Michael Lampis LAMSADE Universite Paris...
Transcript of Michael Lampis LAMSADE Universite Paris...
Hardness of Approximation for the TSP
Michael LampisLAMSADE
Universite Paris Dauphine
Sep 2, 2015
Overview
Parameterized Approximation Schemes 2 / 39
• Hardness of Approximation
• What is it?
• How to do it?
• (Easy) Examples
• The PCP Theorem
• What is it?
• How to use it?
• The Traveling Salesman Problem
• Approximation algorithms
• Strategy for Proving Hardness
• Other tools
• Expander Graphs
• Bounded-Occurrence CSPs
• A full reduction for the TSP
Hardness of Approximation
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
• Reminder:
• We have an (NP-hard) optimization problem
• We want to design an algorithm that gives “good enough” solution
• We want a guarantee of this
For all instances I we haveSOL(I)OPT(I) < r
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
• How close to 1 can we get the approximation ratio r?
• Typical situation:
• An initial algorithm gives some (bad) r.
• Then someone comes up with an improvement
• Repeat. . .
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
• How close to 1 can we get the approximation ratio r?
• Typical situation:
• An initial algorithm gives some (bad) r.
• Then someone comes up with an improvement
• Repeat. . .
• Until we are stuck! Now what?
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
• How close to 1 can we get the approximation ratio r?
• Typical situation:
• An initial algorithm gives some (bad) r.
• Then someone comes up with an improvement
• Repeat. . .
• Until we are stuck! Now what?
Hardness of Approximation
Parameterized Approximation Schemes 4 / 39
• Day Summary: Approximation Algorithms with a performance
guarantee
• How close to 1 can we get the approximation ratio r?
• Typical situation:
• An initial algorithm gives some (bad) r.
• Then someone comes up with an improvement
• Repeat. . .
• Until we are stuck! Now what?
• Goal of the theory of Hardness of Approximation:
Prove the we are not incompetent!
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
The two are related!
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
• The main tool will be algorithmic: Reductions
• Reminder: Basic tool of NP-hardness
• A is NP-hard. Reduce A to B. → B is NP-hard.
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
• The main tool will be algorithmic: Reductions
• Reminder: Basic tool of NP-hardness
• A is NP-hard. Reduce A to B. → B is NP-hard.
• Approximation version: Approximation Preserving Reductions
• Idea: A has no good approximation algorithm.
• We reduce A to B
• Conclusion: B has no good approximation algorithm.
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 5 / 39
• Approximation Algorithms vs. Hardness
• Poly-time algorithms vs. NP-completeness
• Algorithms vs. Complexity
• The main tool will be algorithmic: Reductions
• Reminder: Basic tool of NP-hardness
• A is NP-hard. Reduce A to B. → B is NP-hard.
• Approximation version: Approximation Preserving Reductions
• Idea: A has no good approximation algorithm.
• We reduce A to B
• Conclusion: B has no good approximation algorithm.
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 6 / 39
There are a couple of serious problems with this approach.
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 6 / 39
There are a couple of serious problems with this approach.
• What is the “first” hard to approximate problem?
• Recall: Cook’s theorem gives us a “first” NP-hard problem. Then
we reduce from that.
• Here, we don’t have a problem to begin from. . .
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 6 / 39
There are a couple of serious problems with this approach.
• What is the “first” hard to approximate problem?
• Recall: Cook’s theorem gives us a “first” NP-hard problem. Then
we reduce from that.
• Here, we don’t have a problem to begin from. . .
• How can we prove that a problem does not have a good approximation
algorithm?
• This implies that it does not have a poly-time exact algorithm.
• P6=NP !!
Hardness of Approximation: Can we do it?
Parameterized Approximation Schemes 6 / 39
There are a couple of serious problems with this approach.
• What is the “first” hard to approximate problem?
• Recall: Cook’s theorem gives us a “first” NP-hard problem. Then
we reduce from that.
• Here, we don’t have a problem to begin from. . .
• How can we prove that a problem does not have a good approximation
algorithm?
• This implies that it does not have a poly-time exact algorithm.
• P6=NP !!
Hardness of Approximation: How to do it
Parameterized Approximation Schemes 7 / 39
• We cannot avoid the second problem (without resolving P=NP)
• We will prove all our hardness results assuming P6=NP
Hardness of Approximation: How to do it
Parameterized Approximation Schemes 7 / 39
• We cannot avoid the second problem (without resolving P=NP)
• We will prove all our hardness results assuming P6=NP
• We can solve the first problem using gap-introducting reductions.
