Talks on parts of 4 papers.

1) M. Hajiaghayi, Khandekar and K.

2) M. Cygan, K

3) R Chitnis, M. Hajiaghayi, K

4) M. Hajiaghayi, K and some students of M. Hajiaghayi

Optimal running times for exact solutions and approximated solutions

The 3-SAT problem with n variables and m clauses can not be solved in time

2o(n)

Due to Impagliazzo, Paturi and Zane. FOCS 1998. Do you think its false?

Lemma of Calabro, Impagliazzo and Paturi: The 3-SAT problem with n variables and m

clauses can not be solved in time 2o(m) This is called the Sparsification Lemma.

The Exponential Time Hypotesis

What can we prove under the Exponential Time Hypothesis?

Many problems have “optimum” running time algorithms under this assumption.

We later present such a result in connectivity. Tight lower bound that uses the Exponential Time Hypothesis

The subject of this talk

How can we prove that there is no f(k)poly(n) algorithm for Clique?

The assumption of P=NP implies that f(k) is polynomial in n. To show Clique FPT we need to show PNP.

Instead: assume the much stronger ETH assumption.

Why do we need the ETH?

If you want approximation ratio of for some problem what is the best possible running time?

You need to do two things First give an approximation ratio of in some time t(n).

Then show that approximation of , with time better than t(n) would contradict the ETH. We start with this.

Harder (but natural) subject

In approximation algorithms I do not think somebody tried to show that in linear time you can not get better than 2 ratio for Vertex Cover. Should we create a new subject? If its possible to prove such things.

Using the ETH this may be possible. Needs knowledge far from FPT. Needs a knowledge of almost linear PCP and

about gap reductions and about deep theorems in Inapproximability theory.

See more later.

How do we lower bound the time for approximation?

The directed Steiner problems

s6

3 1 1

6

2

3

5

2

2

1

1

1

3

44

4

Optimum solution with all terminals

The directed Steiner problems

s

3 1 12

3

2

1

1

4

A very important problem in Approximation Algorithms. Key for other problems.

This problem is FPT by the cost of the optimum solution.

It admits n for every .

In the next slide I will give the correct credit for this result. Never done.

What is known

The best approximation algorithm for the problem was designed in SODA 1997 by K,Peleg. The credit (by mistake) is given to Charikar et al. Implies ratio f()n for any .

In SODA 1998 Charikar et al used the same algorithm. Said explicitly that Implies ratio f() n for any for the Directed Steiner tree.

Charikar et al: better f() term. Charikar et al also implied that the

problem has log3 n ratio, time quasi-polynomial in n.

Approximation

At the time such an algorithm was considered as a sign that a polynomial polylogarithmic approximation exists.

A paper by Chandra Chekuri and Martin Pal: under the ETH, PQuasi-P

Conjecture (Kortsarz): Under the ETH there is no polynomial time polylogarithmic ratio approximation for the Directed Steiner Tree problem.

Does this imply that there is polynomial time polylogarithmic ratio?

It turns out that linear reductions are crucial for Fixed Parameter Inapproximability.

This is known for quite some time. This means a reduction from SAT with m

cluases and n variables that creates a gap. The size of the instance of the new problem

is O(m+n) Unfortunately, if the ETH is correct there are

almost no linear reductions.

Linear reductions

Unfortunately, a linear reduction from PCP to Set-Cover implies that ETH fails.

If we had that we could show that Set-Cover admits no (r(k),t(k)) FPT-approximation for any r,t.

There is a linear reduction from SAT to Clique. This does not help because first we need to do a gap reduction from SAT to 3-SAT.

Example for what is not possible

SAT with n variables and m clauses. An almost linear reduction is a

reduction to Label-Cover of size m1+o(1)

Known (Dinur). Reduction of size mpolylog(m) to Label-Cover, gap 2.

The projection game conjecture:

Moskowitz: Reduction to Label-cover of

size m2 log1- m but gap nc for some c.

What almost linear hardness do we know?

W[1]-Hard problem. n ratio approximation This problem is clearly finding a Directed

Steiner tree and a reverse directed Steiner tree.

The Directed Steiner Tree problem is FPT when parameterized by the optimum solution

A rare case in which FPT time improves drastically the approximation ratio.

As we saw, ratio 2 is possible in time t(k)poly(n).

Due to Chitnis, Hajiaghayi, K.

Remark about the Strongly conneceted subgraph problem.

M. Cygan, K If you want a ratio of ln n/2 the time required is roughly 2sqrt{n}log n

Using the ETH we show that this time is optimal (the exponent can not be o(sqrt{n})).

If you want a ratio for Directed Steiner Tree what time is needed?

The upper bound is designing an algorithm.

The problematic part is the lower bound. Relies on Almost Linear PCP, Projection Game conjecture. Different kind of knowledge.

Maybe because of that I found very very few results of this kind.

If you want a ratio for Directed Steiner Tree what time is needed?

