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 Multiterminal Cut (MTC) 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 a set of k edges (respectively, non-terminal vertices), whose removal from G separates each terminal from all the others. Related Problems Multi-Way Cut: to separate the graph into at least l components. Multicut: to separate l pairs of vertices.

4 History Results (Approximation ratio)Authors (2-2/l) for EMTCDahlhaus et al. (STOC 92) 2 for VMTCGarg et al. (ICALP 94) 2 for directed versionNaor & Zosin (FOCS 97) (1.5-1/l) for EMTCCalinescu et al. (STOC 98) 1.34 for EMTCKarger et al. (STOC 99) Approximation algorithms: l=2: the classical minimum (s,t) cut problem. l>2: MTC is NP-hard. (Dahlhaus et al. 1992) NP-hardness of MTC:

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

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 parameterk 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 Farthest minimum isolating cut 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. A minimum isolating cut for terminal t i is a minimum cut that separates t i away from all other terminals T -i. A Minimum isolating cut C i for terminal t i separates the graph into two components: one that contains t i is called the residual of C i and denoted by R i ; the other one contains T -i. The farthest minimum isolating cut for terminal t i is the unique minimum isolating cut that makes the residual of the maximum cardinality. Minimum isolating cuts Farthest minimum isolating cut

8 A Structural Property Lemma: Let C i be the (farthest) minimum isolating cut for terminal t i in G, and G’ be the graph after merging R i into a new terminal t i. Then any minimum multiterminal cut in G’ is a minimum multiterminal cut in G. This lemma holds for both edge and vertex version. CiCi G RiRi G’

9 Rule 1: For each terminal t i, let C i and R i be its farthest minimum isolating cut and the corresponding residual, then we can contract R i in the graph to form terminal t i. Edge Multiterminal Cut Data reduction rules: 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 Proof: Let S be a minimum multiterminal cut and the minimal isolating cut for t i. We have Edge Multiterminal Cut Lemma: is a 2-approximation solution. Rule 4: Let C i be a minimum isolating cut for terminal t i. If, then there is no multiterminal cut with size We are ready to design our algorithm now.

11 Edge Multiterminal Cut Main steps of our recursive algorithm: 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-eG*e …… G*e is the graph obtained by shrinking e in G. 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 We will use a control value to build up a recurrence relation. Edge Multiterminal Cut Analysis of our algorithm 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 G G-eG*e …… is the size of the tree. Lemma: Edge Multiterminal Cut can be solved in time. Corollary: Edge 3-Terminal Cut can be solved in time. Previous result

14 Rule 1: For each terminal t i, let C i and R i be its farthest minimum isolating cut and the corresponding residual, then we can contract R i in the graph to form terminal t i. Vertex Multiterminal Cut Do data reduction rules still hold? 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 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.

15 Vertex Multiterminal Cut Rule 4: Let C i be a minimum isolating cut for terminal t i. If, then there is no multiterminal cut with size Edge version Vertex 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 Vertex Multiterminal Cut The algorithm is almost the same as the algorithm for EMTC. 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. 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 Analysis

17 Vertex Multiterminal Cut We can verify that when l=3,4,5,6, and respectively satisfy (1) and (2). Now we get two relations: (1) (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 The jth layer farthest minimum isolating cut The first layer farthest minimum isolating cut for t i is just the farthest minimum isolating cut for t i. The jth layer farthest minimum isolating cut is the farthest minimum for t i ’, where t i ’ is formed by merge and together. An Alternative Algorithm for VMTC Obviously, Let b be the smallest number such that does not exist or, and 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 Since we get Recursive algorithm: Recursive step: Branching on B by including each vertex in B into the solution. An Alternative Algorithm for VMTC where C(k) is the size of the search tree when our algorithm finds a solution of size≤k. Analysis: Then 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 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. Multicut Objective: to separate l pairs { s i, t i } of vertices (terminals). Measure: the cardinality of the deletion set (solution size not greater then k). G By using our results on MTC, we can also improve previously known results on Mulicut.