Paths, Flowers and Vertex Cover

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1 Paths, Flowers and Vertex Cover Venkatesh Raman, M.S. Ramanujan, and Saket Saurabh Presenting: Hen Sender 1

2 Introduction 2

3 Abstract. It is well known that in a bipartite (and more generally in a Konig) graph, the size of the minimum vertex cover is equal to the size of the maximum matching. 3

4 The question: Is (and if not when) this property still holds in a Konig graph if we insist on forcing one of the two vertices of some of the matching edges in the vertex cover solution? 4

5 We will use FTP For decision problems with input size n, and a parameter k, the goal in parameterized complexity is to design an algorithm with runtime f k n O(1) where f is a function of k alone, as contrasted with a trivial n k+o(1) algorithm. Such algorithms are said to be fixed parameter tractable (FPT). 5

6 The version of classical VC that we are interested in is following: 6

7 Related problems: The only known parameterized algorithm for AGVC is using a parameter preserving reduction to Almost 2-SAT. In Almost 2-SAT, we are given a 2-SAT formula f, a positive integer k and the objective is to check whether there exists at most k clauses whose deletion from f can make the resulting formula satisfiable. The Almost 2-SAT problem was shown to have an O (15 k ) time algorithm. We will now prove that AGVC can be calculate in O (9 k ) time and because of the reduction, so is The Almost 2-SAT. 7

8 Preliminaries 8

9 Preliminaries Let G = (V, E) be a graph, for a subset S of V, the subgraph of G induced by S is denoted by G[S]. By N G (u) we denote (open) neighborhood of u that is set of all vertices adjacent to u. Similarly, for a subset T V, we define N G T = N G (v)\t v T. The vertices of G that are the endpoints of edges in the matching M are said to be saturated by M and we denote the set of these vertices by V(M), - all other vertices are unsaturated by M. 9

10 Preliminaries Lemma 1: A graph G = (V, E) is Konig if and only if there exists a bipartition of V into V 1 V 2, with V 1 a vertex cover of G such that there exists a matching across the cut (V 1, V 2 ) saturating every vertex of V 1. Definition 1: Let Z be a finite set. A function f: 2 Z N is submodular if for all subsets A and B of Z, f A B + f A B f A + f(b) 10

11 Outline of the Algorithm 11

12 Outline of the Algorithm We first make use of a known reduction that allows us to assume that the input graph has a perfect matching. Given an instance (G = V, E, M, k) of AGVC, in polynomial time we obtain an equivalent instance with a perfect matching. That is, if (G, M, k) is an instance of AGVC and G is a graph without a perfect matching, then in polynomial time we can obtain an instance (G, M, k) such that G has a perfect matching M and (G, M, k) is a Yes instance of AGVC if and only if (G, M, k) is a Yes instance of AGVC. 12

13 Iterative Compression for AGVC Given an instance (G = V, E, M, k) of AGVC let M = {m 1,, mn} be a perfect matching for G, 2 where n = V. Define M i = {m 1,, m i } and G i = [V(M i )], 1 i n 2. We iterate through the instances (G i, M i, k) starting from i = k + 1 and for the i th instance, with the help of a known solution S i of size at most M i + k + 1 we try to find a solution S i of size at most M i + k. 13

14 Iterative Compression for AGVC We will reduce the AGVC problem to n 2 k instances of the AGVCC problem. Let I i =(G i, M i,s i, k) be the i th instance. Clearly, the set V(M k+1 ) is a vertex cover of size at most M k+1 + k + 1 for the instance I k+1. It is also easy to see that if S i 1 is a vertex cover of size at most M i 1 + k for instance I i 1, then the set S i 1 V( M i ) is a vertex cover of size at most M i + k + 1 for the instance I i. If, during any iteration, the corresponding instance does not have a vertex cover of the required size, it implies that the original instance is also a No instance. Finally the solution for the original input instance will be Sn. 2 14

15 AGVCC: 15

16 Our algorithm for AGVCC: Let the input instance be I=(G=(V,E),S,M, k). Let M be the edges in M which have both vertices in S. Note that M k + 1. Then, G\V(M ) is a Konig graph and by Lemma 1 has a partition (A, B) such that A is a minimum vertex cover and there is a matching saturating A across the cut (A, B), which in this case is M\M. We guess a subset Y M with the intention of picking both vertices of these edges in our solution. For the remaining edges of M, exactly one vertex from each edge will be part of our eventual solution. 16

17 Our algorithm for AGVCC: For each edge of M \Y, we guess the vertex which is not going to be part of our eventual solution. Let the set of vertices guessed this way as not part of the solution be T. Define L = A N G (T) and R = B N G (T). Clearly our guess forces L R to be part of the solution. We have thus reduced this problem to checking if the instance (G V M\M, A, M\M, k M ) has a vertex cover of size at most M\M + k M which contains L and R. 17