In this paper we define a new way to use the known definition for Fixed Parameter Inapproximability.

We call this method inapproximability in opt

The definition requires k=opt(I) for some I.

The definition was heavily influenced by talks with Cygan and Marx.

Paper Hajiaghayi Khandekar ,K

Since approximation is in opt, inapproximability should also be in opt. This is the logical counter statement.

We were trying to avoid reduction under FPT W[1], FPTW[2].

The ETH implies both statements above.

Far reaching consequences.

Why would we want k=opt(I)?

ETH implies FPT W[1], FPT W[2]. We are given that no time t(k) is enough.

The value is usually k versus k+1 for minimization. Hard to get strong hardness.

The proof above reduces k below any given function. Thus k is not related to any opt(I).

Proofs under FPT W[i]

However, if approximation in opt why not inapproximabiliy in opt?

Also our definition does not throw all problems in the same bin.

Does not seem logical that all prolems behave the same. Completely different problems.

By our definition we get a much richer behavior.

Each problem, its own behavior.

Proofs under FPT W[i]

Start with SAT. A yes instance goes to value X for our problem.

A no instance goes to value larger than X, >1 for our problem.

Important: can produce huge gaps, solving the k versus k+1 issue.

Method: Gap reductions

Polynomial algorithm with ratio implies P=NP

A approximation algorithm with running time 2o(m+n) implies that the ETH fails.

Method: Gap reductions

A good ((opt), t(opt)) ratio needs gap preserving reduction that makes opt very small. Not well understood.

We gave the first super exponential time inapproximability for Clique and Set Cover.

In fact for Clique Almost doubly exponential.

Method: Gap reductions

FPTW[1], FPT W[2] does not imply anything on the optimum solution of any instance.

The problems are not thrown in the same bin.

In fact for every problem we check what kind of gap reduction do we have?

For every problem: is there a gap preserving (increasing, slightly decreasing) reduction that makes opt very small? The latter is the new technical challenge.

Properties

It looks for simple variants of Directed Steiner Network that can be solved exactly.

Its seem that there are not many.The lower bounds do not use almost linear PCP but rather something standard in FPT theory.

Time to show the exact result with optimum time we proved

Let the vertices be 1,2,…..,n Given G(V,E) and a demand dij for every i

and j (could be 0) and cost c(e) for every e. The goal is to select a subgraph G(V,E’) so

that there are dij edge disjoint paths from i to j (separately). Use minimum cost.

Hopeless problem to approximate. For Directed Steiner Forest:

Feldman,K,Nutov gave an O(n3/4) ratio. But that is it.

What are the simplest solvable cases?

The Directed Steiner network problem

Given a graph G(V,E) with unit costs (makes a difference!) and a root s output minimum cost subgraph that contains 2 edge disjoint paths from s to the other terminals.

Our usual trick (Set Families, Uncrossable, Weakly Super Modular functions, Laminar Basic Feasible solution) do not work.

The idea of starting with Directed Steiner tree and then add edges to give two paths from s to all vertices seems to badly fail.

A problem that we do not know anything for

Given a DIRECTED graph G(V,E) and two nodes s and t find a minimum cost graph so that there is a path from s to t and from t to s

The paths may not be edge disjoint. Minimize the number of vertices in the

solution (reduction from the edge case). We generalize this problem, and gave a

tight upper lower bound on the time. Even the solution of the above non trivial.

A “simple” solvabable problem

The solution may be complex

Not shortest path.

The solution may be complex

We will have two tokes both in s. One tokens, f goes on edges in the regular

way. This creates the path from s to t. A second token called b goes in the wrong

direction. This token would create a path from t to s.

Bring the two tokens from (s,s) to (t,t). Due to Jon Feldman.

A token game

Token f moving forward (u,x) (v,x) for an edge with (u,v).

Token b moves backward (creating an t to s path): (x,u)(x,v) for the edge (v,u) adding a back edge in the path from t to s.

If both tokens reach t in the best way, we solved the problem.

How do tockens move?

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

Edges : (s,q), (q,p), (p,x), (x,t) (r,u) (r,y) (u,t), (r,u), (y,r), (t,y) ,(p,x)

An example that does not cause problems

(s,s)

(s,u)

(q,u)

(p,u) (p,r) (x,r) (t,r)

(t,y)

(t,t)

s q p x t

s u r y t

At the moment we enter a vertex, this vertex is declared a dead vertex.

At every moment there must be a path

from the location of f to t using live vertices.

And there must be a back path from b into t of live vertices.

Making sure we do not over count

The backward needs x. The forward needs y.

Getting stuck because f,b dead vertices

s

xy

f

t

b

The following three paths must exist:

Getting stuck because of dead vertices

s

xy

f

t

b

This contradicts f needing y to get to t.

Getting stuck because of dead vertices

s

xy

f

t

b

We now move (x,y) to (y,x). Clearly a must.