Hardness of Approximation: How to do it
Parameterized Approximation Schemes 7 / 39
• We cannot avoid the second problem (without resolving P=NP)
• We will prove all our hardness results assuming P6=NP
• We can solve the first problem using gap-introducting reductions.
• A gap-introducting reduction from SAT to a problem A has the
following properties
• Given a SAT formula φ it produces in polynomial time an instance I
of A
• (Completeness): If φ is satisfiable then OPT(I) > c• (Soundness): If φ is not satisfiable then OPT(I) < s
Hardness of Approximation: How to do it
Parameterized Approximation Schemes 7 / 39
• We cannot avoid the second problem (without resolving P=NP)
• We will prove all our hardness results assuming P6=NP
• We can solve the first problem using gap-introducting reductions.
• A gap-introducting reduction from SAT to a problem A has the
following properties
• Given a SAT formula φ it produces in polynomial time an instance I
of A
• (Completeness): If φ is satisfiable then OPT(I) > c• (Soundness): If φ is not satisfiable then OPT(I) < s
• This establishes that no algorithm can achieve approximation ratio
better than c/s
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
• This proves that P=NP!
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
• Recall: Deciding if a graph can be colored with 3 colors in NP-hard.
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
• Recall: Deciding if a graph can be colored with 3 colors in NP-hard.
• Translation: there is reduction which given a SAT formula φ produces
either a graph that can be 3-colored, or one that needs more colors.
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
• Recall: Deciding if a graph can be colored with 3 colors in NP-hard.
• Translation: there is reduction which given a SAT formula φ produces
either a graph that can be 3-colored, or one that needs more colors.
• Run Bob’s algorithm on this graph.
• If the graph can be 3-colored, the algorithm is guaranteed to
produce a solution with at most 3× 1.1 = 3.3 colors (!!)
• Otherwise, the algorithm will return a solution with at least 4 colors.
• From the number of colors of the solution we can deduce if the formula
was satisfiable!
Gap introduction: An easy example
Parameterized Approximation Schemes 8 / 39
Recall the NP-hard Graph Coloring Problem
• Given a graph G(V,E) we want to find a coloring of the vertices such
that any two neighbors have different colors.
• Objective: Minimize the number of colors used.
• Suppose my friend Bob claims to have designed an algorithm for
Graph Coloring with approximation ratio 1.1.
• Recall: Deciding if a graph can be colored with 3 colors in NP-hard.
• Translation: there is reduction which given a SAT formula φ produces
either a graph that can be 3-colored, or one that needs more colors.
• Run Bob’s algorithm on this graph.
• If the graph can be 3-colored, the algorithm is guaranteed to
produce a solution with at most 3× 1.1 = 3.3 colors (!!)
• Otherwise, the algorithm will return a solution with at least 4 colors.
• From the number of colors of the solution we can deduce if the formula
was satisfiable!
TSP
Parameterized Approximation Schemes 9 / 39
Traveling Salesman Problem:
• Given: Edge-weighted complete graph, weights follow triangle
inequality
• Output: A tour that visits each vertex exactly once
• Objective: Minimize total cost
TSP
Parameterized Approximation Schemes 9 / 39
Traveling Salesman Problem:
• What if we don’t have the triangle inequality?
TSP
Parameterized Approximation Schemes 9 / 39
Traveling Salesman Problem:
• What if we don’t have the triangle inequality?
Reduction from Hamiltonian Cycle
• Ham. Cycle: Given a graph, is there a cycle that visits each vertex
exactly once?
• Given graph G(V,E) construct an instance of TSP
• Each edge ∈ E has weight 1
• Each non-edge has weight w
• YES: There is a TSP tour with weight |V |• NO: Any TSP tour has weight ≥ |V | − 1 + w
• → No algorithm can have ratio better than|V |−1+w
|V |
• We can now set w to something huge! (e.g. w = 2n)
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
We are mainly interested in the case of the problem where each job can
run on two machines.
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
We are mainly interested in the case of the problem where each job can
run on two machines.
Example:
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
We are mainly interested in the case of the problem where each job can
run on two machines.
Example:
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
We are mainly interested in the case of the problem where each job can
run on two machines.