18 A-AGVC: 18

19 Running time of AGVC: Lemma 2: A-AGVC can be solved in O (4 k ) time. (will be proven later) AGVC can be solved in O (9 k ) time: For every 0 i k, for every i sized subset Y, for every guess of T, we run the algorithm for A-AGVC with parameter k i. For each i, there are k + 1 subsets of M of size i, and for every i choice of Y of size i, there are 2 k+1 i choices for T and for every choice of T, running the algorithm for A-AGVC given by Lemma 2 takes time O (4 k i ). Hence, the running time of our algorithm for AGVCC is bounded by O k k + 1 i=0 2 k+1 i 4 k i = O (9 k ). i 19

20 Konig Graphs with Extendable Vertex Covers 20

21 Definition 2: Given a graph G = (V, E) and a matching M, we call a path P = v 1,, v t in the graph, an M-alternating path if the edge (v 1, v 2 ) M, every subsequent alternate edge is in M and no other edge of P is in M. An odd length M-alternating path is called an odd M-path and an even length M-alternating path is called an even M-path. Observation 1: In odd M-paths, the last edge of the path is a matched edge, while in even M- paths, the last edge is not a matching edge. 21

22 Definition 3: Given a graph G and a matching M, we define an M-flower as a walk W = v 1,, v b, v b+1,, v t 1, v t with the following properties: o The vertex v t = v b and all other vertices of W are distinct. o The subpaths P 1 = v 1,, v b and P 2 = v 1,, v t 1 are odd M-paths. o The odd cycle C = v b, v b+1,, v t, which has length t b and contains exactly t b edges from M is 2 called the blossom. o The odd M-path P 1 is called the stem of the flower, the vertex v b is called the base and v 1 is called the root. 22

23 Examples: 23

24 Lemma 3: Let (G = A, B, E, M, L, R, k) be an instance of A-AGVC. a) There cannot be an odd M-path from A to A. b) Any odd M-path from B to B has to contain exactly one edge between two vertices in A. c) There cannot be an R M-flower with its base in B. d) Let P be an R M-flower with base v and let the neighbors of v in the blossom be u 1 and u 2. Then, u 1 B or u 2 B. e) Let P = v 1,, v t be an odd M-path in G and suppose S is a minimum vertex cover of G. If v 1 S, then v t S. f) Let P = v 1,, v t be an odd M-path from B to B. Then there is an edge (u, v) such that u, v A and there is an odd M-path P 1 from v 1 to u and an odd M-path P 2 from v t to v and P 1 and P 2 are subpaths of P. g) Consider a vertex v and an even M-path P from some vertex u to v. Then P does not contain the matching partner of v. 24

25 Lemma 4: Given an instance (G = A, B, E, M, L, R, k) of A-AGVC, G has a minimum vertex cover S such that L R S if and only if there is neither an odd M-path from L R to L R, nor an R M-flower. Lemma 5: Given an instance (G = A, B, E, M, L, R, k) of A-AGVC, if G has a minimum vertex cover containing L R, then one such minimum vertex cover can be computed in polynomial time. 25

26 Important Separators and the Algorithm 26

27 Important Separators and the Algorithm: The overall idea of our algorithm is: we use important separators to eliminate odd M-paths. when no such path exists, we find an edge and recursively try to solve the problem by including one of the end-points in our potential vertex cover. 27

28 Definitions: Given a directed graph D = (V, A), consider a set X V. We denote by δ G + (X), the set of arcs going out of X in D. We define a function f: 2 V N where f X = δ + G (X). It is easy to verify that the function f is submodular. 28

29 Important Separators: Let X, Y V be two disjoint vertex sets: A set S A is called an X Y separator or an arc separator if no vertex in Y is reachable from any vertex in X in D\S. We call S a minimal separator if no proper subset of S is an X Y separator and denote by K X,S the set of vertices reachable from X in the graph D\S. We let λ D (X, Y) denote the size of the smallest X Y separator in D. 29

30 Definition 4. Let X, Y V be two disjoint vertex sets of D and let S, S 1 V be two X Y separators. We say that S 1 dominates S with respect to X if S 1 = S and K X,S K X,S1. We call S an important X Y separator if it is minimal and there is no X Y separator that dominates S with respect to X. 30

31 Observations and Lemmas: Lemma 6: Let D = (V, A) be a directed graph where V = n, X, Y V be two disjoint vertex sets and S be an important X Y separator. 1. For every e = (u, v) S, S\{e} is an important X Y separator in the graph D\{e}. 2. If S is an X Y arc separator for some X X such that X is reachable from X in D X where D X is the subgraph of D induced on the vertices of X, then S is also an important X Y separator. 3. There is a unique important X Y separator S of size λ(x, Y) and it can be found in polynomial time. Furthermore, K S K S. 31

32 Given an instance of A-AGVC we define a directed graph D(G) corresponding to this instance as follows: Remove all the edges in G[A], orient all the edges of M from A to B and all the other edges from B to A. 32

