1. Algorithms for hard problems WQO theory and applications to parameterized complexity Juris Viksna, 2015

2. Quasi-ordering For a given set S and a binary relation ,  is quasi-ordering, if for all x,y,z  S: x  x (reflexivity) x  y and y  z  x  z (transitivity)

3. Quasi-ordering - terminology Let S be a set with quasi ordering . Let a 0,a 1,... be a sequence of elements of S. a 0,a 1,... is chain if i  j  a i  a j a 0,a 1,... is antichain if i  j  a i | a j

4. Quasi-ordering - terminology Let S be a set with quasi ordering  S'  S - filter if x  S' and y  x  y  S' F(A)={y  S |  x  A: x  y} - filter generated by A S'  S - ideal if x  S' and y  x  y  S' I(A)={y  S |  x  A: x  y} - ideal generated by A If F is a filter then S–F is an ideal and vice versa

5. Quasi-ordering - terminology Let S be a set with quasi ordering  S'  S - filter if x  S' and y  x  y  S' F(A)={y  S |  x  A: x  y} - filter generated by A A filter S' if finitely generated if S'=F(A) for a finite set A ‹S,  › has a finite a basis property if F(A) is finitely generated for all A  S.

6. WQO (Well-quasi ordering) A quasi ordering  S,  is well-quasi ordering (WQO) if any of the following holds: 1.for every infinite sequence x 1,x 2,... of S there are indices i and j such that i < j and x i  x j ; [simplest to verify] 2.every infinite sequence x 1,x 2,... of S contains an infinite chain; 3.there are no infinite antichains or infinite decreasing sequences; 4.  S,  has a finite basis property. [most useful] Proof of equivalence of definitions: Try to show that 1  2  3  4  1

7. Examples of WQO  N,  is WQO under usual order relation . For each x 1,x 2,...  i,j: i < j and x i  x j  S,  is WQO for each finite set and equality relation . Each infinite sequence x 1,x 2,... will contain at least one repeated element x i Other examples?

8. Higman’s Lemma Let S be a finite set and S * be a set of all finite sequences of elements of S. Let  be defined on set S * as follows: v  w  v is a subsequence of w. Then  S *,  is WQO.

9. Higman’s Lemma Higman's Lemma - the idea of proof: assume (S*,  ) is not WQO let w 1,w 2,... be the smallest infinite antichain there is a  S, such that infinite number of w i 's starts with a let the sequence of these w i 's be w i 1,w i 2,... denote w ij with the first symbol a removed by w' ij then w 1,...,w i 1 -1,w' i 1,w' i 2,... is antichain and smaller than the smallest one (!!!) Note that the result holds for subsequence relation, not for substring relation! How about subsequences of infinite sequences?

10. Counterexample for infinite sequences Rado Structure:

11. Infinite strings are not WQO...

12. WQO Principle If ABOVE(x) is solvable in polynomial time, then for any filter F of ‹S,  › the decision problem "Is y  F?" is solvable in polynomial time. Suppose ‹S,  › is a WQO ABOVE(x) Instance:y  S Parameter:x  S Question:Is x  y?

13. WQO - Obstruction set Definition (Obstruction set) Let ‹S,  › be a quasi-ordering. Let I be an ideal of ‹S,  ›. We say that a set O  S forms an obstruction set for I if x  I iff  y  O: not y  x That is, O is an obstruction set for I if I is the complement of F(O).

14. WQO - Obstruction Principle Let ‹S,  › be a quasi-ordering. Suppose that ABOVE(x) is in P for all x  S, and that ideal I has a finite obstruction set. Then membership of I is in P too. Obstruction principle If ‹S,  › is WQO and, if ABOVE(x) is in P for all x  S, then every ideal I  S has a P-time membership test.

15. WQO - Obstruction Principle Obstruction principle If ‹S,  › is WQO and, if ABOVE(x) is in P for all x  S, then every ideal I  S has a P-time membership test. A method to prove that problems are in P: Show that problem is characterized by ideal in WQO set with ABOVE(x) in P.

16. CLOSEST STRING problem There is an algorithm that solves the CLOSEST STRING problem in time O(kL+kd d+1 ). The specific algorithm involves both: reduction to kernel and bounded search tree approaches. However from Higman’s lemma we immediately know that the problem is FPT! (Can we find also obstruction sets in this case?)

