New Inapproximability Bounds for TSP Marek Karpinski, Michael Lampis - - PowerPoint PPT Presentation

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New Inapproximability Bounds for TSP Marek Karpinski, Michael Lampis - - PowerPoint PPT Presentation

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New Inapproximability Bounds for TSP Marek Karpinski, Michael Lampis and Richard Schmied ISAAC 2013 The Traveling Salesman Problem Input: An edge-weighted graph G ( V, E ) Objective: Find an ordering of the vertices v 1 , v 2 , . . . , v n

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SLIDE 1

New Inapproximability Bounds for TSP

Marek Karpinski, Michael Lampis and Richard Schmied

ISAAC 2013

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SLIDE 2

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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 on

G

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SLIDE 3

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 4

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 5

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 6

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 7

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 8

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 9

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 10

The Traveling Salesman Problem

New Inapproximability Bounds for TSP 2 / 20

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SLIDE 11

TSP Approximations – Upper bounds

New Inapproximability Bounds for TSP 3 / 20

  • 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)
  • For ATSP the best ratio is O(log n/ log log n)

(Asadpour et al. SODA ’10)

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SLIDE 12

TSP Approximations – Lower bounds

New Inapproximability Bounds for TSP 4 / 20

  • Problem is APX-hard (Papadimitriou and Yannakakis

’93)

  • TSP

5381 5380-inapproximable,

ATSP

2805 2804

(Engebretsen STACS ’99)

  • TSP 3813

3812-inapproximable (B¨

  • ckenhauer et al.

STACS ’00)

  • TSP 220

219-inapproximable, ATSP 117 116 (Papadimitriou and

Vempala STOC ’00, Combinatorica ’06)

  • TSP 185

184-inapproximable (L. APPROX ’12)

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SLIDE 13

TSP Approximations – Lower bounds

New Inapproximability Bounds for TSP 4 / 20

  • Problem is APX-hard (Papadimitriou and Yannakakis

’93)

  • TSP

5381 5380-inapproximable,

ATSP

2805 2804

(Engebretsen STACS ’99)

  • TSP 3813

3812-inapproximable (B¨

  • ckenhauer et al.

STACS ’00)

  • TSP 220

219-inapproximable, ATSP 117 116 (Papadimitriou and

Vempala STOC ’00, Combinatorica ’06)

  • TSP 185

184-inapproximable (L. APPROX ’12)

This talk: Theorem It is NP-hard to approximate TSP better than 123

122 and ATSP

better than 75

74.

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SLIDE 14

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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

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SLIDE 15

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

First, design some gadgets to represent the clauses

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SLIDE 16

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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

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SLIDE 17

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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

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SLIDE 18

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

. . . and another path for its negative appearances

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SLIDE 19

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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SLIDE 20

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

A truth assignment dictates a general path

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SLIDE 21

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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SLIDE 22

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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SLIDE 23

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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

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SLIDE 24

Reduction Technique

New Inapproximability Bounds for TSP 5 / 20

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

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SLIDE 25

How to ensure consistency

New Inapproximability Bounds for TSP 6 / 20

  • Basic idea here: consistency would be easy if each variable occurred

at most c times, c a constant.

  • Cheating would only help a tour ”fix” a bounded number of clauses.
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SLIDE 26

How to ensure consistency

New Inapproximability Bounds for TSP 6 / 20

  • Basic idea here: consistency would be easy if each variable occurred

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: “amplifier graph” constructions due to Berman and

Karpinski.

  • We introduce a new bi-wheel amplifier.
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SLIDE 27

How to ensure consistency

New Inapproximability Bounds for TSP 6 / 20

  • Basic idea here: consistency would be easy if each variable occurred

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: “amplifier graph” constructions due to Berman and

Karpinski.

  • We introduce a new bi-wheel amplifier.
  • Result: modular proof, improved bounds
  • Potential for further improvements: parts of the reduction have no
  • verhead!
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SLIDE 28

Overview

New Inapproximability Bounds for TSP 7 / 20

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. Problem known to be 2-inapproximable (H˚ astad ’01)

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SLIDE 29

Overview

New Inapproximability Bounds for TSP 7 / 20

We use a new version of the Berman-Karpinski wheel amplifier: the bi-wheel. We obtain an instance where each variable appears exactly 3 times (and most equations have size 2).

