A constant-factor approximation algorithm for the asymmetric travelling salesman problem

Transcription

1 A constant-factor approximation algorithm for the asymmetric travelling salesman problem London School of Economics Joint work with Ola Svensson and Jakub Tarnawski cole Polytechnique F d rale de Lausanne

2 Travelling salesman problem Given n cities and their pairwise distances, find a shortest tour visiting all n cities. One of the best known NP-hard optimization problems Studied since the 19 th century Symmetric TSP:, =,, Asymmetric TSP:,, is possible UK pub tour [Cook et al., 2015] Triangle inequality:,, +,,,

3 Symmetric vs Asymmetric TSP Symmetric TSP 1.5-approximation algorithm [Christofides 76] Graphic TSP: unweighted graph shortest path metric Current best 1.4 following

4 Symmetric vs Asymmetric TSP Asymmetric TSP log -approximation algorithm [Frieze, Galbiati & Maffioli 82] 0.99 log 0.84 log [Kaplan, Lewenstein, Shafrir & Sviridenko 03] 0.67 log [Feige & Singh 07] lo g lo glo g [Asadpour, Goemans, M dry, Oveis Gharan & Saberi 10] via thin trees.

5 Asymmetric TSP recent developements log log bound on integrality gap of LP [Anari & Oveis Gharan 15] Constant-factor approximations: Bounded genus graphs [Oveis Gharan & Saberi 11] Node-weighted graphs [Svensson 15] Graphs with 2 edge weights [Svensson, Tarnawski & V. 16] Our result: constant-factor approximation for general ATSP with respect to the Held-Karp relaxation.

6 ATSP Graphic formulation Input: directed graph =,, edge weights : R + Find a minimum weight tour. Tour = closed walk visiting every vertex at least once = = Eulerian and connected edge multiset Eulerian: = Subtour = closed walk (not necessarily connected) 4 In-degree & out-degree in F

7 ATSP Graphic formulation Input: directed graph =,, edge weights : R + Find a minimum weight tour. Tour = closed walk visiting every vertex at least once = = Eulerian and connected edge multiset Eulerian: = Subtour = closed walk (not necessarily connected) 4 In-degree & out-degree in F

8 Held-Karp relaxation Input: =,, edge weights : R +. Variables : E R + : multiplicity of selecting edge. minimize subject to =, Eulerian degree constraints Subtour elimination constraints Undirected degree: = +

9 Held-Karp relaxation Input: =,, edge weights : R +. Variables : E R + : multiplicity of selecting edge. minimize subject to =, Can be solved in polynomial time Integrality gap [Charikar, Goemans & Karloff 06]

10 Pick any two integral cycle cover ATSP spanning tree Eulerian Held-Karp connected

11 Pick any two integral cycle cover ATSP spanning tree Eulerian Held-Karp connected

12 Repeated cycle cover algorithm [Frieze, Galbiati & Maffioli 82] Relaxing connectivity: 1. Find minimum weight cycle cover 2. Contract and repeat Each cycle cover has cost Overall log rounds log approximation

13 Node-weighted case [Svensson 15] Directed graph =,, node weights h: R +, = h + h, Local-Connectivity ATSP: relaxing connectivity constraints to local -light algorithm for Local-Connectivity ATSP Theorem [Svensson 15] There exists a polytime + )- approximation for node-weighted ATSP. + -approximation for ATSP

14 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

15 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

16 Dual of the Held-Karp relaxation minimize subject to = maximize subject to :, +,, Dual can be solved in polynomial time. One can efficiently find an optimal, such that the support of is a laminar family of sets. Efficient uncrossing [Karzanov 96]

17 Laminarly weighted ATSP: I =, L,, minimize subject to = maximize subject to :, +,, : directed graph L: laminar family of sets : feasible Held-Karp solution tight on every set in L: = L : L R +

18 Laminarly weighted ATSP: I =, L,, minimize subject to = maximize subject to :, +,, : directed graph L: laminar family of sets : feasible Held-Karp solution tight on every set in L: = L : L R + Induced weight function:, = :,