17. Ordering relations on graphs Subgraph relation: A graph H=(V 1,E 1 ) is a subgraph of G=(V 2,E 2 ) if there is an injective mapping  : V 1  E 1, such that {x,y}  E 1  {  (x),  (y)}  E 2. H  S G  H is a subgraph of G. SS SS

18. Ordering relations on graphs Topological embedding relation: For a graph G let P(G) denote a set of all simple paths in G. A graph H=(V 1,E 1 ) is topologically embedded in G=(V 2,E 2 ) if there is an mapping  : E 1  P(G), such that for all pairs of distinct edges e 1,e 2  E 1 the corresponding paths  (e 1 ) and  (e 2 ) are vertex-disjoint. H  T G  there is a topological embedding of H in G.

19. Ordering relations on graphs Graph minor relation: A graph H=(V 1,E 1 ) is a minor of G=(V 2,E 2 ) if H is a subgraph of graph G' obtained from G by a sequence of edge contractions. H  M G  H is minor of G. K 4  M K 3,3 Edge contraction Edge contraction

20. Ordering relations on graphs Graph immersion relation: A graph H=(V 1,E 1 ) is immersion in G=(V 2,E 2 ) if there is an mapping  : E 1  P(G), such that for all pairs of distinct edges e 1,e 2  E 1 the corresponding paths  (e 1 ) and  (e 2 ) are edge- disjoint. H  I G  H is immersion in G. II

21. Graph operations and QO on graphs Topological order  {E,V,T}, minor order  {E,V,C}, immersion order  {E,V,L}.

22. Are any of these orderings WQO? subgraph relation  S is not WQO topological embedding  T is not WQO Ordering relations on graphs......

23. graph minor relation  M is WQO graph immersion relation  I is WQO Theorem [N.Robertson, P.Seymour] Every infinite set of graphs contains two graphs such that one is a minor of another. Theorem [N.Robertson, P.Seymour] Every infinite set of graphs contains two graphs such that one is immersed in another. Ordering relations on graphs The proof is spread over 23 articles published in Journal of Algorithms from 1983 to 2009

24. Graph minor theorem

25. Testing graph minor and immersion properties MINOR ORDER TEST is FPT-solvable in time O(n 3 ) [Robertson and Seymour 1995] IMMERSION ORDER TEST is FPT-solvable in polynomial time poly(n) [Fellows and Langston 1988]

26. WQO - Obstruction Principle MINOR ORDER TEST Instance:Graphs G=(V,E) and H=(V',E') Parameter:A graph H Question:Is H a minor of G? MINOR ORDER TEST is FPT-solvable in time O(|V| 3 ) [Robertson and Seymour 1995] Finite graphs are WQO by  minor [Robertson and Seymour "papers"] So, to prove that a graph problem is in P, it is sufficient to show that it is characterized by ideal with respect  minor ordering (!)

27. WQO - Graph planarity GRAPH PLANARITY Instance:Graph G=(V,E) Question:Is G planar? To prove that a graph problem is in P, it is sufficient to show that it is characterized by ideal with respect  minor ordering. Observation If G is planar and H is G minor, then H is planar. Thus, planar graphs form an ideal. Graph planarity is in P However, Graph Minor Theorem is intrinsically nonconstructive - there is no algorithm, which for a given ideal computes its obstruction set...

28. WQO - Graph planarity GRAPH PLANARITY Instance:Graph G=(V,E) Question:Is G planar? To prove that a graph problem is in P, it is sufficient to show that it is characterized by ideal with respect  minor ordering. This already gives a P-time algorithm for graph planarity problem Kuratowski theorem {K 3,3, K 5 } is an obstruction set for the ideal of planar graphs under  minor ordering. [Adapted from R.Downey and M.Fellows]

29. WQO - Vertex cover Observation If G has vertex cover of size k and H is G minor, then H has vertex cover of size k. Thus, graphs having vertex cover of size k form an ideal. VERTEX COVER Instance:A graph G=(V,E) Parameter:A positive integer k Question:Is there a subset S  V, such that |S|=k and for all {x,y}  E either x  S or y  S? For fixed k VERTEX COVER is in P VERTEX COVER  FPT

30. WQO - Graph genus GRAPH GENUS Instance:A graph G=(V,E) Parameter:A positive integer k Question:Does graph G have genus k? [Adapted from R.Downey and M.Fellows] Easy to see that the problem is in FPT. But can we effectively find an obstruction set for any given k? Graphs with genus k has bounded treewidth. This gives (totally unpractical) algorithm for computation of obstruction set for each k.

