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Improved Inapproximability for TSP Michael Lampis KTH Royal Institute of Technology August 15, 2012 The Traveling Salesman Problem Input: An edge-weighted graph G ( V, E ) Objective: Find an ordering of the vertices v 1 , v 2 , . . . ,

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Improved Inapproximability for TSP

Michael Lampis KTH Royal Institute of Technology

August 15, 2012

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The Traveling Salesman Problem

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Input:

  • An edge-weighted graph G(V, E)

Objective:

  • Find an ordering of the vertices v1, v2, . . . , vn

such that d(v1, v2) + d(v2, v3) + . . . + d(vn, v1) is minimized.

  • d(vi, vj) is the shortest-path distance of vi, vj
  • n G
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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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The Traveling Salesman Problem

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TSP Approximations – Upper bounds

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  • 3

2 approximation (Christofides 1976)

For graphic (un-weighted) case

  • 3

2 −ǫ approximation (Oveis Gharan et al. FOCS

’11)

  • 1.461 approximation (M¨
  • mke and Svensson

FOCS ’11)

  • 13

9 approximation (Mucha STACS ’12)

  • 1.4 approximation (Seb¨
  • and Vygen arXiv ’12)
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SLIDE 12

TSP Approximations – Lower bounds

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  • Problem is APX-hard (Papadimitriou and Yannakakis

’93)

  • 5381

5380-inapproximable (Engebretsen STACS ’99)

  • 3813

3812-inapproximable (B¨

  • ckenhauer et al. STACS ’00)
  • 220

219-inapproximable

(Papadimitriou and Vempala STOC ’00, Combinatorica ’06)

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TSP Approximations – Lower bounds

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  • Problem is APX-hard (Papadimitriou and Yannakakis

’93)

  • 5381

5380-inapproximable (Engebretsen STACS ’99)

  • 3813

3812-inapproximable (B¨

  • ckenhauer et al. STACS ’00)
  • 220

219-inapproximable

(Papadimitriou and Vempala STOC ’00, Combinatorica ’06) This talk: Theorem There is no

185 184-approximation algorithm for TSP

, unless P=NP .

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Reduction Technique

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We reduce some inapproximable CSP (e.g. MAX-3SAT) to TSP .

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Reduction Technique

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First, design some gadgets to represent the clauses

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Reduction Technique

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Then, add some choice vertices to represent truth assignments to variables

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Reduction Technique

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For each variable, create a path through clauses where it appears positive

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Reduction Technique

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. . . and another path for its negative appearances

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Reduction Technique

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Reduction Technique

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A truth assignment dictates a general path

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Reduction Technique

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Reduction Technique

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Reduction Technique

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We must make sure that gadgets are cheaper to traverse if corresponding clause is satisfied

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Reduction Technique

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For the converse direction we must make sure that ”cheating” tours are not optimal!

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How to ensure consistency

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  • Papadimitriou and Vempala design a gadget

for Parity.

  • They eliminate variable vertices altogether.
  • Consistency is achieved by hooking up gad-

gets ”randomly”

  • In fact gadgets that share a variable are

connected according to the structure dic- tated by a special graph

  • The graph is called a ”pusher”.

Its ex- istence is proved using the probabilistic method.

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How to ensure consistency

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  • Basic idea here: consistency would be easy if each variable
  • ccurred at most c times, c a constant.
  • Cheating would only help a tour ”fix” a bounded number of

clauses.

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How to ensure consistency

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  • Basic idea here: consistency would be easy if each variable
  • ccurred at most c times, c a constant.
  • Cheating would only help a tour ”fix” a bounded number of

clauses.

  • We will rely on techniques and tools used to prove inapproximability

for bounded-occurrence CSPs.

  • Main tool: an ”amplifier graph” construction due to Berman and

Karpinski.

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How to ensure consistency

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  • Basic idea here: consistency would be easy if each variable
  • ccurred at most c times, c a constant.
  • Cheating would only help a tour ”fix” a bounded number of

clauses.

  • We will rely on techniques and tools used to prove inapproximability

for bounded-occurrence CSPs.

  • Main tool: an ”amplifier graph” construction due to Berman and

Karpinski.