Example:
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 10 / 39
Graph Balancing / Scheduling with restricted Assignment
• Given: n machines and m jobs. Each job has a duration and a set of
machines it is allowed to run on.
• Output: An assignment of jobs to machines.
• Objective: Minimize makespan (time needed for last machine to finish
all its jobs).
We are mainly interested in the case of the problem where each job can
run on two machines.
Example:
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 11 / 39
• Target Theorem: There is no approximation algorithm for Graph
Balancing with ratio better than 3/2.
Plan:
• Gap-Introducing reduction from 3-SAT
• Satisfiable formula → maximum load 2
• Unsatisfiable formula → maximum load ≥ 3.
Gap Introduction: A non-trivial example
Parameterized Approximation Schemes 11 / 39
• Target Theorem: There is no approximation algorithm for Graph
Balancing with ratio better than 3/2.
Plan:
• Gap-Introducing reduction from 3-SAT
• Satisfiable formula → maximum load 2
• Unsatisfiable formula → maximum load ≥ 3.
Thm: 3-OCC-3-SAT is NP-hard
• This is the version of 3-SAT where each variable appears at most 3
times and each literal at most twice.
• Proof:
• Replace each appearance of variable x with a fresh variable
x1, x2, . . . , xn• Add the clauses (x1 → x2) ∧ (x2 → x3) ∧ . . . ∧ (xn → x1)
Example continued
Parameterized Approximation Schemes 12 / 39
Reduction: 3-OCC-3-SAT → Graph Balancing
Example continued
Parameterized Approximation Schemes 12 / 39
Reduction: 3-OCC-3-SAT → Graph Balancing
For each variable create an edge of weight 2 and two vertices
Example continued
Parameterized Approximation Schemes 12 / 39
Reduction: 3-OCC-3-SAT → Graph Balancing
For each create a vertex and connect it with its literals, here
c1 = (x1 ∨ ¬x2 ∨ x3)
Example continued
Parameterized Approximation Schemes 12 / 39
Reduction: 3-OCC-3-SAT → Graph Balancing
A truth assignment orients heavy edges towards the false literal
Example continued
Parameterized Approximation Schemes 12 / 39
Reduction: 3-OCC-3-SAT → Graph Balancing
In order to achieve load=2 we must find a true literal in each clause
Recap
Parameterized Approximation Schemes 13 / 39
• Gap-introducing reductions
• Reduce an NP-hard problem to instances of our problem which are
very different in the YES/NO cases.
• This implies hardness of approximation for our problem
• Next step, reduct to other problems. . .
• Unfortunately, direct gap-introducing reductions are very rare.
• Usually work for problems of the form Max-Min
• Does not work for Min-Avg, Min-Sum, . . .
• How to prove that such problems are hard?
The PCP Theorem
Min-Max or Min-Sum?
Parameterized Approximation Schemes 15 / 39
• Consider the MAX-3-SAT problem
• Given: 3-SAT formula
• Objective: Find assignment that satisfies most clauses
• We can try the same trick to prove it’s hard to approximate
• YES: OPT(I) = m• NO: OPT(I) ≤ m− 1• No approximation better than m−1
m
Min-Max or Min-Sum?
Parameterized Approximation Schemes 15 / 39
• Consider the MAX-3-SAT problem
• Given: 3-SAT formula
• Objective: Find assignment that satisfies most clauses
• We can try the same trick to prove it’s hard to approximate
• YES: OPT(I) = m• NO: OPT(I) ≤ m− 1• No approximation better than m−1
m
• Unfortunately, this ratio is basically 1. . .
• Generally, direct gap-introducing reductions are hard to do for
problems where a bad instance needs to have “many” problems.
• To prove that such problems are hard we generally need the famous
PCP theorem.
The PCP theorem: approximation view
Parameterized Approximation Schemes 16 / 39
Theorem: There is a polynomial-time reduction from 3-SAT to 3-SAT with
the following properties
• If the original formula φ is satisfiable then the new formula φ′ is also
• If the original formula is not satisfiable, then any assignment satisfies
at most an r fraction of φ′, where r < 1 a constant indepent of φ.
The PCP theorem: approximation view
Parameterized Approximation Schemes 16 / 39
Theorem: There is a polynomial-time reduction from 3-SAT to 3-SAT with
the following properties
• If the original formula φ is satisfiable then the new formula φ′ is also
• If the original formula is not satisfiable, then any assignment satisfies
at most an r fraction of φ′, where r < 1 a constant indepent of φ.