19 Laminarly weighted ATSP: I =, L,, minimize subject to = maximize subject to :, +,, : directed graph L: laminar family of sets : feasible Held-Karp solution tight on every set in L: = L 23 : L R + Induced weight function:, = :,

20 Reduction to laminarly weighted ATSP Start with any and. Compute Held-Karp optimal solution and dual supported on laminar family L Delete all edges with =. maximize subject to :, +,, Observations: Optimal solutions and optimum value are the same for and for, =, + All remaining edges have, = :,

21 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

22 Vertebrate pairs Vertebrate pair I, I =, L,, instance : backbone = subtour that crosses every nonsingleton set in L

23 Vertebrate pairs We will reduce general ATSP to solving ATSP for a vertebrate pair I, with = (more or less ) Solve Local-Connectivity ATSP on such instances, and apply [Svensson 15]

24 Local-Connectivity ATSP [Svensson 15] Instance I =, L,, with induced weights : R + Lower bound function lb: R + with lb = Input: partition of the vertex set =

25 Local-Connectivity ATSP [Svensson 15] Instance I =, L,, with induced weights w: R + Lower bound function lb: R + with lb = Input: partition of the vertex set = Output: Eulerian edge set with > for each

26 Local-Connectivity ATSP [Svensson 15] Instance I =, L,, with induced weights w: R + Lower bound function lb: R + with lb = Input: partition of the vertex set = Output: Eulerian with > for each -light algorithm: for every component of, lb Every component pays for itself locally

27 Local-Connectivity ATSP [Svensson 15] -light algorithm for Local-Connectivity ATSP Theorem [Svensson 15] There exists a polytime + )- approximation for node-weighted ATSP. + -approximation for ATSP

28 Local-Connectivity ATSP: node-weighted case Instance I = Define lb, L,,, with L containing only singletons (ignore ), = {} + {} = {} Partition = all strongly connected Modify and, and solve an integer circulation problem

29 Local-Connectivity ATSP: node-weighted case Instance I = Define lb, L,,, with L containing only singletons (ignore ), = {} + {} = {} Partition = all strongly connected Modify and, and solve an integer circulation problem For each, create auxiliary vertex Reroute 1 fractional unit of incoming and outgoing flow to Solve integer circulation problem routing =1 unit through each Map back to original

30 Local-Connectivity ATSP: node-weighted case The rerouted is feasible to the circulation problem of weight Flow integrality: there exists integer solution of weight After mapping back, every vertex with > has in-degree For a component, =, {} + {} {} lb = {} 2-light algorithm

31 Local-Connectivity ATSP: one nonsingular set in L Vertebrate pair I,. Assume L has a single non-singleton component. Thus,, = {} + +, {} + {}, Define {} lb = lb =, since = )

32 Local-Connectivity ATSP: one nonsingular set in L By assumption, = =1 Backbone property: there is a node Simple flow argument: we can route the incoming 1 unit of flow to to s

33 Local-Connectivity ATSP: one nonsingular set in L Partition = Add backbone into Eulerian set. Via flow splitting, force all edges entering to proceed to Create auxiliary vertices as before Solve integral circulation problem, and add solution to.

34 Local-Connectivity ATSP: one nonsingular set in L Analysis For all components not crossing S, /lb exactly as in the node-weighted case Giant component containing. Contains all edges crossing Has lower bound lb lb = ) ) Therefore solution is -light. Same approach extends to arbitrary L: enforce that every subtour crossing a set in L must intersect the backbone.

