Master 1 Informatique ISTIC
Module AA
2016
Advanced Algorithmics
Exercises on Approximation
Algorithms
Inapproximability
Exercise 1 (Inapproximability of TSP)
We recall
Travelling Salesman (TSP)
Input : A graph G = (V, E), for each e ∈ E, an integer de > 0.
Output : A cycle that passes through every vertex exactly once
with minimal cost.
and its decision problem version
Budget Travelling Salesman (budgetTSP)
Input : A graph G = (V, E), for each e ∈ E, an integer de > 0, and a budget b ≥ 0.
Output : Does there exist a cycle that passes through every vertex exactly once whose cost
is at most b?
We let G = (V, E) be an input of TSP, we write opt(G) the value of an optimal tour1 .
1) For a start, we suppose that G is such that the triangular inequality for the distances de
holds2 , and aim at designing a 2-approximation algorithm.
1. With such an assumption on G, show that for any minimum spanning tree MST
of G, we have
opt(G) ≥ cost(MST)
2. Exploit the Euclidean assumption to build a tour from a (efficient) minimum spanning tree computation, and show it is a 2-aproximation (Hint: see Figures 1 and 2).
1
2
We take the convention that opt(G) = +∞ if G has no Hamiltonian cycle
that is the Euclidean TSP problem.
Advanced Algorithmics
2
Exercises on Approximation Algorithms
Wichita
Albuquerque
Wichita
Tulsa
Amarillo
Albuquerque 8
Little
Rock
11
9
Amarillo
5
6
7
Tulsa
12
Little
Rock
14
10
Dallas
3
Dallas
15
El Paso
El Paso
4
13
16
Houston
2
Houston
1
San Antonio
San Antonio
Figure 1: TSP and MST.
Wichita
Tulsa
Albuquerque
Little
Rock
Amarillo
Dallas
El Paso
Houston
San Antonio
Figure 2: Exploit the triangular inequality.
2) We suppose that G is arbitrary, and we write n its mumber of vertices.
We recall the classic Hamiltonian cycle decision problem:
Hamiltonian cycle (HAMCYCL)
Input : A graph G = (V, E).
Output : Is there a cycle that visits each vertex exactely once?
1. Given an arbitrary instance G = (V, E) of the Hamiltonian Cycle and C ∈ IN,
I(G, C) of TSP as the very graph G with the following distances: d(u,v) = 1 if
(u, v) ∈ E, and 1 + C otherwise.
Justify the following two assertions (recall that n is the number of vertices):
(a) If G has an Hamiltonian cycle, then opt(I(G, C)) = n, the number of vertices
in G.
(b) If G has no Hamiltonian cycle, then opt(I(G, C)) ≥ n + C.
Also check that the distances of the graph I(G, 1) (case where C = 1) verify the
triangular inequality.
2. From the reduction above and its properties, we show that an efficient solution for
HAMCYCL can be deduce from an approximation algorithm for TSP.
Advanced Algorithmics
Exercises on Approximation Algorithms
3
Assume there exists a (polynomial-time) approximation algorithm A for TSP with
ratio ρ. Use the reduction above to design a polynomial algorithm that answers
the decision problem HAMCYCL.
(Hint: C must be chosen big because of the Remark above, try out C = nρ).
3. What can you conclude regarding the existence of an approximation algorithm for
TSP in the general case?
Exercise 2 (Minimum Vertex Cover and Maximal Matching)
Vertex Cover (VC)
Input : An undirected graph G = (V, E) and k > 0.
Output : Does there exist a subset of vertices S ⊆ V with |S| ≤ k
that touches every edge.
The Minimum Vertex Cover problem is to minimize the size of the vertex cover, we shall write
opt the value of its cardinal.
Maximal matchings will help us find good vertex covers. We consider an instance G = (V, E)
of VC.
1. Define a matching of G as a subset M ⊆ E of edges that have no vertices in common. A
matching is maximal if no more edges can be added to it while remaining a matching.
2. Show that maximal matching are easy to generate.
3. Show that any matching M ⊆ E provides a lower bound on opt, the optimal value for
the minimum vertex cover problem.
4. Show how to build a vertex cover from a maximal matching. What is its cardinal?
Deduce an upper bound for opt.
5. Use all previous questions to design an approximation algorithm for the minimal vertex
cover problem. What is the ratio ρ?
6. Show that this ratio is optimal, in the sense that there are arbitrarily big instances
where your algorithm computes a vertex cover computed whose size is ρopt.