• Translation: The PCP theorem gives a gap-introducing reduction to
MAX-3-SAT.
• This produces a “starting problem” from which we can do reductions to
show that other problems are hard.
• In this way, the PCP theorem is to approximation hardness what
Cook’s theorem is to NP-completeness.
The PCP theorem: approximation view
Parameterized Approximation Schemes 16 / 39
Theorem: There is a polynomial-time reduction from 3-SAT to 3-SAT with
the following properties
• If the original formula φ is satisfiable then the new formula φ′ is also
• If the original formula is not satisfiable, then any assignment satisfies
at most an r fraction of φ′, where r < 1 a constant indepent of φ.
• Translation: The PCP theorem gives a gap-introducing reduction to
MAX-3-SAT.
• This produces a “starting problem” from which we can do reductions to
show that other problems are hard.
• In this way, the PCP theorem is to approximation hardness what
Cook’s theorem is to NP-completeness.
• But it is also much more. . .
The PCP theorem: proof-checking view
Parameterized Approximation Schemes 17 / 39
• Problems in NP: ∃ a short proof for YES instances
• E.g. SAT, 3-Coloring
• Mathematical Theorems themselves !?!?
The PCP theorem: proof-checking view
Parameterized Approximation Schemes 17 / 39
• Problems in NP: ∃ a short proof for YES instances
• E.g. SAT, 3-Coloring
• Mathematical Theorems themselves !?!?
• Given such a proof/certificate, how can we verify it’s correct?
• We have to read it, of course.
• All of it ???
The PCP theorem: proof-checking view
Parameterized Approximation Schemes 17 / 39
• Problems in NP: ∃ a short proof for YES instances
• E.g. SAT, 3-Coloring
• Mathematical Theorems themselves !?!?
• Given such a proof/certificate, how can we verify it’s correct?
• We have to read it, of course.
• All of it ???
• PCP theorem (informal statement):
• There is a way to write the proof so that its size stays roughly the
same but it can be verified with high probability by reading a
constant number of bits.
The PCP theorem: proof-checking view
Parameterized Approximation Schemes 17 / 39
• Problems in NP: ∃ a short proof for YES instances
• E.g. SAT, 3-Coloring
• Mathematical Theorems themselves !?!?
• Given such a proof/certificate, how can we verify it’s correct?
• We have to read it, of course.
• All of it ???
• PCP theorem (informal statement):
• There is a way to write the proof so that its size stays roughly the
same but it can be verified with high probability by reading a
constant number of bits.
• This is unbelievable! (and it made the NY Times)
The PCP theorem: implications
Parameterized Approximation Schemes 18 / 39
• Equivalence of two forms: a 3-SAT formula for which it is easy to verify
a certificate (assignment) is a formula for which every assignment
makes many clauses false.
• Using the PCP theorem we have some (tiny) constant for the hardness
of MAX-3-SAT
Is this all?
The PCP theorem: implications
Parameterized Approximation Schemes 18 / 39
• Equivalence of two forms: a 3-SAT formula for which it is easy to verify
a certificate (assignment) is a formula for which every assignment
makes many clauses false.
• Using the PCP theorem we have some (tiny) constant for the hardness
of MAX-3-SAT
Is this all?
[Hastad 2001]
• There is no better than 7/8-approximation for MAX-E3-SAT
• There is no better than 1/2-approximation for MAX-E3-LIN2
• In MAX-E3-LIN2 we are given equations of the form x⊕ y ⊕ z = 0and want to satisfy as many as possible.
• MAX-E3-LIN2 is a common starting point for inapproximability
reductions.
These results match the performance of the trivial algorithm!
The Traveling Salesman Problem
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
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) is
minimized.
