Simple and Improved Parameterized Algorithms for Multiterminal Cuts

1. Simple and Improved Parameterized Algorithms for Multiterminal Cuts Mingyu Xiao The Chinese University of Hong Kong Hong Kong SAR, CHINA CSR 2008 Presentation, Moscow, Russia, June 2008

2. Outline • Problems— Definitions of Multiterminal Cuts • History— Previous results and our results • Methodology — Parameterized algorithm • Important structural results— Farthest minimum isolating cut and others • Edge Multiterminal Cut— An simple algorithm • Vertex Multiterminal Cut—Two algorithms

3. Edge (Vertex) Multiterminal Cut:Given an unweighted graph G=(V,E) and a subset of l terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find aset of k edges (respectively, non-terminal vertices), whose removal fromG separates each terminal from all the others. Multiterminal Cut (MTC) Related Problems Multi-Way Cut: to separate the graph into at least l components.Multicut: to separate l pairs of vertices.

4. History NP-hardness of MTC: l=2: the classical minimum (s,t) cut problem.l>2: MTC is NP-hard. (Dahlhaus et al. 1992) Approximation algorithms:

5. Results Authors for EMTC in planar Dahlhaus et al. (STOC 92) for EMTC in planar Hartvigsen. (D.A.M. 98) for EMTC in planar Yeh (J. ALG 01) for EMTC in a special case Chen&Wu (Algorithmica 03) for VMTC Marx (TCS 06) for VMTC Chen et al. (Algorithmica, to appear) Our results in this paper: for EMTC (To be exact, ) and for VMTC and for Vertex {3,4,5,6}-TC where T(n,m) is the running time for finding a max flow in an unweighted graph. History Exact algorithms:

6. Parameterized Algorithm What is parameterized algorithm? • Exact algorithm. • The exponential part of the running time is only related to one or more parameters, but not the input size. k is the parameter k and l are the parameters Some (parameterized) problems are unlike to have any parameterized algorithms, such as the k-clique problem with parameter k. Those kinds of problems are called W[1]-hard in Parameterized Complexity. Readers are referred to “Parameterized Complexity” by Downey and Fellow for more details about parameterized algorithms.

7. Techniques All of our algorithms are based on a simple technique:Branching at an edge (a vertex) in a farthest minimum isolating cut: including it in the solution or excluding it from the solution. Farthest minimum isolating cut A minimum isolating cut for terminal ti is a minimum cut that separates ti away from all other terminals T-i. A Minimum isolating cut Ci for terminal ti separates the graph into two components: one that contains ti is called the residual of Ci and denoted by Ri; the other one contains T-i. The farthestminimum isolating cut for terminal ti is the unique minimum isolating cut that makes the residual of the maximum cardinality. Minimum isolating cuts Farthest minimum isolating cut

8. Ci Ri G’ G A Structural Property Lemma: Let Ci be the (farthest) minimum isolating cut for terminal ti in G, and G’ be the graph after merging Ri into a new terminal ti. Then any minimum multiterminal cut in G’ is a minimum multiterminal cut in G. This lemma holds for both edge and vertex version.

9. Edge Multiterminal Cut Data reduction rules: Rule 1: For each terminal ti, let Ciand Ribe its farthest minimum isolating cut and the corresponding residual, then we can contract Ri in the graph to form terminal ti. Rule 2: We can remove all the edges that connect two terminals from the graph and put them into the solution. Rule 3: If , then is a multiterminal cut with size at most k, where satisfying A solution

10. Lemma: is a 2-approximation solution. Rule 4: Let Cibe a minimum isolating cut for terminal ti. If , then there is no multiterminal cut with size Edge Multiterminal Cut Proof: Let S be a minimum multiterminal cut and the minimal isolating cut for ti. We have We are ready to design our algorithm now.

11. Step 1: applying the 4 reduction rules to reduce the input size.Step 2: Let , branching at an edge e in B by including it in the solution or excluding it from the solution. G G-e G*e …… …… …… G*e is the graph obtained by shrinking e in G. Edge Multiterminal Cut Main steps of our recursive algorithm: Does this simple algorithm work efficiently? How to analyze the running time?

12. Note: The notation system is different. Here l denotes the solution size.

13. G G-e G*e …… …… …… is the size of the tree. Previous result Lemma: Edge Multiterminal Cut can be solved in time. Corollary: Edge 3-Terminal Cut can be solved in time. Edge Multiterminal Cut Analysis of our algorithm We will use a control value to build up a recurrence relation. It is easy to see that in Step 1 (applying reduction rules), p will not increase. We can further prove that in each branch of Step 2, p decreases by at least 1. Then we get It is easy to see that satisfies it. If , we will find a solution when applying Rule 3. Else we have

14. Vertex Multiterminal Cut Do data reduction rules still hold? Rule 1: For each terminal ti, let Ciand Ribe its farthest minimum isolating cut and the corresponding residual, then we can contract Ri in the graph to form terminal ti. Rule 2’: There is no solution if one terminal is adjacent to another terminal. We can remove all the vertices that are common neighbors of two terminals and put them into the solution. Rule 2: We can remove all the edges that connect two terminals from the graph and put them into the solution. Rule 3: If , then is a multiterminal cut with size at most k, where satisfying

15. Vertex Multiterminal Cut Rule 4: Let Cibe a minimum isolating cut for terminal ti. If , then there is no multiterminal cut with size Vertex version Edge version EMTC: every edge in S will appear in exactly two isolating cuts.VMTC: a vertex in S will appear in up to l isolating cuts.

16. Step 1: applying the 4 reduction rules to reduce the input size.Step 2: Let , branching at a vertex v in B by including it in the solution or excluding it from the solution. Vertex Multiterminal Cut The algorithm is almost the same as the algorithm for EMTC. Analysis Let be the control value. In Step 1, p will not increase. In Step 2, when v is included into the solution, p will decrease by l-1; when v is excluded from the solution, p will decrease by 1. We get recurrence relation If , we will find a solution when applying Rule 3. Else we have

17. Vertex Multiterminal Cut Now we get two relations: (1) (2) We can verify that when l=3,4,5,6, and respectively satisfy (1) and (2). Lemma: Vertex {3,4,5,6}-Terminal Cut can be solved in and time respectively. Furthermore, we can prove that Lemma: Vertex Multiterminal Cut can be solved in time. The exponential part is related to l and k.

18. Obviously, Let b be the smallest number such that does not exist or , and An Alternative Algorithm for VMTC The jth layer farthest minimum isolating cut The first layer farthest minimum isolating cut for ti is just the farthest minimum isolating cut for ti. The jth layer farthest minimum isolating cut is the farthest minimum for ti’, where ti’ is formed by merge and together. Claim: If there is a solution (a multiterminal cut with size ≤k), then at least one vertex in B is contained in a solution.

19. An Alternative Algorithm for VMTC Recursive algorithm: Recursive step: Branching on B by including each vertex in B into the solution. Analysis: Since we get Then where C(k) is the size of the search tree when our algorithm finds a solution of size≤k. To compute B, we need at most b<k farthest minimum isolating cut computations. Lemma: Vertex Multiterminal Cut can be solved in time. The exponential part is only related to k.

20. G Multicut • Objective: to separate l pairs {si, ti} of vertices (terminals). • Measure: the cardinality of the deletion set (solution size not greater then k). We present a simple reduction from Multicut to Multierminal Cut:For each instance of Multicut, we can reduce it to at most instances of Muliterminal Cut with at most terminals. By using our results on MTC, we can also improve previously known results on Mulicut.