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SLIDE 30

Overview

New Inapproximability Bounds for TSP 7 / 20

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SLIDE 31

Overview

New Inapproximability Bounds for TSP 7 / 20

From this instance we construct a TSP/ATSP graph instance.

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SLIDE 32

Amplifiers and Bounded Occurrences

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SLIDE 33

Amplifiers

New Inapproximability Bounds for TSP 9 / 20

What is an amplifier?

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SLIDE 34

Amplifiers

New Inapproximability Bounds for TSP 9 / 20

What is an amplifier?

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SLIDE 35

Amplifiers

New Inapproximability Bounds for TSP 9 / 20

An amplifier is a graph with edge expansion 1 for a subset of its vertices.

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SLIDE 36

Amplifiers

New Inapproximability Bounds for TSP 9 / 20

An amplifier is a graph with edge expansion 1 for a subset of its vertices. 3-regular wheel amplifier [Berman Karpinski 01]

  • Start with a cycle on 7n vertices.
  • Every seventh vertex is a contact vertex.

Other vertices are checkers.

  • Take

a random perfect matching

  • f

checkers.

  • Crucial Property: whp any partition cuts

more edges than the number of contact vertices on the smaller set.

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SLIDE 37

How to use amplifiers

New Inapproximability Bounds for TSP 10 / 20

  • Input: MAX-E3-LIN2, variables appear B times.
  • For each variable x construct an amplifier.
  • For each vertex construct a variable xi, yi
  • For each edge of the amplifier make an equality constraint

(yi + yj = 0).

  • Use the xi’s in the original constraints.
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SLIDE 38

How to use amplifiers

New Inapproximability Bounds for TSP 10 / 20

  • Input: MAX-E3-LIN2, variables appear B times.
  • For each variable x construct an amplifier.
  • For each vertex construct a variable xi, yi
  • For each edge of the amplifier make an equality constraint

(yi + yj = 0).

  • Use the xi’s in the original constraints.
  • Inconsistent assignments → partition of vertices
  • But cut edges → violated equalities
  • Large cut → Flipping the minority part is always good
  • → Consistent assignment is optimal
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SLIDE 39

How to use amplifiers

New Inapproximability Bounds for TSP 10 / 20

  • Input: MAX-E3-LIN2, variables appear B times.
  • For each variable x construct an amplifier.
  • For each vertex construct a variable xi, yi
  • For each edge of the amplifier make an equality constraint

(yi + yj = 0).

  • Use the xi’s in the original constraints.
  • Inconsistent assignments → partition of vertices
  • But cut edges → violated equalities
  • Large cut → Flipping the minority part is always good
  • → Consistent assignment is optimal
  • Problem: New equations are pure overhead! (always satisfiable)
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SLIDE 40

The reduction

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SLIDE 41

TSP and Euler tours

New Inapproximability Bounds for TSP 12 / 20

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SLIDE 42

TSP and Euler tours

New Inapproximability Bounds for TSP 12 / 20

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SLIDE 43

TSP and Euler tours

New Inapproximability Bounds for TSP 12 / 20

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SLIDE 44

TSP and Euler tours

New Inapproximability Bounds for TSP 12 / 20

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

posed with edges of G.

  • An Eulerian multi-graph composed with edges of

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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SLIDE 45

Gadget – Forced Edges

New Inapproximability Bounds for TSP 13 / 20

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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SLIDE 46

Gadget – Forced Edges

New Inapproximability Bounds for TSP 13 / 20

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

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SLIDE 47

Gadget – Forced Edges

New Inapproximability Bounds for TSP 13 / 20

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

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SLIDE 48

Gadget – Forced Edges

New Inapproximability Bounds for TSP 13 / 20

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

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SLIDE 49

Gadget – Forced Edges

New Inapproximability Bounds for TSP 13 / 20

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

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SLIDE 50

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

We can encode x + y = 1 with two parallel forced edges

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SLIDE 51

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

These are a connected component in any tour

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SLIDE 52

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

This is a good and honest assignment

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SLIDE 53

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

This is a bad and honest assignment

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SLIDE 54

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

This is a PROBLEM!

