This document summarizes a research paper about maximizing a nonnegative, monotone, submodular function subject to matching constraints. It shows that this constrained submodular maximization matching (CSM-Matching) problem is NP-hard by reducing the maximum k-cover problem to it. It then presents a 3-approximation greedy algorithm for CSM-Matching. It further reduces CSM-Matching to the problem of maximizing a submodular function over two matroids, for which there is a (2+ε)-approximation algorithm called LSV2. Using LSV2 as a black box, the document shows how to find a (4+ε)-approximate solution to CSM-

1. Maximizing a Nonnegative, Monotone, Submodular Function
Constrained to Matchings
Sagar Kale∗
Department of Computer Science, Dartmouth College,
Hanover, NH 03755, USA
sag@cs.dartmouth.edu
December 31, 2012
Abstract
Submodular functions have many applications. Matchings have many applications. The
bitext word alignment problem can be modeled as the problem of maximizing a nonnegative,
monotone, submodular function constrained to matchings in a complete bipartite graph where
each vertex corresponds to a word in the two input sentences and each edge represents a potential
word-to-word translation. We propose a more general problem of maximizing a nonnegative,
monotone, submodular function defined on the edge set of a complete graph constrained to
matchings; we call this problem the CSM-Matching problem. CSM-Matching also generalizes
the maximum-weight matching problem, which has a polynomial-time algorithm; however, we
show that it is NP-hard to approximate CSM-Matching within a factor of e/(e − 1) by reducing
the max k-cover problem to it. Our main result is a simple, greedy, 3-approximation algorithm
for CSM-Matching. Then we reduce CSM-Matching to maximizing a nonnegative, monotone,
submodular function over two matroids, i.e., CSM-2-Matroids. CSM-2-Matroids has a (2 + ǫ)-
approximation algorithm—called LSV2 . We show that we can find a (4+ǫ)-approximate solution
to CSM-Matching using LSV2 . We extend this approach to similar problems.
1 Introduction
Submodular functions generalize the notion of diminishing returns. A function f : 2E → R is a
submodular function if for all A ⊆ B ⊂ E and† e ∈ E − B,
f (A ∪ {e}) − f (A) f (B ∪ {e}) − f (B) ,
or if for all X, Y ⊆ E,
f (X) + f (Y ) f (X ∪ Y ) + f (X ∩ Y ) .
Both definitions are equivalent. Loosely speaking, adding an element to a smaller subset yields
more increase in the function value than that by adding it to the bigger set, which is the law of
∗
This paper is based on the work done while the author was in Indian Institute of Technology Bombay, Mumbai,
India.
†
We use e to denote the mathematical constant that is the base of the natural logarithm or to denote an element
in the set E (which usually denotes the edge set of a graph). The meaning should be clear from the context.
1

2. diminishing returns. Examples of submodular functions are the cut function in a graph, the rank
function of a matroid, and the coverage function for a collection of subsets of a finite set. Many
problems in areas such as machine learning and natural language processing can be reduced to
optimizing a submodular function subject to some constraints [3, 6]. Inspired by the problem of
optimizing a submodular function over cuts in a graph [2], we propose the following problem: Given
a complete graph G = (V, E) and a nonnegative, monotone, submodular function defined on the
edges of the graph, say f : 2E → R+ , we want to find a matching that has the maximum function
value, i.e., we want to find
argmax f (M ) .
M is a matching
We call this problem the CSM-Matching problem—constrained submodular maximization with
respect to matchings. To the best of our knowledge, nobody has worked on this problem directly.
In Section 2, we show that CSM-Matching is NP-hard, and hence, we seek polynomial-time
approximation algorithms for CSM-Matching. We also give a simple, greedy, 3-approximation
algorithm for CSM-Matching.
In Section 3, we reduce CSM-Matching to submodular maximization over multiple matroids. A
matroid is a pair M = (E, I) such that E is a finite set, I is a collection of subsets of E, and the
following conditions hold:
1. ∅ ∈ I,
2. if I ∈ I and J ⊆ I, then J ∈ I (in other words, I is closed under the subset operation),
3. if I, J ∈ I and |J| < |I|, then there exists an element e ∈ I − J such that J + e ∈ I.
Here, I + e means I ∪ {e}; I − e means I − {e}. The set E is sometimes called the ground set, and
a subset in the collection I is sometimes called an independent set. The problem of nonnegative,
monotone, submodular maximization over multiple matroids has approximation algorithms [5],
which we use for finding an approximate solution to CSM-Matching.