• d(vi, vj) is the shortest-path distance of vi, vj on
G
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
The Traveling Salesman Problem
Parameterized Approximation Schemes 20 / 39
TSP Approximations – Upper bounds
Parameterized Approximation Schemes 21 / 39
• 32 approximation (Christofides 1976)
For graphic (un-weighted) case
• 32 − ǫ approximation (Oveis Gharan et al. FOCS
’11)
• 1.461 approximation (Momke and Svensson
FOCS ’11)
• 139 approximation (Mucha STACS ’12)
• 1.4 approximation (Sebo and Vygen arXiv ’12)
• For ATSP the best ratio is O(logn/ log logn)(Asadpour et al. SODA ’10)
TSP Approximations – Lower bounds
Parameterized Approximation Schemes 22 / 39
• Problem is APX-hard (Papadimitriou and Yannakakis
’93)
• 53815380 -inapproximable, ATSP 2805
2804 (Engebretsen STACS
’99)
• 38133812 -inapproximable (Bockenhauer et al. STACS ’00)
• 220219 -inapproximable, ATSP 117
116 (Papadimitriou and Vem-
pala STOC ’00, Combinatorica ’06)
Current best (Karpinski, L., Schmied):
Theorem
It is NP-hard to approximate TSP better than 123122 and ATSP
better than 7574 .
TSP Approximations – Lower bounds
Parameterized Approximation Schemes 22 / 39
• Problem is APX-hard (Papadimitriou and Yannakakis
’93)
• 53815380 -inapproximable, ATSP 2805
2804 (Engebretsen STACS
’99)
• 38133812 -inapproximable (Bockenhauer et al. STACS ’00)
• 220219 -inapproximable, ATSP 117
116 (Papadimitriou and Vem-
pala STOC ’00, Combinatorica ’06)
Current best (Karpinski, L., Schmied):
Theorem
It is NP-hard to approximate TSP better than 123122 and ATSP
better than 7574 .
Notice the huge distance between the best algorithm (50%error) and hardness (0.8% error). . .
Reduction Technique
Parameterized Approximation Schemes 23 / 39
We reduce some inapproximable CSP (e.g. MAX-3SAT) to TSP.
Reduction Technique
Parameterized Approximation Schemes 23 / 39
First, design some gadgets to represent the clauses
Reduction Technique
Parameterized Approximation Schemes 23 / 39
Then, add some choice vertices to represent truth assignments to
variables
Reduction Technique
Parameterized Approximation Schemes 23 / 39
For each variable, create a path through clauses where it appears positive
Reduction Technique
Parameterized Approximation Schemes 23 / 39
. . . and another path for its negative appearances
Reduction Technique
Parameterized Approximation Schemes 23 / 39
Reduction Technique
Parameterized Approximation Schemes 23 / 39
A truth assignment dictates a general path
Reduction Technique
Parameterized Approximation Schemes 23 / 39
Reduction Technique
Parameterized Approximation Schemes 23 / 39
Reduction Technique
Parameterized Approximation Schemes 23 / 39
We must make sure that gadgets are cheaper to traverse if corresponding
clause is satisfied
Reduction Technique
Parameterized Approximation Schemes 23 / 39
If a clause is not satisfied, we will pay more. We need many clauses to be
unsatisfied in a No instance to have a big gap. (PCP theorem)
Reduction Technique
Parameterized Approximation Schemes 23 / 39
For the converse direction we must also make sure that ”cheating” tours
are not optimal!
How to ensure consistency
Parameterized Approximation Schemes 24 / 39
• 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.
How to ensure consistency
Parameterized Approximation Schemes 24 / 39
• 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.
• This is where expander graphs are important.
• Main tool: “amplifier graph” constructions due to Berman and
Karpinski.
How to ensure consistency
Parameterized Approximation Schemes 24 / 39
• 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.
• This is where expander graphs are important.
• Main tool: “amplifier graph” constructions due to Berman and
Karpinski.
• Expander graphs are a generally useful tool, so let’s take a look at
what they are. . .
Expander and Amplifier Graphs
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• Definition:
A graph G(V,E) is an expander if
• For all S ⊆ V with |S| ≤ |V |2 we have for some constant c
|E(S, V \ S)|
|S|≥ c
• The maximum degree ∆ is bounded
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A complete bipartite graph is well-connected
but not sparse.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A complete bipartite graph is well-connected
but not sparse.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A complete bipartite graph is well-connected
but not sparse.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A grid is sparse but not well-connected.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A grid is sparse but not well-connected.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
A grid is sparse but not well-connected.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
An infinite binary tree is a good expander.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
An infinite binary tree is a good expander.
Expander Graphs
Parameterized Approximation Schemes 26 / 39
• Informal description:
An expander graph is a well-connected and sparse graph.
• In any possible partition of the vertices into two sets, there are
many edges crossing the cut.
• This is achieved even though the graph has low degree, therefore
few edges.