  • Result: an easier hardness proof that can be broken down into

independent pieces, and also gives an improved bound.

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Overview

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We start from an instance of MAX-E3-LIN2. Given a set of linear equations (mod 2) each of size three satisfy as many as possible. Known to be 2-inapproximable (H˚ astad).

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Overview

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We use the Berman-Karpinski amplifier construction to obtain an instance where each variable appears exactly 5 times (and most equations have size 2).

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Overview

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Overview

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A simple trick reduces this to the 1in3 predicate.

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Overview

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From this instance we construct a graph.

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Overview

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From this instance we construct a graph. Rest of this talk: some more details about the construction.

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1in3-SAT

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Input: A set of clauses (l1 ∨ l2 ∨ l3), l1, l2, l3 literals. Objective: A clause is satisfied if exactly one of its literals is true. Satisfy as many clauses as possible.

  • Easy to reduce MAX-LIN2 to this problem.
  • Especially for size two equations (x + y = 1) ↔ (x ∨ y).
  • Naturally gives gadget for TSP
  • In TSP we’d like to visit each vertex at least once, but not more

than once (to save cost)

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TSP and Euler tours

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TSP and Euler tours

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TSP and Euler tours

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TSP and Euler tours

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  • A TSP tour gives an Eulerian multi-graph com-

posed with edges of G.

  • An Eulerian multi-graph composed with edges
  • f G gives a TSP tour.
  • TSP ≡ Select a multiplicity for each edge

so that the resulting multi-graph is Eulerian and total cost is minimized

  • Note: no edge is used more than twice
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Gadget – Forced Edges

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We would like to be able to dictate in our construction that a certain edge has to be used at least once.

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Gadget – Forced Edges

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If we had directed edges, this could be achieved by adding a dummy intermediate vertex

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Gadget – Forced Edges

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Here, we add many intermediate vertices and evenly distribute the weight w among them. Think of B as very large.

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Gadget – Forced Edges

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At most one of the new edges may be unused, and in that case all others are used twice.

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Gadget – Forced Edges

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In that case, adding two copies of that edge to the solution doesn’t hurt much (for B sufficiently large).

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1in3 Gadget

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Let’s design a gadget for (x ∨ y ∨ z)

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1in3 Gadget

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First, three entry/exit points

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1in3 Gadget

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Connect them . . .

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1in3 Gadget

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. . . with forced edges

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1in3 Gadget

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The gadget is a con- nected component. A good tour visits it

  • nce.
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1in3 Gadget

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. . . like this

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1in3 Gadget

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This corresponds to an unsatisfied clause

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1in3 Gadget

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This corresponds to a dishonest tour

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1in3 Gadget

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The dishonest tour pays this edge twice. How expensive must it be before cheating becomes suboptimal? Note that w = 10 suffices, since the two cheating variables appear in at most 10 clauses.

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Construction

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High-level view: con- struct an origin s and two terminal vertices for each variable.

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Construction

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Connect them with forced edges

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Construction

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Add the gadgets

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Construction

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An honest traversal for x2 looks like this

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Construction

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A dishonest traversal looks like this. . .

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Construction

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. . . but there must be cheating in two places There are as many doubly-used forced edges as affected variables → w ≤ 5

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Construction

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. . . but there must be cheating in two places There are as many doubly-used forced edges as affected variables → w ≤ 5 In fact, no need to write off affected clauses. Use random assignment for cheated variables and some of them will be satisfied

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Under the carpet

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  • Many details missing
  • Dishonest variables are set randomly but

not independently to ensure that some clauses are satisfied with probability 1.

  • The structure of the instance (from BK am-

plifier) must be taken into account to calcu- late the final constant.

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Under the carpet

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  • Many details missing
  • Dishonest variables are set randomly but

not independently to ensure that some clauses are satisfied with probability 1.

  • The structure of the instance (from BK am-

plifier) must be taken into account to calcu- late the final constant. Theorem: There is no 185

184 approximation algorithm for TSP

, unless P=NP .

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Conclusions – Open problems

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  • A simpler reduction for TSP and a better inapproximability threshold
  • But, constant still very low!

Future work

  • Better amplifier constructions?
  • Get rid of 1in3 SAT?
  • ATSP
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The end

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Questions?