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The Traveling Salesman Problem in Theory & Practice Lecture 14: More on Maximum TSP Problems 29 April 2014 David S. Johnson dstiflerj@gmail.com http://davidsjohnson.net Seeley Mudd 523, Tuesdays and Fridays

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

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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 2014. 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 0.795454..) (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: 0.5000 ➟ 0.6250 ➟ 0.6666.. ➟ 0.75000

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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

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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.

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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.

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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.

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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.

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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)

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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

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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

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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

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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.

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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).

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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 }.

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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).

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

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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.

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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]

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Course Review

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

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

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

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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

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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

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

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

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

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

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

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

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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

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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

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

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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

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

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

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

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

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

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

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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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