Example:
An infinite binary tree is a good expander.
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expander graphs have a number of applications
• Proof of PCP theorem
• Derandomization
• Error-correcting codes
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expander graphs have a number of applications
• Proof of PCP theorem
• Derandomization
• Error-correcting codes
• . . . and inapproximability of bounded occurrence CSPs!
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expanders and inapproximability
• Consider the standard reduction from 3-SAT to 3-OCC-3-SAT
• Replace each appearance of variable x with a fresh variable
x1, x2, . . . , xn• Add the clauses (x1 → x2) ∧ (x2 → x3) ∧ . . . ∧ (xn → x1)
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expanders and inapproximability
• Consider the standard reduction from 3-SAT to 3-OCC-3-SAT
• Replace each appearance of variable x with a fresh variable
x1, x2, . . . , xn• Add the clauses (x1 → x2) ∧ (x2 → x3) ∧ . . . ∧ (xn → x1)
Problem: This does not preserve inapproximability!
• We could add (xi → xj) for all i, j.• This ensures consistency but adds too many clauses and does not
decrease number of occurrences!
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expanders and inapproximability
• We modify this using a 1-expander [Papadimitriou Yannakakis 91]
• Recall: a 1-expander is a graph s.t. in each partition of the vertices
the number of edges crossing the cut is larger than the number of
vertices of the smaller part.
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Expanders and inapproximability
• We modify this using a 1-expander [Papadimitriou Yannakakis 91]
• Replace each appearance of variable x with a fresh variable
x1, x2, . . . , xn• Construct an n-vertex 1-expander.
• For each edge (i, j) add the clauses (xi → xj) ∧ (xj → xi)
Applications of Expanders
Parameterized Approximation Schemes 27 / 39
Why does this work?
• Suppose that in the new instance the optimal assignment sets some of
the xi’s to 0 and others to 1.
• This gives a partition of the 1-expander.
• Each edge cut by the partition corresponds to an unsatisfied clause.
• Number of cut edges > number of minority assigned vertices =number of clauses lost by being consistent.
Hence, it is always optimal to give the same value to all xi’s.
• Also, because expander graphs are sparse, only linear number of
clauses added.
• This gives some inapproximability constant.
Limits of expanders
Parameterized Approximation Schemes 28 / 39
• Expanders sound useful. But how good expanders can we get?
We want:
• Low degree – few edges
• High expansion (at least 1).
These are conflicting goals!
Limits of expanders
Parameterized Approximation Schemes 28 / 39
• Expanders sound useful. But how good expanders can we get?
We want:
• Low degree – few edges
• High expansion (at least 1).
These are conflicting goals!
• The smallest ∆ for which we currently know we can have expansion 1
is ∆ = 6. [Bollobas 88]
Limits of expanders
Parameterized Approximation Schemes 28 / 39
• Expanders sound useful. But how good expanders can we get?
We want:
• Low degree – few edges
• High expansion (at least 1).
These are conflicting goals!
• The smallest ∆ for which we currently know we can have expansion 1
is ∆ = 6. [Bollobas 88]
• Problem: ∆ = 6 is too large, ∆ = 5 probably won’t work. . .
Amplifiers
Parameterized Approximation Schemes 29 / 39
• Amplifiers are expanders for some of the vertices.
• The other vertices are thrown in to make consistency easier to
achieve.
• This allows us to get smaller ∆.
Amplifiers
Parameterized Approximation Schemes 29 / 39
• Amplifiers are expanders for some of the vertices.
• The other vertices are thrown in to make consistency easier to
achieve.
• This allows us to get smaller ∆.
5-regular amplifier [Berman Karpinski 03]
• Bipartite graph. n vertices on left, 0.8n vertices
on right.
• 4-regular on left, 5-regular on right.
• Graph constructed randomly.
• Crucial Property: whp any partition cuts more
edges than the number of left vertices on the
smaller set.
Amplifiers
Parameterized Approximation Schemes 29 / 39
• Amplifiers are expanders for some of the vertices.
• The other vertices are thrown in to make consistency easier to
achieve.
• This allows us to get smaller ∆.
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 of
checkers.
Back to the Reduction
Overview
Parameterized Approximation Schemes 31 / 39
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 (Hastad)
Overview
Parameterized Approximation Schemes 31 / 39
We use the Berman-Karpinski amplifier construction to obtain an instance
where each variable appears exactly 5 times (and most equations have
size 2).