We construct two matroids from the given complete graph and show that the intersection of these
two matroids contains all matchings; we call this construction the 2-Matroids construction. We
convert an approximate solution to submodular maximization over the intersection of 2-Matroids
to a matching without losing its function value by much. Specifically, using an algorithm that we
call LSV2 [5] as a black box, we give a procedure for finding a (4 + ǫ)-approximate solution to
CSM-Matching.
In Section 4, using the 2-Matroids construction and the LSV2 algorithm, we give a (4 + ǫ)-
approximation algorithm for CSM-Cycle and CSM-Cycle-Cover (defined analogously to CSM-
Matching). Since CSM-Cycle generalizes the NP-hard problem maximum-weight cycle, CSM-Cycle
is also NP-hard. We do not know whether CSM-Cycle-Cover is in P or is NP-hard. We also give
a 2-Matroids construction for directed graphs, and using LSV2 algorithm, we can find a (4 + ǫ)-
approximate solution to the directed versions of CSM-Cycle and CSM-Cycle-Cover.
2 CSM-Matching
Given a complete graph G = (V, E) and a nonnegative, monotone, submodular function f: 2E → R+ ,
we want to find a matching M in the graph G that maximizes f (M ), where monotone means for
all A ⊆ B ⊆ E, we have f (A) f (B). Indeed, the maximum-matching problem is a special case of
CSM-Matching; although maximum matching has a polynomial-time algorithm, CSM-Matching is
NP-hard. Hence, the techniques used in solving maximum-matching might not work when used for
2

3. S6 S1 S6 S1 S6 S1
S2 S2 S2
S5 S5 S5
S4 S3 S4 S3 S4 S3
Figure 1: An input graph to CSM-Matching constructed using the input to max k-cover, where
F = {S1 , S2 , . . . , S6 } and k = 3. Shown edges map to the subsets in their labels, and others map
to the empty set; call this mapping h. For a subset A ⊆ E, the function value in CSM-Matching is
the size of the union of subsets in h(A).
solving CSM-Matching. We will show in this section that CSM-Matching is NP-hard by reducing
the NP-hard problem max k-cover to it. Then we give a greedy 3-approximation algorithm and its
tight analysis.
Given a set S = {1, . . . , n}, a collection F of subsets of S (say |F| = m), and an integer k m,
the max k-cover problem is to find at most k subsets from F with largest union, i.e., to find
argmax B .
A⊆F ,|A| k B∈A
Given a max k-cover instance, we will construct a CSM-Matching instance, i.e., a complete graph
and a nonnegative, monotone, submodular function defined on the edges of that graph.
Proof outline: We need to select at most k subsets and maximize the cardinality of their union,
and so we construct a graph such that any matching in that graph corresponds to a subcollection
of F of size at most k. See Figure 1. Specifically, we construct a graph G = (V, E) that is union
of k identical star graphs, and each star graph has m edges that correspond to m subsets in F.
A matching in G will contain at most one edge from each star graph, and it will correspond to
a subcollection of F of size at most k. The submodular function on the edge set of G will be
the coverage function—a nonnegative, monotone, submodular function—on the collection F, i.e.,
g : 2F → N+ , such that for any A ⊆ F, we have g(A) = | ∪T ∈A T |, and by definition of the coverage
function, maximizing g is same as maximizing the size of the union of the subcollection.
Theorem 1. CSM-Matching is NP-hard.
Proof. We reduce the max k-cover problem to CSM-Matching. The input to the max k-cover
problem is a set S = {1, . . . , n}, a collection F of subsets of S (say |F| = m), and an integer k m.
Given an instance of the max k-cover problem, we construct the input to CSM-Matching as follows.
Construct a union of k star graphs, each with m edges, and a bijection from the set of edges in a
star graph to the collection F; collectively, call this mapping h. Add non-star edges to the union
of k star graphs to make it a complete graph, say G = (V, E). Assume that h maps non-star edges
to ∅. We can assume without loss of generality that ∅ ∈ F. Now we have a complete mapping
h : E → F. We will overload h as follows. For any A ⊆ E, define h(A) = {h(e) : e ∈ A}. Recall
that we defined the coverage function g : 2F → N+ as, for A ⊆ F, we have g(A) = | ∪T ∈A T |.
