Spring 2010 CMSC 858Y: Homework 1 Samir Khuller
Due in class: No official due date. Complete it before Mar 1..
(1) A maximal matching is a matching Msuch that no edges can be added to the matching,
without destroying the property that it is a matching. In other words, for any edge e,e∪Mis
not a matching. First give a linear time algorithm for finding a maximal matching in a graph.
Then prove that the size of the maximal matching is at least half the size of a maximum
matching.
(2) Consider a set of costs cij ≥0 on the edges of a complete graph Knwith nnodes, and let
S⊆ {v1, . . . , vn}. We wish to find a subgraph of Knthat has odd degree at nodes in S, an
even degree (possibly zero) at all other nodes, and as low cost as possible. Show that this is a
minimum weight matching problem. (Min weight matching can be solved in polynomial time
in graphs that are not necessarily bipartite.)
(3) We are given a connected graph G= (V, E) and a cost function c:E→Z+. We wish to
find a walk, traversing each edge of Eat least once, such that the total cost of the walk (with
multiple traversals of an edge charged multiply) is as small as possible. Give a polynomial
time algorithm for this problem. (You may assume a solution to problem (2) even if you did
not solve problem (2).)
(4) The b−matching problem is the following: given a bipartite graph B= (V, U, E) and a
function b:V∪U→Z+, is there a subset M⊆Esuch that for each v∈V∪U,vis incident
upon exactly b(v) edges in M? How would one design a strongly polynomial algorithm for this
problem? (An algorithm is strongly polynomial if its running time depends only on the input
size (i.e., number of vertices or edges) and not on the particular numbers/weights. In this case
the running time should not depend on the bvalues.)
(5) Let G= (X, Y , E) be a bipartite graph. Assume for simplicity that |X|=|Y|.
A subset S⊆(X∪Y) is called a vertex cover if for each edge e= (x,y ) in Eeither x∈Sor
y∈S(or both).
Let M∗be a maximum matching in G.
(a) Prove that if Sis a vertex cover then |S| ≥ |M∗|.
(b) Give a polynomial time algorithm to find the minimum cardinality vertex cover. Give a
proof for its correctness.
(6) Design a polynomial time algorithm for the following problem. You have a directed network,
with two special vertices sand t. Each edge ehas a positive capacity c(e) and a positive
delay d(e). The time to ship σunits of data from sto ton path Pis Pe∈Pd(e) + σ
Cwhere
C= mine∈Pc(e). Design a polynomial time algorithm to find the path that will minimize the
shipping time? You should also give an argument that shows why your method works.
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