Improved Inapproximability for TSP - LAMSADEmlampis/slides/APPROX12.pdf · Improved...
Transcript of Improved Inapproximability for TSP - LAMSADEmlampis/slides/APPROX12.pdf · Improved...
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Improved Inapproximability for TSP
Michael LampisKTH Royal Institute of Technology
August 15, 2012
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The Traveling Salesman Problem
Improved Inapproximability for TSP – APPROX 2012 2 / 16
Input:
• An edge-weighted graph G(V,E)
Objective:
• Find an ordering of the vertices v1, v2, . . . , vnsuch that d(v1, v2)+ d(v2, v3)+ . . .+ d(vn, v1) isminimized.
• d(vi, vj) is the shortest-path distance of vi, vjon 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
Improved Inapproximability for TSP – APPROX 2012 2 / 16
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The Traveling Salesman Problem
Improved Inapproximability for TSP – APPROX 2012 2 / 16
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The Traveling Salesman Problem
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The Traveling Salesman Problem
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The Traveling Salesman Problem
Improved Inapproximability for TSP – APPROX 2012 2 / 16
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TSP Approximations – Upper bounds
Improved Inapproximability for TSP – APPROX 2012 3 / 16
•3
2approximation (Christofides 1976)
For graphic (un-weighted) case
•3
2−ǫ approximation (Oveis Gharan et al. FOCS
’11)
• 1.461 approximation (Momke and SvenssonFOCS ’11)
•13
9approximation (Mucha STACS ’12)
• 1.4 approximation (Sebo and Vygen arXiv ’12)
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TSP Approximations – Lower bounds
Improved Inapproximability for TSP – APPROX 2012 4 / 16
• Problem is APX-hard (Papadimitriou and Yannakakis’93)
•5381
5380-inapproximable (Engebretsen STACS ’99)
•3813
3812-inapproximable (Bockenhauer et al. STACS ’00)
•220
219-inapproximable (Papadimitriou and Vempala
STOC ’00, Combinatorica ’06)
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TSP Approximations – Lower bounds
Improved Inapproximability for TSP – APPROX 2012 4 / 16
• Problem is APX-hard (Papadimitriou and Yannakakis’93)
•5381
5380-inapproximable (Engebretsen STACS ’99)
•3813
3812-inapproximable (Bockenhauer et al. STACS ’00)
•220
219-inapproximable (Papadimitriou and Vempala
STOC ’00, Combinatorica ’06)
This talk:
TheoremThere 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
Improved Inapproximability for TSP – APPROX 2012 5 / 16
First, design some gadgets to represent the clauses
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
Then, add some choice vertices to represent truth assignments tovariables
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
For each variable, create a path through clauses where it appears positive
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
. . . and another path for its negative appearances
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
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
Improved Inapproximability for TSP – APPROX 2012 5 / 16
We must make sure that gadgets are cheaper to traverse if correspondingclause is satisfied
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Reduction Technique
Improved Inapproximability for TSP – APPROX 2012 5 / 16
For the converse direction we must make sure that ”cheating” tours arenot optimal!
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How to ensure consistency
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• Papadimitriou and Vempala design a gadgetfor Parity.
• They eliminate variable vertices altogether.
• Consistency is achieved by hooking up gad-gets ”randomly”
• In fact gadgets that share a variable areconnected according to the structure dic-tated by a special graph
• The graph is called a ”pusher”. Its ex-istence is proved using the probabilisticmethod.
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How to ensure consistency
Improved Inapproximability for TSP – APPROX 2012 7 / 16
• Basic idea here: consistency would be easy if each variableoccurred at most c times, c a constant.
• Cheating would only help a tour ”fix” a bounded number ofclauses.
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How to ensure consistency
Improved Inapproximability for TSP – APPROX 2012 7 / 16
• Basic idea here: consistency would be easy if each variableoccurred at most c times, c a constant.
• Cheating would only help a tour ”fix” a bounded number ofclauses.
• We will rely on techniques and tools used to prove inapproximabilityfor bounded-occurrence CSPs.
• Main tool: an ”amplifier graph” construction due to Berman andKarpinski.
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How to ensure consistency
Improved Inapproximability for TSP – APPROX 2012 7 / 16
• Basic idea here: consistency would be easy if each variableoccurred at most c times, c a constant.