We use the function g to define f : E → N+ as for T ⊆ E, we have f (T ) = g(h(T )). Since g is
nonnegative, monotone, and submodular, so is f .
3

4. We claim that a matching in G is mapped by h to a subcollection of F of size at most k, and
vice versa. If M is a matching in G, then we know that it contains at most one edge from each star
graph, and we select the subsets corresponding to those edges as the solution to max k-cover. Since
non-star edges map to the empty set, we can ignore them because adding them will not increase the
cardinality of the union of the subsets in h(M ). Now f (M ) = g(h(M )), and by definition, g(h(M ))
is the size of the union of the subsets in h(M ). Analogously, if A ⊆ F such that |A| k, then we
can construct a matching, say M , such that h(M ) = A; for each subset T ∈ A, select one edge,
say eT , from a star graph such that h(eT ) = T ; such an edge exists because there is a bijection
from the set of edges in each star graph to the collection F. Hence, we have
f (M ) = g(h(M )) = g(A) = | ∪T ∈A T | .
Therefore, an optimum matching corresponds to an optimum subcollection. We can construct G in
polynomial time, and G has size polynomial in n and m. Hence, since max k-cover is NP-hard [1],
so is CSM-Matching.
In addition, since it is NP-hard to approximate max k-cover within the approximation ratio of
e/(e − 1), we immediately get the following result.
Theorem 2. It is NP-hard to approximate CSM-Matching within a factor of e/(e − 1).
2.1 A Greedy Algorithm for CSM-Matching
Now we will give a simple greedy algorithm for CSM-Matching and a bad input for the greedy
algorithm.
Consider the following algorithm. Start with M = ∅. Add an edge to M that gives the maximum
increase in f value while maintaining the property that M is a matching, i.e., add
argmax f (M + e) − f (M ) .
e∈E, M +e is a matching
Repeat the previous step until M becomes a maximal matching. We give a bad input to this algo-
rithm that achieves the approximation ratio of 3 − ǫ. We will need a particular type of submodular
function for the bad input, called unique-weights-sum submodular function. The bad input leads
us to the analysis that the greedy algorithm is indeed a 3-approximation algorithm.
Unique-weights-sum submodular function fu
Given a finite set E and a weight function w : E → R+ , for any A ⊆ E, let U (A) = {w(a) : a ∈ A}
(i.e., U (A) is the set of unique weights in A). Define a function fu : 2E → R+ such that for any
A ⊆ E, we have fu (A) = r∈U (A) r.
Lemma 3. The function fu is nonnegative, monotone, and submodular.
Proof. For any element e ∈ E, we have w(e) 0; hence fu is nonnegative. For any A ⊆ B ⊆ E,
we have U (A) ⊆ U (B); therefore fu (A) fu (B), and so fu is monotone.
For any A, B ⊆ E,
fu (A) + fu (B) = r + r
r∈U (A) r∈U (B)
r + r
r∈U (A∩B) r∈U (A∪B)
= fu (A ∩ B) + fu (A ∪ B) ,
where the argument for the inequality in the second step is as follows. Let x = r∈U (A) r +
4

5. 2
α
1 3
α−ǫ α α − 2ǫ
6 4
5
Figure 2: An example that achieves approximation ratio of (3 − ǫ) for the greedy algorithm. The
greedy algorithm outputs the edge {1, 3} (shown as a thin edge), with f value α, but the optimum
matching is {{1, 6}, {2, 5}, {3, 4}} (shown as thick edges) with f value 3(α − ǫ).
r∈U (B) r and y = r∈U (A∩B) r + r∈U (A∪B) r. A number r ∈ U (A ∩ B) contributes 2r to both
x and y. If r ∈ U (A ∪ B) − U (A ∩ B), then there exists an element e ∈ (A − B) ∪ (B − A) such
that w(e) = r, so that it contributes r to y and either r or 2r to x. Hence, x y, and fu is
submodular.
The greedy algorithm gives a ratio of 3 on the example in Figure 2. Consider the complete
graph G = (V, E), where V = {1, 2, 3, 4, 5, 6}. Define a weight function w by assigning each edge
a nonnegative number as shown in the figure, where α is a large positive number and ǫ is a very
small positive number. For the edges not shown in the figure, assign weight 0. Define a unique-
weights-sum submodular function fu : 2E → R+ using the weight function w.
If the greedy algorithm is given the input (G, f ), then it gives an output matching containing
the edge {1, 3} with f value α, but the optimum matching is {{1, 6}, {2, 5}, {3, 4}} with f value
3(α − ǫ).