33 Observations and Lemmas: Observation 2: There is a path from L to R in D(G) if and only if there is an odd M-path from L to R in G. Lemma 7: Given an instance (G = A, B, E, M, L, R, k), any important L R separator in D(G) comprises precisely arcs corresponding to some subset of M. Lemma 8: Let (G = A, B, E, M, L, R, k) be an instance of A-AGVC and let D = D G = (V, A) be defined as above. Then the number of important L R separators of size at most k is bounded by 4 k and these can be enumerated in time O (4 k ). 33

34 Observations and Lemmas: Lemma 9: Let (G = A, B, E, M, L, R, k) be an instance of A-AGVC. If (G = A, B, E, M, L, R, k) is a Yes instance, then it has a solution S which contains an important L R separator in D G. Lemma 10: Let (G = A, B, E, M, L, R, k) be an instance of A-AGVC such that there are no odd M-paths from L to R in G. If there is either an odd M-path P from R to R or an R M-flower P, then there is an edge (u, v) such that u, v A\L and there is an odd M-path from u to R and an odd M-path from v to R. Moreover, this edge can be found in polynomial time. 34

35 The Algorithm: (A-AGVC) 35

36 The Algorithm: If there is an odd M-path from L to R in G Lemma 9 we know that if there is a solution, there is one which contains an important L R separator in D(G). Hence we branch on a choice of an important L R separator. 36

37 The Algorithm: The size of the minimum L R separator in D(G) is a lower bound on the solution size. 37

38 The Algorithm: If there is an important separator: Compute an important separator with the smallest size, remove an arc from it, and try to solve recursively without it 38

39 The Algorithm: If there are no odd M-paths from L to R, but there is either an odd M-path from R to R or an R M-flower, Lemma 10 we get an edge (u, v) between two vertices in A and guess the vertex which covers this edge and continue. 39

40 The Algorithm: If neither of these two cases occur, then Lemma 4 the graph has a minimum vertex cover containing L R. Such a minimum vertex cover will not contain both end points of any edge of the perfect matching and hence the algorithm returns the empty set. 40

41 Running Time: (Proof of Lemma 2) We define the search tree T(G, M, L, R, k) resulting from a call to Solve-AAGV C (G, M, L, R, k) inductively as follows. The tree T(G, M, L, R, k) is a rooted tree whose root node corresponds to the instance (G, M, L, R, k). If Solve-AAGV C (G, M, L, R, k) does not make a recursive call, then (G, M, L, R, k) is said to be the only node of this tree. If it does make recursive calls, then the children of (G, M, L, R, k) correspond to the instances given as input to the recursive calls made inside the current procedure call. The subtree rooted at a child node (G, M, L, R, k ) is the search tree T(G, M, L, R, k ). 41

42 Running Time: (Proof of Lemma 2) We ll define μ I = 2k λ D G (L, R). Given an instance I = (G, M, L, R, k), we prove by induction on μ I that the number of leaves of the tree T(I) is bounded by max 2 2μ(I), 1. In the base case, if μ I < k, then λ L, R > k in which case the number of leaves is 1. Assume that μ I = k and our claim holds for all instances I such that μ I < μ I. 42

43 Running Time: (Proof of Lemma 2) Suppose λ L, R = 0. In this case, the children I 1 and I 2 of this node correspond to the recursive calls made in Steps 6 and 8. By Lemma 10 there are odd M-paths from u to R and from v to R. Hence, λ L {u}, R > 0 and λ L {v}, R > 0. This implies that μ I 1, μ I 2 <μ I. By the induction hypothesis, the number of leaves in the search trees rooted at I 1 and I 2 are at most 2 μ I 1 and 2 μ I 2 respectively. Hence the number of leaves in the search tree rooted at I is at most 2 2 μ I 1 = 2 μ(i). 43

44 Running Time: (Proof of Lemma 2) Suppose λ L, R > 0. In this case, the children I 1 and I 2 of this node correspond to the recursive calls made in Steps 15 and 17. But in these two cases, as seen in the proof of Lemma 8, μ I 1, μ I 2 <μ I and hence applying induction hypothesis on the two child nodes and summing up the number of leaves in the sub trees rootes at each, we can bound the number of leaves in the sub tree ofi by 2 μ I. 44

45 Running Time: (Proof of Lemma 2) Hence the number of leaves of the search tree T rooted at the input instance I = (G, M, L, R, k) is 2 μ(i) 2 2k = 4 k. The time spent at a node I is bounded by the time required to compute the unique smallest L R separator in D(G) which is polynomial(lemma 6). Along any path from the root to a leaf, at any internal node, the size of the set L increases or an edge is removed from the graph. Hence the length of any root to leaf path is at most n 2. Therefore the running time of the algorithm is O (4 k ). 45

46 Conclusion 46

47 Conclusion In this paper we obtained a faster FPT algorithm for AGVC through a structural characterization of Konig graphs in which a minimum vertex cover is forced to contain some vertices. This also led to faster FPT algorithms for Almost 2SAT, Almost 2 SAT(Variable) and Konig Vertex Deletion. 47

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