Getting stuck because of dead vertices

s

xy

f

t

b

We move from (x,y) to (y,x) but with one edge.

Getting stuck because of dead vertices

s

xy

f

t

b

alive

alive

We make the graph with all pairs vertices and edges as discussed.

We add edges from (x,y) to (w,y) with cost c: the cost of the shortest path

from x to w. Since direct edge move, dead vertices do not present a problem.

We apply the Dijkstra’s algorithm to find the shortest path on the graph of pairs, finding the optimum

The shortest path algorithm

We have a structural lemma Pity: even for (2,2) does not work (we give

an example that the structural lemma is false)

We solve this generalization in time nO(k). We show that under the ETH there is no

f(k)no(k)

Quite complex in my opinion. Uses the grid tiling problem.

We solve: s,t, k disjoint paths for s to t one from t to s

Chen et al showed a nice result about k-clique: no exact solution in time f(k) no(k) for any f.

Marx: reduction from k-clique to Grid Tiling.

This reduction has surprising number of applications.

Many in planar graphs.

How do we get the hardness

We reduce from Grid Tiling to our problem with linear blowup

The time lower bound follows. This is also a W[1]-hardness reduction.

Are there other problems that use a reduction from Grid Tiling for W[1] hardness?

Seems a complex reduction to me.

How do we get the hardness

Do not prove FPT approximation unless the problem is both W[i]-hard for i≥1 and has poor approximation. If one of the two statements is not true, what is the point?

Reductions should have only super exponential time in opt (or k). Otherwise we just translate a hardness to FPT terms.

Also, makes no sense to apply FPT-inapproximability if the optimum is a constant.

Some rules we suggest

We feel that what we called inapproximability in opt is the right counter statement to “approximation in opt”. In our opinion “hardness in opt” is better.

Gives more interesting behavior. Gives large gaps/hardness. Needs knowledge in proving hardness of approximation.

FPT theory people: study approximation!

Given a problem, FPT people usually know if it is W[i]-hard for i≥1.

But what about hardness? Thus you have to either know the approximation lower bound if exists, or prove an inapproximability result.

There are excellent books and lecture notes in Approximation Algorithms.

FPT people study approximation!

When you study a new problem, check if it is in FPT.

Or perhaps it is W[i]-hard for i≥1. Fortunately: Nice slides by Marx. A new state of the art book by Cygan et

al. People in approximation can make papers

on the topic of my talk: “optimal” time for a required approximation.

People who work in Approximation: study FPT!

Kernelization, Crown Reduction, Sunflower, Lemma, Bounded Tree search, Branching Vectors, all can give FPT algorithms for Vertex cover.

Forbidden subgraphs (Triangle Free Graphs) Iterative compression (Bipartite Deletion)

Graph Minors (k-leaves spanning tree) , Color Coding (k length paths), Dynamic Programming (Steiner tree), Important Separators (Multiway Cut), Treewidth………

Some tools used in FPT proofs

Assume the ETH in hardness. Example M. Cygan, K. Consider a graph with

costs and profits on the verticess. A cost bound B: Find a tree of cost at most B

and maximize the profit. Conjectured to have O(1) approximation

(Moss, Rabani). Due to Set Coverage. Nutov,K: loglog n hardness Cygan, K: log n hardness under ETH. Tight! An O(log n) ratio exists.

In inapproximability assume the ETH!

Fellows conjecture: Clique And Set-Cover admit no ((k),t(k)) approximation for any ,t.

I believe this conjecture (even in opt). May require a Parameterized PCP I talked to Dinur, Khot and other

experts. All told me in a very polite way:

More open problems than known results

Fellows conjecture: Clique And Set-Cover admit no ((k),t(k)) approximation for any ,t.

I believe this conjecture. May require a Parameterized PCP I talked to Dinur, Khot and other

experts. All told me in a very polite way:

Please get a hobby and leave us alone.

More open problems than known results

Fellows conjecture: Clique And Set-Cover admit no ((k),t(k)) approximation for any ,t.

I believe this conjecture. May require a Parameterized PCP I talked to Dinur, Khot and other

experts. All told me in a very polite way: Please

get a hobby and leave us alone. However there is now a simple PCP and

simple parallel repetition theorems.

More open problems than known results

Say that opt=logloglog n. Is the Set-Cover and Clique NPC under this value?

What about if opt=log* n. Better results: can we show an

inapproximability for opt=log* n. According to the Fellows conjecture we should

not be able to give any approximation thus the above value of opt do not matter.

Current PCP even the best possible (and not known) gives double exponential time in opt lower bound.

A problem I do not know anything about

How can we prove my conjecture? No polynomial time polylogarithmic ratio algorithm under ETH.

The Directed Steiner tree has roughly log2 n lower bound. Can we get exact running times for c log2 n approximations?

A log3 n inapproximability I have no idea how to prove.

What do I do with the directed Steiner tree problem?

Any questions?

Thank You

Any questions?