35 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

36 Motivation: reducing by contraction All sets in the family L are singletons: node-weighted ATSP Would like to reduce the problem by contracting nonsingleton sets in L

37 Motivation: reducing by contraction All sets in the family L are singletons: node-weighted ATSP Would like to reduce the problem by contracting nonsingleton sets in L

38 Motivation: reducing by contraction All sets in the family L are singletons: node-weighted ATSP Would like to reduce the problem by contracting nonsingleton sets in L

39 Irreducible instances Irreducible set S L: There exists, such that the shortest path between and inside visits almost all sets, L Irreducible instance I =, L,, : all sets in L are irreducible

40 Irreducible instances Irreducible set S L: There exists, such that the shortest path between and inside visits almost all sets, L Irreducible instance I =, L,, : all sets in L are irreducible

41 Irreducible instances Reducible set S L: For every pair,, there is a cheap path connecting them (if they are connected). Reducible sets can be contracted. Theorem: polytime -approximation for irreducible instances polytime -approximation for arbitrary instances

42 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

43 Vertebrate pairs Vertebrate pair I, I =, L,, instance : backbone = subtour that crosses every nonsingleton set in L

44 Finding a vertebrate pair in an irreducible instance I =, L,, 1. Obtain a node-weighted instance by contracting all maximal sets in L 2. Use [Svensson 15] to find a tour here, and blow it back to a subtour in the original instance I in a pessimistic way: inside each maximal S L, crosses. 3. If it crosses every set in L, then I, is a vertebrate pair 4. Otherwise, recurse by contracting all maximal sets in L not crossed by. This works because their total weight is.i

45 General ATSP Roadmap LP duality + uncrossing Laminarly weighted ATSP Graph theory: contractions Irreducible instances Node weighted algorithm + contractions Vertebrate pairs O(1)-light lcatsp algorithm in vertebrate pairs Local-connectivity ATSP [Svensson 15]

46 Summary Via all these reductions, we obtain an approximation algorithm for ATSP. Squeezing the arguments a bit more and opening up black boxes, can be probably decreased to a few hundreds. Still very far from lower bound 2 on the integrality gap of Held-Karp Open questions Improve to a constant < Thin tree conjecture is still open. Bottleneck ATSP. Better than 3/2 approximation for symmetric TSP.

47 SCALEOPT Scaling Methods for Discrete and Continuous Optimization ERC Starting Grant Openings for post docs and PhD students Thank you!

48 Simplifying assumption for the talk Assumption: all sets in the family L are strongly connected in. Not true in general, but the connected components have a nice path structure:

49 Paths traversing a set How much is the weight of connecting an incoming and an outgoing edge in a set S L?, =

50 Paths traversing a set How much is the weight of connecting an incoming and an outgoing edge in a set S L?, = :, 6

51 Paths traversing a set How much is the weight of connecting an incoming and an outgoing edge in a set S L?, = +, :, Min weight path inside

52 Paths traversing a set How much is the weight of connecting an incoming and an outgoing edge in a set S L?, = :, +, + :, =

53 Paths traversing a set How much is the weight of connecting an incoming and an outgoing edge in a set S L?, = :, +, + :, = Lemma:, =

54 Irreducible instances Reducible set S L: Max,, Irreducible instance I =, L,, : no set S L is reducible Lemma:, = Theorem: polytime -approximation for irreducible instances polytime -approximation for arbitrary instances

55 Recursive algorithm via contractions Instance I =, L,, I = I = =Held-Karp optimum : minimal reducible set in L. -approximation for I = -approximation on instance by contracting + -approximation of irreducible instance inside

56 Recursive algorithm via contractions Instance I =, L,, I = I = =Held-Karp optimum : minimal reducible set in L. I = I/: contract in I. {} = + 8 I = I = I = +

57 Recursive algorithm via contractions Inductive assumption: We have a polytime -approximation for smaller instances Apply recursively on I to obtain tour I I = I = = I = +

58 Contracting Inductive assumption: We have a polytime -approximation for smaller instances Apply recursively on I to obtain tour I = I I = I = Map back to subtour in I with = +

59 Inducing on We add a tour inside S, using the -approximation on irreducible instances. I = I : remove S, and contract to, with { } = / I is irreducible. I = 12=/2

60 Inducing on We add a tour inside S, using the -approximation on irreducible instances. I = I : remove S, and contract to, with { } = / I is irreducible. Find tour in I with weight I = 12=/2

61 Inducing on Find tour in I with weight I = Map back to in I with is a tour in I I + = I I = 12=/2