Chapter
Set cover and its application to
shortest sup erstring
Understanding the area of approximation algorithms involves identifying cornerstone problems
problems whose study leads to discovering techniques that b ecome general principles in the area
and problems that are general enough that other problems can b e reduced to them Problems such
as matching maximum ow shortest path and minimum spanning tree are cornerstone problems
in the design of exact algorithms In approximation algorithms the picture is less clear at present
Even so problems such as minimum set cover and minimum Steiner tree can already b e said to
o ccupy this p osition
In this chapter we will rst analyse a natural greedy algorithm for the minimum set cover
problem We will then show an unexp ected use of set cover to solve the minimum sup erstring
problem An algorithm with a much b etter approximation guarantee will b e presented in Chapter
for the latter problem the p oint here is to illustrate the wide applicabili ty of the set cover problem
Minimum set cover
Problem Minimum set cover Given a universe U of n elements and a collection of
subsets of U S S with non negative costs sp eci ed the minimum set cover problem asks for
1 k
a minimum cost collection of sets whose union is U
Perhaps the rst algorithm that comes to mind for this problem is one based on the greedy
strategy of iteratively picking the most cost e ective set and removing the covered elements until
all elements are covered Let C b e the set of elements already covered at the b eginning of an
iteration During this iteration de ne the cost e ectiveness of a set S to b e the average cost at
cost(S )
De ne the price of an element to b e the average cost at which it covers new elements i e
jS C j
which it is covered Equivalently when a set S is picked we can think of its cost b eing distributed
equally among the new elements covered to set their prices
Set cover and its application to shortest superstring
Algorithm Greedy set cover algorithm
C
while C U do
Find the most cost e ective set in the current iteration say S
cost(S )
i e the cost e ectiveness of S Let
jS C j
Pick S and for each e S C price e
Output the picked sets
Numb er the elements of U in the order in which they were covered by the algorithm resolving
ties arbitrarily Let e e b e this numb ering
1 n
OPT
Lemma For each k f ng price e
k
n k +1
Pro of In any iteration the left over sets of the optimal solution can cover the remaining
elements at a cost of at most OPT Therefore there must b e a set having cost e ectiveness at most
OPT
In the iteration in which element e was covered C contained at least n k elements
k
jC j
Since e was covered by the most cost e ective set in this iteration it follows that
k
OPT OPT
price e
k
n k
jC j
2
From Lemma we immediately obtain
Theorem The greedy algorithm is an H factor approximation algorithm for the minimum set
n
1 1
cover problem where H
n
2 n
Pro of Since the cost of each set picked is distributed among the new elements covered the
P
n
total cost of the set cover picked is equal to price e By Lemma this is at most
k
k =1
1 1
OPT 2
2 n
Example Following is a tight example
... 1+ε
1/n 1/(n-1) 1
The greedy algorithm outputs the cover consisting of the n singleton sets since in each iteration
some singleton is the most cost e ective set So the algorithm outputs a cover of cost
H
n
n n
On the other hand the optimal cover has a cost of 2
Surprisingly enough the obvious algorithm given ab ove is essentially the b est one can hop e for
for the minimum set cover problem it is known that an approximation guarantee b etter than ln n
is not p ossible assuming P NP
In Chapter we p ointed out that nding a go o d lower b ound on OPT is a basic starting p oint
in the design of an approximation algorithm for a minimization problem At this p oint the reader
may b e wondering whether there is any truth to this claim We will show in Chapter that the
correct way to view the greedy set cover algorithm is in the setting of LP duality theory this
will not only provide the lower b ound on which this algorithm is based but will also help obtain
algorithms for several generalizations of this problem
Exercise The maximum coverage problem is the following Given a universe U of n elements
with non negative weights sp eci ed a collection of subsets of U S S and an integer k pick k
1 l
sets so as to maximize the weight of elements covered Show that the obvious algorithm of greedily
picking the b est set in each iteration until k sets are picked achieves an approximation factor of
k
k e
Solving shortest sup erstring via set cover
Let us motivate the shortest sup erstring problem The human DNA can b e viewed as a very
long string over a four letter alphab et Scientists are attempting to decipher this string Since it
is very long several overlapping short segments of this string are rst sequenced Of course the
lo cations of these segments on the original DNA are not known It is hyp othesised that the shortest
string which contains these segments as substrings is a go o d approximation to the original DNA
string
Problem Shortest sup erstring Given a nite alphab et and a set of n strings
S fs s g nd a shortest string s that contains each s as a substring Without loss
1 n i
of generality we may assume that no string s is a substring of another string s j i
i j
This problem is NP hard Perhaps the rst algorithm that comes to mind for nding a short
sup erstring is the following greedy algorithm De ne the overlap of two strings s t as the
maximum length of a su x of s that is also a pre x of t The algorithm maintains a set of strings
T initially T S At each step the algorithm selects from T two strings that have maximum
overlap and replaces them with the string obtained by overlapping them as much as p ossible After
n steps T will contain a single string Clearly this string contains each s as a substring
i
This algorithm is conjectured to have an approximation factor of To see that the approximation
k k
factor of this algorithm is no b etter than consider an input consisting of strings ab b c and
k +1
b If the rst two strings are selected in the rst iteration the greedy algorithm pro duces the
k k +1 k +1
string ab cb This is almost twice as long as the shortest sup erstring ab c
We will obtain a H factor approximation algorithm by a reduction to the minimum set cover
n
problem The set cover instance denoted by S is constructed as follows For s s S and k
i j
if the last k symb ols of s are the same as the rst k symb ols of s let b e the string obtained
i j ij k
by overlapping these k p ositions of s and s Let I contain the strings for all valid choices of
i j ij k
i j k The set cover instance S consists of S as the universal set The sp eci ed subsets of S are
for each string S I de ne set fsj s S s is a substring of g The cost of this set is