Talks%20on%20parts%20of%204%20papers.

1
Optimal running times for exact solutions and
approximated solutions

• 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

2
The Exponential Time Hypotesis

• 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.

3
The subject of this talk

• 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

4
Why do we need the ETH?

• How can we prove that there is no f(k)?poly(n)
  algorithm for Clique?
• The assumption of PNP implies that f(k) is
  polynomial in n. To show Clique? FPT we need to
  show P?NP.
• Instead assume the much stronger ETH assumption.

5
Harder (but natural) subject

• 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.

6
How do we lower bound the time for approximation?

• 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.

7
The directed Steiner problems
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8
The directed Steiner problems

• Optimum solution with all terminals

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2
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2
4
9
What is known

• 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.

10
Approximation

• 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.

11
Does this imply that there is polynomial time
polylogarithmic ratio?

• 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, P?Quasi-P
• Conjecture (Kortsarz) Under the ETH there is no
  polynomial time polylogarithmic ratio
  approximation for the Directed Steiner Tree
  problem.

12
Linear reductions

• 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(mn)
• Unfortunately, if the ETH is correct there are
  almost no linear reductions.

13
Example for what is not possible

• 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.

14
What almost linear hardness do we know?

• SAT with n variables and m clauses.
• An almost linear reduction is a reduction to
  Label-Cover of size m1o(1)
• Known (Dinur). Reduction of size m?polylog(m) to
  Label-Cover, gap 2.
• The projection game conjecture
• Moskowitz Reduction to Label-cover of size m?2
  log1-? m but gap nc for some c.

15
Remark about the Strongly conneceted subgraph
problem.

• W1-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.

16
If you want a ratio ? for Directed Steiner Tree
what time is needed?

• M. Cygan, K
• If you want a ratio of ln n/2 the time required
  is roughly 2sqrtn?log n
• Using the ETH we show that this time is optimal
  (the exponent can not be o(sqrtn)).

17
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.

18
Paper Hajiaghayi Khandekar ,K

• 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 kopt(I) for some I.
• The definition was heavily influenced by talks
  with Cygan and Marx.

19
Why would we want kopt(I)?

• 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?
  W1, FPT?W2.
• The ETH implies both statements above.
• Far reaching consequences.

20
Proofs under FPT ?Wi

• ETH implies FPT? W1, FPT? W2.
• We are given that no time t(k) is enough.
• The value is usually k versus k1 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).

21
Proofs under FPT ?Wi

• 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.

22
Method Gap reductions

• Start with SAT. A yes instance goes to value X
  for our problem.
• A no instance goes to value larger than ??X, ?gt1
  for our problem.
• Important can produce huge gaps, solving the k
  versus k1 issue.

23
Method Gap reductions

• Polynomial algorithm with ratio ? implies PNP
• A ? approximation algorithm with running time
  2o(mn) implies that the ETH fails.

24
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.

25
Properties

• FPT?W1, FPT ? W2 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.

26
Time to show the exact result with optimum time
we proved

• 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.

27
The Directed Steiner network problem

• 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?

28
A problem that we do not know anything for

• 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.

29
A simple solvabable problem

• 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.

30
The solution may be complex
31
The solution may be complex

• Not shortest path.

32
A token game

• 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.

33
How do tockens move?

• 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.

34
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
35
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
36
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
37
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
38
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
39
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
40
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
41
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
42
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
(t,y)
43
An example that does not cause problems

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

(s,u)
(s,s)
(p,u)
(p,r)
(x,r)
(t,t)
(t,r)
(q,u)
q
p
s
x
t
(t,y)
s
u
r
y
t
44
Making sure we do not over count

• 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.

45
Getting stuck because f,b dead vertices

• The backward needs x. The forward needs y.

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f
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b
y
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Getting stuck because of dead vertices

• The following three paths must exist

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f
b
x
y
t
47
Getting stuck because of dead vertices

• This contradicts f needing y to get to t.

s
f
b
x
y
t
48
Getting stuck because of dead vertices

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

s
f
b
x
y
t
49
Getting stuck because of dead vertices

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

s
f
b
x
y
alive
alive
t
50
The shortest path algorithm

• 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 Dijkstras algorithm to find the
  shortest path on the graph of pairs, finding the
  optimum

51
We solve s,t, k disjoint paths for s to t
one from t to s

• 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.

52
How do we get the hardness

• 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.

53
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 W1-hardness reduction.
• Are there other problems that use a reduction
  from Grid Tiling for W1 hardness?
• Seems a complex reduction to me.

54
Some rules we suggest

• Do not prove FPT approximation unless the problem
  is both Wi-hard for i1 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-inapproximabilit
  y if the optimum is a constant.

55
FPT theory people study approximation!

• 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.

56
FPT people study approximation!

• Given a problem, FPT people usually know if it
  is Wi-hard for i1.
• 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.

57
People who work in Approximation study FPT!

• When you study a new problem, check if it is in
  FPT.
• Or perhaps it is Wi-hard for i1.
• 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.

58
Some tools used in FPT proofs

• 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

59
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 (even in opt).
• May require a Parameterized PCP
• I talked to Dinur, Khot and other experts.
• All told me in a very polite way

60
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.

61
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.

62
A problem I do not know anything about

• Say that optlogloglog n. Is the Set-Cover and
  Clique NPC under this value?
• What about if optlog n.
• Better results can we show an inapproximability
  for optlog 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.

63
What do I do with the directed Steiner tree
problem?

• 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.

64
Any questions?
65
Any questions?

• Thank You