2.2 The Greedy Algorithm is 3-Approximation Algorithm
Now we will show that the greedy algorithm is a 3-approximation algorithm.
Theorem 4. A 3-approximate solution to CSM-Matching can be found in polynomial time using
the greedy algorithm.
Proof. Let MOP T be an optimum matching, M = {e1 , e2 , . . . , ek } be an output from the greedy
algorithm, and e1 , e2 , . . . , ek be the order in which edges were added by the greedy algorithm. Let
Mi = {e1 , e2 , . . . , ei }, and let ri (e) be the increase in f obtained by adding the edge e to the
matching Mi−1 , i.e., let ri (e) = f (Mi−1 + e) − f (Mi−1 ). Let e′ and e′′ be the edges in MOP T − M
i i
that intersect with the edge ei and were candidates for being added to Mi−1 , i.e., Mi−1 + e′ and i
Mi−1 + e′′ are matchings; call such edges ith-candidates. If an edge in MOP T − M intersects with
i
two edges in M , then it is an ith-candidate for the edge with lower index only. For example,
if e ∈ MOP T − M intersects with e5 and e9 , M8 + e will not be a matching, and hence, e is
a 5th-candidate but not a 9th-candidate. Note that for each ei , there can be zero, one, or two
ith-candidates, which we denote by e′ and e′′ . We will show that f (M ∪ MOP T ) 3f (M ).
i i
The edge ei was added over e′ and e′′ because ri (ei )
i i ri (e′ ) and ri (ei )
i r(e′′ ). Since f is
i
submodular, adding ei or ei to any superset of Mi−1 will increase the f value by at most ri (e′ ) or
′ ′′
i
5

6. ri (e′′ ), respectively. In particular,
i
f (M + e′ ) − f (M )
i f (Mi−1 + e′ ) − f (Mi−1 ) = ri (e′ )
i i ri (ei ) ,
f (M + e′′ ) −
i f (M ) f (Mi−1 + e′′ ) −
i f (Mi−1 ) = ri (e′′ )
i ri (ei ) .
Thus, for all i ∈ {1, . . . , k}, if we add an ith-candidate to M , it will increase the f value by at
most ri (ei ). We keep on adding ith-candidates to M , and by submodularity of f , we can bound
the increase as follows:
k k
f (M ∪ MOP T ) = f M∪ {e′ , e′′ }
i i since M is maximal, MOP T − M = {e′ , e′′ }
i i ,
i=1 i=1
k
f (M ) + 2 ri (ei ) ( because f is submodular ) ,
i=1
k
f (M ) + 2f (M ) because ri (ei ) = f (M ) − f (∅) f (M ) ,
i=1
= 3f (M ) .
In the third step, k ri (ei ) = k f (Mi ) − f (Mi−1 ) = f (Mk ) − f (∅) f (M ) because f (∅) 0.
i=1 i=1
Since f is monotone, f (MOP T ) f (M ∪ MOP T ) 3f (M ). Clearly, the greedy algorithm runs in
polynomial time, and the theorem is proved.
3 Matroid Construction
We reduce CSM-Matching to nonnegative, monotone, submodular maximization constrained to
the intersection of two matroids (i.e., CSM-2-Matroids). We use a (2 + ǫ)-approximate solution
to CSM-2-Matroids given by an algorithm (which we call LSV2 ) to obtain a (4 + ǫ)-approximate
solution to CSM-Matching.
Given a nonnegative, monotone, submodular function f : 2E → R+ and k 2 matroids
M1 = (E, I1 ), M2 = (E, I2 ), . . . , Mk = (E, Ik ), the CSM-k-Matroids problem is to find
argmax f (S) .
S∈I1 ∩I2 ∩···∩Ik
The following theorem is due to a result by Lee, Sviridenko, and Vondr´k [5] and a similar
a
result by Lee, Mirrokni, Nagarajan, and Sviridenko [4].
Theorem 5. For k 2 and for any constant ǫ > 0, CSM-k-Matroids has a polynomial-time,
deterministic, local-search, (k + ǫ)-approximation algorithm.
Call this algorithm LSVk (M1 , M2 , . . ., Mk , f ).
A matroid M = (E, I) is a partition matroid if, for a partition E1 , E2 , . . . , El of E and some
constants k1 , k2 , . . . , kl , we have
I = {S ⊆ E : |S ∩ Ei | ki for all i ∈ {1, 2, . . . , l}} .