Overview
Parameterized Approximation Schemes 31 / 39
Overview
Parameterized Approximation Schemes 31 / 39
A simple trick reduces this to the 1in3 predicate.
Overview
Parameterized Approximation Schemes 31 / 39
From this instance we construct a graph.
1in3-SAT
Parameterized Approximation Schemes 32 / 39
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)
TSP and Euler tours
Parameterized Approximation Schemes 33 / 39
TSP and Euler tours
Parameterized Approximation Schemes 33 / 39
TSP and Euler tours
Parameterized Approximation Schemes 33 / 39
TSP and Euler tours
Parameterized Approximation Schemes 33 / 39
• 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
Gadget – Forced Edges
Parameterized Approximation Schemes 34 / 39
We would like to be able to dictate in our construction that a certain edge
has to be used at least once.
Gadget – Forced Edges
Parameterized Approximation Schemes 34 / 39
If we had directed edges, this could be achieved by adding a dummy
intermediate vertex
Gadget – Forced Edges
Parameterized Approximation Schemes 34 / 39
Here, we add many intermediate vertices and evenly distribute the weight
w among them. Think of B as very large.
Gadget – Forced Edges
Parameterized Approximation Schemes 34 / 39
At most one of the new edges may be unused, and in that case all others
are used twice.
Gadget – Forced Edges
Parameterized Approximation Schemes 34 / 39
In that case, adding two copies of that edge to the solution doesn’t hurt
much (for B sufficiently large).
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
Let’s design a gadget
for (x ∨ y ∨ z)
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
First, three entry/exit
points
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
Connect them . . .
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
. . . with forced edges
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
The gadget is a con-
nected component.
A good tour visits it
once.
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
. . . like this
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
This corresponds to
an unsatisfied clause
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
This corresponds to a
dishonest tour
1in3 Gadget
Parameterized Approximation Schemes 35 / 39
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.
Construction
Parameterized Approximation Schemes 36 / 39
High-level view: con-
struct an origin s and
two terminal vertices
for each variable.
Construction
Parameterized Approximation Schemes 36 / 39
Connect them with
forced edges
Construction
Parameterized Approximation Schemes 36 / 39
Add the gadgets
Construction
Parameterized Approximation Schemes 36 / 39
An honest traversal for
x2 looks like this
Construction
Parameterized Approximation Schemes 36 / 39
A dishonest traversal
looks like this. . .
Construction
Parameterized Approximation Schemes 36 / 39
. . . but there must be
cheating in two places
There are as many doubly-used forced edges as affected variables
→ w ≤ 5
Construction
Parameterized Approximation Schemes 36 / 39
. . . 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
Under the carpet
Parameterized Approximation Schemes 37 / 39
• 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 ampli-
fier) must be taken into account to calculate
the final constant.
Under the carpet
Parameterized Approximation Schemes 37 / 39
• 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 ampli-
fier) must be taken into account to calculate
the final constant.
Theorem:
There is no 185184 approximation algorithm for TSP, unless P=NP.
Under the carpet
Parameterized Approximation Schemes 37 / 39
• 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 ampli-
fier) must be taken into account to calculate
the final constant.
Theorem:
There is no 185184 approximation algorithm for TSP, unless P=NP.
Can we do better?
Under the carpet
Parameterized Approximation Schemes 37 / 39
• 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 ampli-
fier) must be taken into account to calculate
the final constant.
Theorem:
There is no 185184 approximation algorithm for TSP, unless P=NP.
Can we do better?
Summary
Parameterized Approximation Schemes 38 / 39
• Hardness of Approximation theory is the evil twin
of the theory of approximation algorithms.
• It relies on some deep mathematical tools
• PCP theorem, expander graphs, . . .
• We discussed some general common patterns
• Local vs. global errors, gaps, . . .
Summary
Parameterized Approximation Schemes 38 / 39
• Hardness of Approximation theory is the evil twin
of the theory of approximation algorithms.
• It relies on some deep mathematical tools
• PCP theorem, expander graphs, . . .
• We discussed some general common patterns
• Local vs. global errors, gaps, . . .
• Area still under construction!
• Still far from answer for TSP and many other
prominent problems!
• For Graph Balancing the answer is between
1.5 and 1.75.
• Can we make more progress?
The end
Parameterized Approximation Schemes 39 / 39
Questions?