31. WQO -graph linking number GRAPH LINKING NUMBER Instance:A graph G=(V,E) Parameter:A positive integer k Question:Can G be embedded in 3-space such that the maximum size of a collection of topologically linked disjoint cycles is bounded by k? [Adapted from R.Downey and M.Fellows] Again, it is easy to see that the problem is in FPT. In this case however there is no known algorithm that computes obstruction set for each k...

32. Uniformity of FTP VERTEX COVER: A single algorithm  running in time 2 k |G| for each k [Downey and Fellows 1992] GRAPH GENUS: A single algorithm  which for each fixed k determines whether graph G has genus k in time O(|G| 3 ) [Fellows and Langston 1988] GRAPH LINKING NUMBER: For each k there is an algorithm  k, which determines whether graph G has linking number k in time O(|G| 3 ) [Fellows and Langston 1988]

33. Uniformity of FTP A problem L is uniformly FPT, if there is an algorithm , a constant c and a function f, such that: - the running time of  (‹x,k › ) is at most f(k)|x| c - ‹x,k›  A iff  (‹x,k›) = 1. A problem L is strongly uniformly FPT, if L is uniformly FPT via some  and f, such that f is recursive. A problem L is nonuniformly FPT if there is a constant c, a function f and a collection of algorithms {  k } k  N, such that for each k: - the running time of  k (‹x,k›) is at most f(k)|x| c - ‹x,k›  A iff  k (‹x,k›) = 1.

34. Ordering relations on graphs Graph immersion relation: A graph H=(V 1,E 1 ) is immersion in G=(V 2,E 2 ) if there is an mapping  : E 1  P(G), such that for all pairs of distinct edges e 1,e 2  E 1 the corresponding paths  (e 1 ) and  (e 2 ) are edge- disjoint. H  I G  H is immersion in G. II

35. Immersion vs graph minor relation Graph immersion relation: A graph H=(V 1,E 1 ) is immersion in G=(V 2,E 2 ) if there is an mapping  : E 1  P(G), such that for all pairs of distinct edges e 1,e 2  E 1 the corresponding paths  (e 1 ) and  (e 2 ) are edge- disjoint. H  I G  H is immersion in G. Are there problems which can be shown FPT using immersion but not graph minor relation (and vice versa)?

36. Immersion vs graph minor relation Are there problems which can be shown FPT using immersion but not minor relation (and vice versa)? GRAPH GENUS Invariant for both graph minor and immersion relations. VERTEX COVER Invariant for graph minor but not immersion relation.

37. Immersion vs graph minor relation

38. Finding of obstruction set is algorithmically unsolvable problem... [Adapted from R.Downey and M.Fellows]

39. Self reduction Assuming we have an efficient oracle which for a given graph answers yes/no to the above decision problem, if the answer is yes we can also efficiently find a Hamiltonian cycle as follows: for each e  E: if G with e removed still has Hamiltonian cycle, remove e from G The edges that will be left in E will form a Hamiltonian cycle. HAMILTONICITY Instance:A graph G=(V,E) Question:Does graph G has a Hamiltonian cycle?

40. Self reduction Assume that to prove the existence of a parameterized algorithm for deciding membership to F we have used WQO method – i.e. we know that there exists a finite basis B for F complement. In certain cases this still allows us to obtain a concrete algorithm! For given input x: 1.start with B=  2.check whether there exists y  B with y  x 3.if such y exists the answer is no 4.otherwise try to self-reduce, if solution is found the answer is yes 5.if solution isn’t found, generate the next element from complement of F that doesn’t have a minor in B, add it to B and go to Step 1.

41. Self reduction

43. Self reduction - Hamilton circuit

44. Self reduction

49. Self reduction -planar diameter improvement