Given a CSM-Matching instance, we create two partitions of the edge set E, and we construct
two partition matroids M and M ′ such that an independent set in the intersection of M and M ′ can
be used to obtain a matching without losing much of its f value. Then we use the LSV2 algorithm.
That is, given a complete graph G = (V, E) and a nonnegative, monotone, submodular function
f : 2E → R+ , we construct M = (E, I), and M ′ = (E, I ′ ) such that
6

7. • the intersection I ∩ I ′ contains all matchings, and
• if I ∈ I ∩ I ′ , then I can be partitioned into two matchings.
Note that an independent set in M or M ′ is a subgraph of G.
3.1 2-Matroids
Consider any arbitrary order of vertices of G, say v1 , v2 , . . . , vn . Define two partitions of E, which
′ ′ ′
are, {E1 , E2 , . . . , En } and {E1 , E2 , . . . , En }, as for all i ∈ {1, . . . , n},
Ei = {{vi , vj } : i < j n} ,
′
Ei = {{vi , vj } : i > j 1} .
That is,
E1 = {{v1 , v2 }, {v1 , v3 }, {v1 , v4 }, . . . , {v1 , vn }} ,
E2 = {{v2 , v3 }, {v2 , v4 }, . . . , {v1 , vn }} ,
.
.
.
En−1 = {{vn−1 , vn }} ,
En = ∅.
And analogously,
′
E1 = ∅,
′
E2 = {{v2 , v1 }} ,
.
.
.
′
En−1 = {{vn−1 , vn−2 }, {vn−1 , vn−3 }, . . . , {vn−1 , v1 }} ,
′
En = {{vn , vn−1 }, {vn , vn−2 }, {vn , vn−3 }, . . . , {vn , v1 }} .
We use these two partitions to define two partition matroids M = (E, I) and M ′ = (E, I ′ ) as
I ∈ I if and only if for all j ∈ {1, 2, . . . , n}, we have |I ∩ Ej | 1,
I ∈ I ′ if and only if for all j ∈ {1, 2, . . . , n}, we have |I ∩ Ej |
′
1.
We call this construction the 2-Matroids construction. If I is an independent set in the intersection
of the two matroids M and M ′ , then for each vertex vi in the graph corresponding to I, there is at
most one edge from vi to a higher-indexed vertex and at most one edge from vi to a lower-indexed
vertex. In other words, if we order the vertices vertically from v1 down to vn , then each vertex in I
has at most one edge going up and at most one edge going down. Therefore, we have the following
lemma.
Lemma 6. If I ∈ I ∩ I ′ , then I is a union of vertex-disjoint paths.
Since a matching is a union of vertex-disjoint paths, we immediately get the following theorem.
Lemma 7. The intersection of 2-Matroids, i.e., I ∩ I ′ , contains all matchings.
Now we can use LSV2 because the ground set for the matroids and the edge set of the input
graph is the same, and the output of LSV2 will be a union of vertex-disjoint paths.
7

8. Theorem 8. A (4 + ǫ)-approximate solution to CSM-Matching can be found in polynomial time
using LSV2 .
Proof. Let (G, f ) be the input to CSM-Matching and MOP T be an optimum matching. Construct
2-Matroids M and M ′ , and run LSV2 on these matroids, i.e., let I = LSV2 (M, M ′ , f ). Using
Lemma 7 and Theorem 5, we have f (I) 1/(2 + ǫ′ )f (MOP T ) for a small constant ǫ′ . Since I is a
union of vertex-disjoint paths (by Lemma 6), we can partition I into two matchings P and P ′ . By
submodularity and nonnegativity of f , we have
f (P ) + f (P ′ ) f (I) + f (∅) f (I) .
Hence,
f (I) f (MOP T ) f (MOP T )
max{f (P ), f (P ′ )} ′)
= .
2 2(2 + ǫ 4+ǫ
Therefore, one of the matchings P and P ′ is a (4 + ǫ)-approximate solution to CSM-Matching.
4 2-Matroids Construction for CSM-Cycle and CSM-Cycle-Cover
We define CSM-Cycle and CSM-Cycle-Cover the way we defined CSM-Matching. Given a complete
graph and a nonnegative, monotone, submodular function on the edges of the graph, we want to
find a cycle (respectively, cycle cover) that maximizes the function value. Recall that since CSM-
Cycle generalizes the NP-hard problem maximum-weight cycle, CSM-Cycle is also NP-hard. But
we do not know whether CSM-Cycle-Cover is in P or is NP-hard. In the rest of the section, we do
not mention CSM-Cycle-Cover, but the results hold even if we replace CSM-Cycle by CSM-Cycle-
Cover.