• Cheating would only help a tour ”fix” a bounded number ofclauses.
• We will rely on techniques and tools used to prove inapproximabilityfor bounded-occurrence CSPs.
• Main tool: an ”amplifier graph” construction due to Berman andKarpinski.
• Result: an easier hardness proof that can be broken down intoindependent pieces, and also gives an improved bound.
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Overview
Improved Inapproximability for TSP – APPROX 2012 8 / 16
We start from an instance of MAX-E3-LIN2. Given a set of linearequations (mod 2) each of size three satisfy as many as possible. Knownto be 2-inapproximable (Hastad).
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Overview
Improved Inapproximability for TSP – APPROX 2012 8 / 16
We use the Berman-Karpinski amplifier construction to obtain an instancewhere each variable appears exactly 5 times (and most equations havesize 2).
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Overview
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Overview
Improved Inapproximability for TSP – APPROX 2012 8 / 16
A simple trick reduces this to the 1in3 predicate.
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Overview
Improved Inapproximability for TSP – APPROX 2012 8 / 16
From this instance we construct a graph.
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Overview
Improved Inapproximability for TSP – APPROX 2012 8 / 16
From this instance we construct a graph.
Rest of this talk: some more details about the construction.
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1in3-SAT
Improved Inapproximability for TSP – APPROX 2012 9 / 16
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 manyclauses 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 morethan 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
Improved Inapproximability for TSP – APPROX 2012 10 / 16
• A TSP tour gives an Eulerian multi-graph com-posed with edges of G.
• An Eulerian multi-graph composed with edgesof G gives a TSP tour.
• TSP ≡ Select a multiplicity for each edgeso that the resulting multi-graph is Eulerianand total cost is minimized
• Note : no edge is used more than twice
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Gadget – Forced Edges
Improved Inapproximability for TSP – APPROX 2012 11 / 16
We would like to be able to dictate in our construction that a certain edgehas 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 dummyintermediate vertex
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Gadget – Forced Edges
Improved Inapproximability for TSP – APPROX 2012 11 / 16
Here, we add many intermediate vertices and evenly distribute the weightw among them. Think of B as very large.
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Gadget – Forced Edges
Improved Inapproximability for TSP – APPROX 2012 11 / 16
At most one of the new edges may be unused, and in that case all othersare used twice.
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Gadget – Forced Edges
Improved Inapproximability for TSP – APPROX 2012 11 / 16
In that case, adding two copies of that edge to the solution doesn’t hurtmuch (for B sufficiently large).
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
Let’s design a gadgetfor (x ∨ y ∨ z)
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
First, three entry/exitpoints
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
Connect them . . .
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
. . . with forced edges
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
The gadget is a con-nected component.A good tour visits itonce.
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
. . . like this
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
This corresponds toan unsatisfied clause
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
This corresponds to adishonest tour
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1in3 Gadget
Improved Inapproximability for TSP – APPROX 2012 12 / 16
The dishonest tourpays this edge twice.How expensive mustit be before cheatingbecomes suboptimal?
Note that w = 10 suffices, since the two cheating variables appear in atmost 10 clauses.
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
High-level view: con-struct an origin s andtwo terminal verticesfor each variable.
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
Connect them withforced edges
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
Add the gadgets
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
An honest traversal forx2 looks like this
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
A dishonest traversallooks like this. . .
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
. . . but there must becheating in two places
There are as many doubly-used forced edges as affected variables→ w ≤ 5
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Construction
Improved Inapproximability for TSP – APPROX 2012 13 / 16
. . . but there must becheating 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 forcheated variables and some of them will be satisfied
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Under the carpet
Improved Inapproximability for TSP – APPROX 2012 14 / 16
• Many details missing
• Dishonest variables are set randomly butnot independently to ensure that someclauses 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
Improved Inapproximability for TSP – APPROX 2012 14 / 16
• Many details missing
• Dishonest variables are set randomly butnot independently to ensure that someclauses 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
184approximation algorithm for TSP, unless P=NP.
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Conclusions – Open problems
Improved Inapproximability for TSP – APPROX 2012 15 / 16
• 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
Improved Inapproximability for TSP – APPROX 2012 16 / 16
Questions?