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SLIDE 55

Gadget for Inequality

New Inapproximability Bounds for TSP 14 / 20

Good news: Making this edge expensive fixes the problem. Bad news: making this edge expensive adds overhead to the construction. What is the smallest possible W?

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SLIDE 56

The problem with inequality

New Inapproximability Bounds for TSP 15 / 20

  • We want to use an inequality gadget to represent the matching edges
  • f the amplifier.
  • Normally, amplifier edges become equalities.
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SLIDE 57

The problem with inequality

New Inapproximability Bounds for TSP 15 / 20

  • We want to use an inequality gadget to represent the matching edges
  • f the amplifier.
  • Normally, amplifier edges become equalities.

We want cycle edges to remain equalities.

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SLIDE 58

The problem with inequality

New Inapproximability Bounds for TSP 15 / 20

  • We want to use an inequality gadget to represent the matching edges
  • f the amplifier.
  • Normally, amplifier edges become equalities.

Solution: the bi-wheel!

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SLIDE 59

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality

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SLIDE 60

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality Consider two vertices consecutive in one cycle (x, z)

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SLIDE 61

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality Suppose that their matching gadgets are honest

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SLIDE 62

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality Then if one is traversed as True. . .

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SLIDE 63

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality . . . the other is also!

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SLIDE 64

Free equations!

New Inapproximability Bounds for TSP 16 / 20

Main idea: honesty gives equality . . . the other is also!

  • In other words, we extract an assignment for x by setting it to 1 iff both

its incident non-forced edges are used.

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SLIDE 65

Some handwaving

New Inapproximability Bounds for TSP 17 / 20

What is the cost of the forced edges?

  • In case of dishonest traversal we must make the tour pay for all

unsatisfied equations.

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SLIDE 66

Some handwaving

New Inapproximability Bounds for TSP 17 / 20

What is the cost of the forced edges?

  • In case of dishonest traversal we must make the tour pay for all

unsatisfied equations.

  • There are 5 affected equation.
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SLIDE 67

Some handwaving

New Inapproximability Bounds for TSP 17 / 20

What is the cost of the forced edges?

  • In case of dishonest traversal we must make the tour pay for all

unsatisfied equations.

  • There are 5 affected equation.
  • We can always satisfy 3.
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SLIDE 68

Some handwaving

New Inapproximability Bounds for TSP 17 / 20

What is the cost of the forced edges?

  • In case of dishonest traversal we must make the tour pay for all

unsatisfied equations.

  • There are 5 affected equation.
  • We can always satisfy 3.
  • Hence, cost of forced edges is 2.
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SLIDE 69

More handwaving

New Inapproximability Bounds for TSP 18 / 20

  • For size-three equations we come up with some

gadget (not shown).

  • Some work needs to be done to ensure connec-

tivity.

  • Similar ideas can be used for ATSP

.

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SLIDE 70

More handwaving

New Inapproximability Bounds for TSP 18 / 20

  • For size-three equations we come up with some

gadget (not shown).

  • Some work needs to be done to ensure connec-

tivity.

  • Similar ideas can be used for ATSP

. Theorem: There is no 123

122 − ǫ approximation algorithm for TSP

, unless P=NP . There is no 75

74 − ǫ approximation algorithm for ATSP

, unless P=NP .

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SLIDE 71

Conclusions – Open problems

New Inapproximability Bounds for TSP 19 / 20

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

Future work

  • Applications to other problems (Steiner Tree, Max 3-DM)
  • Better amplifier constructions?
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SLIDE 72

Conclusions – Open problems

New Inapproximability Bounds for TSP 19 / 20

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

Future work

  • Applications to other problems (Steiner Tree, Max 3-DM)
  • Better amplifier constructions?
  • . . . Reasonable inapproximability for TSP?
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SLIDE 73

The end

New Inapproximability Bounds for TSP 20 / 20

Questions?