Again, we use the 2-Matroids construction to solve CSM-Cycle. We state without proof the
following.
Lemma 9. Let M = (E, I) and M ′ = (E, I ′ ) denote the matroids constructed using the 2-Matroids
construction. If COP T is an optimal solution to CSM-Cycle, then COP T can be partitioned into two
subgraphs H1 and H2 such that H1 , H2 ∈ I ∩ I ′ . Moreover, by submodularity and nonnegativity
of f , we have max{f (H1 ), f (H2 )} (1/2)f (COP T ).
Now we can prove the approximation guarantee.
Theorem 10. A (4 + ǫ)-approximate solution to CSM-Cycle can be found in polynomial time using
LSV2 .
Proof. By Lemma 6, the output of the LSV2 algorithm, say I = LSV2 (M, M ′ , f ), is a union of
vertex-disjoint paths. Since the input graph is complete, we can arbitrarily augment I to get a
cycle, say C, such that f (C) f (I) (because f is monotone). Hence,
f (C) f (I)
max{f (H1 ), f (H2 )}
(because H1 , H2 ∈ I ∩ I ′ per Lemma 9) ,
2 + ǫ′
f (COP T )
(by Lemma 9, max{f (H1 ), f (H2 )} (1/2)f (COP T )) .
4+ǫ
Therefore, the cycle C is a (4 + ǫ)-approximate solution to CSM-Cycle.
8

9. 4.1 CSM-Cycle and CSM-Cycle-Cover for Directed Graphs
We construct 2-Matroids using a directed, complete graph without loops as follows. Define two
← ← ← → → →
partitions of E, {E1 , E2 , . . . , En } and {E1 , E2 , . . . , En } as for all i ∈ {1, . . . , n},
→
Ei = {(vi , vj ) : i = j} ,
←
Ei = {(vj , vi ) : i = j} .
Define two partition matroids M → = (E, I → ) and M ← = (E, I ← ) using the above partitions as
follows:
I ∈ I → if and only if for all j ∈ {1, 2, . . . , n}, we have |I ∩ Ej | 1 ,
→
I ∈ I ← if and only if for all j ∈ {1, 2, . . . , n}, we have |I ∩ Ej |
←
1.
In words, a subset I of E is an independent set in the matroid M → if and only if all the vertices
in I have outdegree at most one, and I is an independent set in the matroid M ← if and only if all
the vertices in I have indegree at most one. Therefore, we have the following lemma.
Lemma 11. If I is a cycle (or a cycle cover), then I ∈ I → ∩ I ← . If I ∈ I → ∩ I ← , then I is a
union of vertex-disjoint cycles and paths.
Using the above lemma and the LSV2 -algorithm, we get a (4 + ǫ)-approximate solution to CSM-
Cycle and CSM-Cycle-Cover for directed graphs. The procedure is similar to that in the undirected
case. Hence, we state the following without proof.
Theorem 12. A (4 + ǫ)-approximate solution for CSM-Cycle and CSM-Cycle-Cover for directed
graphs can be found in polynomial time using LSV2 .
5 Summary and Future Work
We saw that it is NP-hard to approximate CSM-Matching within a factor of e/(e − 1), and we gave
a 3-approximation algorithm with tight analysis. This gap can be narrowed by either showing a
better hardness or giving a better algorithm. Also, we used the 2-Matroids construction and showed
that we can find a (4 + ǫ)-approximate solution to CSM-Matching, CSM-Cycle, and CSM-Cycle-
Cover using the LSV2 algorithm. Since we used LSV2 as a black box, we did not get an insight into
the structure of these problems. Thus, another question is whether we can give a better analysis
by considering how LSV2 works.
Bipartite matching is a special case of general matching, and it has many applications such as
bitext word alignment. We can also study the CSM-Bipartite-Matching to see whether we can get
results better than those in this paper.
Acknowledgments
I thank my M. Tech. project advisor Prof. Sundar Vishwanathan, my examiner Prof. Ajit Diwan,
Prof. Thomas Cormen for his suggestions about style and usage, and Indian Institute of Technology
Bombay for the financial support.
9

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