimization PLS-Completeness and the TSP Genetic Algorithms Simulated Annealing Iterated Lin-Kernighan

31 Lecture 8: ATSP Heuristics Algorithm Descriptions and Results of Testing the Algorithms Using Various Instance Generators, Several Based on Purported Applications.

32 Lecture 9: Pruned Exhaustive Search Evaluating All Permutations With Successive Layers of Pruning, Including Lagrangean Relaxation via p-Values And Greater and Greater Speedups with Each Step

33 Lecture 10: Cutting Planes and B & B The Cutting Plane Approach Heuristics and Exact Algorithms for Finding Violated Subtour Constraints Satisfying the Degree-2 Constraints Digression: Computing the HK Bound in Polynomial Time

34 Lecture 10: Cutting Planes and B & B Comb Inequalites Other Classes of Cutting Planes Dantzig, Fulkerson, & Johnson’s Success Branch & Bound

35 Lecture 11: Branch & Cut & Concorde Branch & Cut Implementation Issues Shrinking Safe Edges for Cut Finding More Subtour Constraint Heuristics Restricting to Core Edge Sets for LP Solving

36 Lecture 11: Branch & Cut & Concorde Managing Cuts Branching Strategies Solving the LP’s Quickly & Accurately Solving pla85900 (and Random Euclidean Instances)

37 Lecture 12: The Random Euclid. Constant The BHH Theorem The Toroidal Metric Early Estimates Standard Deviations of Values and Differences

38 Lecture 12: The Random Euclid. Constant Experimental Data Extrapolations Convergences Final Estimate

39 Lecture 12: The Random Euclid. Constant Potential Explanations Results for other Problems Small N Anomalies Approximating Planar Euclidean with Other Topologies

40 Lecture 13: The Maximum TSP General Approximation Algorithms Exact and Asymptotically Optimal Algorithms for Polyhedral Norms Polyhedral Norms Approximating the Euclidean Norm Polyhedrally

41 Lecture 14: Asymmetric MaxTSP, etc. Course Summary The Maximum Scatter TSP Asymmetric Maximum TSP The Bottleneck TSP

42 What should have been covered in more detail? Mihalis Yannakakis’s proof that the TSP cannot be described by a polynomial-size symmetric LP, and the recent strengthening of the result to cover asymmetric LPs ? The (Gödel-Prize-Winning) Arora/Mitchell Approximation Schemes for the Euclidean TSP ? The details of Helsgaun’s variant on Lin-Kernighan ? Proof of the Beardwood, Halton, & Hammersley Theorem ? (Your suggestions here…) What should have been covered in less detail?


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