COMP 260: Advanced Algorithms
Tufts University
Prof. Lenore Cowen
Spring 2016
HW 2: due Monday, March 7
1. A certain commodity is produced at two factories x1 and x2 . The commodity is to be shipped to markets y1 , y2 and y3 through the network
shown below. What is the maximum ammount that can be shipped
from the factories to the markets?
y1
18
x1
4
7
5
a
3
e
24
8
7
16
y2
13
12
24
c
2
7
22
2
b
19
d
15
6
x2
4
y3
2. Let G be a directed graph with source s and sink t. Suppose the
capacities are specified not on the edges of G but on the vertices (other
than s, t); for each vertex there is a fixed limit on the total flow through
it. There is no restriction on flow through the edges. Show how to use
the ordinary network flow theory to determine the maximum capacity
of a feasible flow from s to t in the vertex-capacitated graph G.
3. A vertex cover of a graph is a set of vertices C, such that every edge
has at least one endpoint in C. Use network flows to prove the KóningEgeváry theorem, i.e. if G is bipartite, then the size of the maximal
matching is equal to the size of the minimum vertex cover.
4. Consider the s-t path version of the metric TSP (where we need to
find a minimum length tour that starts at s and ends at t. Show that
the natural variant of Christofides algorithm for the cycle version of
the metric TSP produces a 5/3*OPT approximation algorithm for this
problem.
5. The fault-tolerant version of the k-center problem with triangle inequality has an additional input α ≤ k which specifies the number of centers
that each vertex must be connected to. In other words, we assume that
up to α − 1 centers might be closed, and so the fault-tolerant cost for
a vertex is its distance to its αth closest center. The problem is to
pick k centers so that the maximum fault-tolerant cost of a vertex is
minimized. A set S ⊆ V in an undirected graph H = (V, E) is an αdominating set if each vertex v ∈ V is adjacent to at least α vertices in
S (we consider a vertex to be adjacent to itself). Let domα (H) denote
the size of a minimum cardinality α-dominating set in H.
(a) Let I be an independent set in H 2 . Show that α|I| ≤ domα (H).
(b) Give a factor 3 approximation algorithm for the fault-tolerant kcenter problem (Hint: Compute a maximal independent set Mi in
G2i , for 1 ≤ i ≤ m. Find the smallest index i such that |Mi | ≤ b αk c,
and moreover, the degree of each vertex of Mi in Gi is ≥ α − 1.)