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日期:2022-06-03 04:10

1213-1 COMPSCI 720 (21/06/2021 17:30) Adv Design and Analysis of Alg (Exam)

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1213-1 COMPSCI 720 (21/06/2021 17:30) Adv Design and Analysis of Alg (Exam)

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Maximum marks: 60

Question 1

Attached

1. Combinatorial Objects.

To test your understanding about the bijection between labeled trees and Pru¨fer sequences, for

this question submit a solution (program) for the following problem to the automarker at: http:

//www.automarker.cs.auckland.ac.nz.

The input will be a single line of 1 < n ≤ 50000 (white-space separated) sequence of integers

S = s0 s1 · · · sn?1, where 0 ≤ si < n and si 6= i. This denotes a labeled tree of n nodes (labeled

0 to n? 1) that is constructed as follows. The integer si in position 0 ≤ i < n denotes the label of

an adjacent node of node i (that is, {si, i} is a tree edge). Note that since a tree has only n?1 edges

one of these edges will be duplicated on the input line, as {si = x, i = y)} = {sj = y, j = x}.

You can assume the edge connections generate a connected acyclic graph (a tree).

The output of your program should be one line of n? 2 integers denoting its corresponding Pru¨fer

sequence over {0, 1, . . . , n? 1}. There will be ten (10) input cases, each worth one mark.

Sample Input:

2. Treewidth and Pathwidth.

(a) For the following graph compute its pathwidth and treewidth and give a path decomposition

and a tree decomposition that corresponds to the smallest width(s).

(6 marks)

(b) Show that you can use a DFS (Depth-First-Search) of a graph to find an approximation tree-

decomposition of width at most the distance any back-edge reaches up the tree (e.g. its

treewidth is bounded above by the longest cycle length detected).

(4 marks)

2

3. t-parse Algorithms.

For this question we assume we have the simple variation of the Weighted Independent Set problem

that was a part of your Assignment 2.

Problem: WEIGHTED INDEPENDENT SET (WIS)

Input: (undirected) Graph G = (V,E) with vertices labeled 0, 2, . . . , |V | ? 1.

Question: Let the weight of each vertex be its graph index (label).

What is the largest sum of vertex weights over all independent sets of vertices?

That is, maxV ′?V

v∈V ′ int(v) where for all {v1, v2} ? V ′ we have (v1, v2) 6∈ E.

(a) For the following two t-parses, give the corresponding graph adjacency lists, as we interpreted

them in class.

2 ( 10 20 0 10 21 1 21 10 )

2 ( ( 10 20 0 10 21 1 21 10 ) ( 20 0 20 0 10 0 ) 1 10 21 )

(3 marks)

(b) What is the answer to the WIS problem on the two graphs given in part (a)?

(2 marks)

(c) Assume we have a pathwidth t-parse of a graph G. Describe a linear-time algorithm that can

solve the WIS problem.

(3 marks)

(d) Assume we have a treewidth t-parse of a graph G. Show how one can extend your algorithm

of part (c) for the WIS problem.

(2 marks)

3

4. Parameterized Algorithms.

(a) In your own words, describe what it means for a reduction rule to be safe.

(2 marks)

(b) Let G be a graph with vertex set V . Let H and I be subsets of V with H ∩ I = ?, I is an

independent set of G, and H precisely contains all neighbors of vertices in I . Assume that

|I| < |H|. Is it possible for the vertices in H ∪ I to form a crown of G? Explain your answer.

(2 marks)

(c) Recall the INDEPENDENT SET problem which is formally stated as follows.

Instance. A graph G = (V,E) and a non-negative integer k.

Question. Is there a subset S ? V with |S| ≥ k such that, for all pairs of two distinct ele-

ments u and v in S, we have that {u, v} is not an edge in E?

Is INDEPENDENT SET restricted to graphs for which the maximum degree d of a vertex is a

constant fixed-parameter tractable when parameterized by k? Explain your answer.

(3 marks)

(d) Consider the TRIANGLE VERTEX DELETION problem which is formally stated as follows.

Instance. A graph G = (V,E) and a non-negative integer k.

Question. Are there k vertices in V whose deletion from G results in a graph that has no

cycle of length three?

Show that TRIANGLE VERTEX DELETION is fixed-parameter tractable when parameterized

by k. To this end, describe a depth-bounded search tree algorithm to solve any given instance

of TRIANGLE VERTEX DELETION and give the algorithm’s running time.

(3 marks)

4

5. Approximation Algorithms.

(a) Is the 2-approximation algorithm for the (unweighted) VERTEX COVER problem that we

discussed in class tight? Explain your answer.

(2 marks)

(b) What does it mean for a problem P to have an r-approximation with r = 2?

(2 mark)

(c) Consider the 3-MATCHING problem which is formally stated as follows.

Instance. A set S = {s1, s2, . . . , sn} and a set T = {T1, T2, . . . , Tm}, where each element

Ti with i ∈ {1, 2, . . . ,m} is a set that contains exactly 3 distinct elements of S.

Goal. Find a maximum 3-matching, that is a maximum size subset of T that contains each

element of S at most once.

Example. Let S = {1, 2, 3, 4, 5, 6}, and let T1 = {2, 3, 4}, T2 = {1, 2, 3}, and T3 = {4, 5, 6}.

Then a maximum 3-matching for this example is {T2, T3}. There also exists a maximal 3-

matching of size one for the example because {T1} is a matching and neither T2 nor T3 can

be added to it.

i. Give an instance of 3-MATCHING with |S| = 9 and a set T = {T1, T2, . . . , Tm} such

that the instance has a maximum matching of size 3 and a maximal matching of size 1.

You are free to choose m.

(2 marks)

ii. Describe a 13 -approximation algorithm for finding a maximum 3-matching and explain

why your algorithm has approximation ratio 13 .

(4 marks)

5

6. Algorithms for Problems in Computational Biology.

(a) Give two rooted binary phylogenetic X-trees T and T ′ with |X| = 8 such that h(T , T ′) >

drSPR(T , T ′). Explicitly say what h(T , T ′) and drSPR(T , T ′) are for the two trees that you

have chosen.

(2 marks)

(b) Are there two rooted binary phylogeneticX-trees T and T ′ such that h(T , T ′) < drSPR(T , T ′).

If yes, give an example for two such trees. If no, explain why.

(2 marks)

(c) Describe three differences between the two kernelization algorithms to compute h(T , T ′) and

drSPR(T , T ′) for two rooted binary phylogenetic X-trees T and T ′.

(3 marks)

(d) Let T and T ′ be two rooted binary phylogenetic X-trees, and let F be a maximum acyclic

agreement forest for T and T ′. Let Tρ be the element in F that contains the root ρ, and let

L(Tρ) be the leaf set of Tρ. Explain why L(Tρ) ∩X 6= ?.

(3 marks)


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