Weighted Matching
By: Ehsan Yazdi
Maximal Matching
 We emphasize again that the term maximal matching means a matching
of maximal cardinality, and that an unextendable matching is not
necessarily maximal!
 A subtree T of G with root r is called an alternating tree if r is an exposed
vertex and if every path starting at r is an alternating path.
 The vertices in layers 0, 2, ... are called
outer vertices, and the vertices in
layers 1, 3, ... are inner vertices of T.
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Constructing Alternating Tree
 Let x be a vertex in layer 2i, and let y≠mate(x) be a vertex adjacent to x.
There are four possible cases:
 Case 1: y is exposed (and not yet contained in T). Then we have found an
augmenting path.
 Case 2: y is not exposed, and neither y nor mate(y) are contained in T.
Then we put y into layer 2i+1 and mate(y) into layer 2i+2.
 Case 3: y is already contained in T as an inner vertex. Note that adding the
edge xy to T would create a cycle of even length in T. we may ignore this
case.
 Case 4: y is already contained in T as an outer vertex. Note that adding the
edge xy to T would create a cycle of odd length 2k+1 in T for which k edges
belong to M. Such cycles are called blossoms; these blossoms -which
cannot occur in the bipartite case- cause difficulties!
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Constructing Alternating Tree
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Maximal Matching for bipartite graphs
 Let G=(V,E) be a bipartite graph with respect to the partition V=SŮS’,
where S={1, ..., n} and S’={1’, ..., m’}; we assume n ≤ m. The algorithm
constructs a maximal matching M described by an array mate. The
function p(y) gives, for y∈S’, the vertex in S from which y was accessed.
 Theorem 13.3.3. Let G=(V,E) be a bipartite graph with respect to the
partition V=SŮS’. Then Algorithm BIMATCH determines with complexity
O(|V||E|) a maximal matching of G.
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Bipartite Matching Algorithm
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Weighted Matching
 Let G = (V,E) be a bipartite graph with weight function w:E→R. the weight
w(M) of a matching M of G is defined by:
w(M) = ∑e∈M w(e).
 A matching M is called a maximal weighted matching if w(M) ≥ w(M’)
holds for every matching M’ of G.
 A matching M is called a minimal weighted matching if w(M) ≤ w(M’)
holds for every matching M’ of G.
 We will restrict our attention to the problem of determining a perfect
matching of maximal weight with respect to a nonnegative weight
function w in a complete bipartite graph Kn,n. We call such a matching an
optimal matching of (Kn,n, w).
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Weighted Matching
 Let w be a nonnegative weight function for Kn,n. Then an optimal matching
can be determined with complexity O(n3). It is the best known result for
positive weight functions on complete bipartite graphs Kn,n.
 The best known complexity for determining an optimal matching in a
general bipartite graph is O(|V||E|+|V|2 log |V|) [FrTa87].
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The Hungarian algorithm
 An algorithm for finding an optimal matching in a complete bipartite
graph. This algorithm is due to Kuhn and is based on ideas of Konig and
Egervary, so that Kuhn named it the Hungarian algorithm.
 Thus let G=(V,E) be the complete bipartite graph Kn,n with V=SŮT, where
S = {1, ... , n} and T = {1’, ... , n’}, and with a nonnegative weight function w
described by a matrix W = (wij) : the entry wij is the weight of the edge {i, j}.
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Node-Weighting
 A pair of real vectors u = (u1, ... , un) and v = (v1, ... , vn) is called a feasible
node-weighting if the following condition holds:
ui + vj ≥ wij
for all i,j =1, … , n (14.1)
 We will denote the set of all feasible node-weightings (u, v) by F and the
weight of an optimal matching by D.
 Lemma 14.2.1. For each feasible node-weighting (u, v) and for each
perfect matching M of G, we have:
w(M) ≤ D ≤ ∑i=1n(ui + vi)
(14.2)
 If we can find a feasible node-weighting (u, v) and a perfect matching M
for which equality holds in (14.2), then M has to be optimal.
 It is always possible to achieve equality in (14.2); the Hungarian algorithm
will give a constructive proof for this fact.
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Equality Subgraph
 We now characterize the case of equality in (14.2). For a given feasible
node-weighting (u, v), let Hu,v be the subgraph of G with vertex set V
whose edges are precisely those ij for which ui+vj=wij holds; Hu,v is called
the equality subgraph for (u, v).
 Lemma 14.2.2. Let H = Hu,v be the equality subgraph for (u, v) ∈ F. Then
∑i=1n(ui + vi) = D
holds if and only if H has a perfect matching. In this case, every perfect
matching of H is an optimal matching for (G,w).
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Proof
 : let ∑(ui+vi) = D and suppose that H does not contain a perfect
matching. By Theorem 7.2.5, there exists a subset J of S with |Γ(J)| < |J|.
 Put δ = min{ui + vj − wij : i ∈ J, j  Γ(J)}
 Then (u’, v’) is again a feasible node-weighting. contradiction:
D ≤ ∑(u’i + v’j ) = ∑(ui + vj) − δ|J| + δ|Γ(J)| = D − δ(|J| − |Γ(J)|) < D.
 Conversely, suppose that H contains a perfect matching M. Then ui+vj=wij
for each edge of M, and summing ui+vj=wij over all edges of M yields
equality in (14.2). This argument also shows that every perfect matching of
H is an optimal matching for (G,w).
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The Hungarian algorithm
 The Hungarian algorithm starts with an arbitrary feasible node-weighting
(u, v) ∈ F; usually:
v1 = . . . = vn = 0
ui = max{wij : j=1, ... , n} (for i = 1, . . . , n)
 If the corresponding equality subgraph contains a perfect matching, our
problem is solved. Otherwise, the algorithm determines a subset J of S
with |Γ(J)| < |J| and changes the feasible node-weighting (u, v) in
accordance with the proof of Lemma 14.2.2.
 This decreases the sum ∑(ui +vi) and adds at least one new edge ij’ with i∈J
and j‘ Γ(J) (with respect to Hu,v) to the new equality subgraph Hu’,v’ .
 This procedure is repeated until the partial matching in H is no longer
maximal. Finally, we get a graph H containing a perfect matching M, which
is an optimal matching of G as well.
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The Hungarian algorithm
 The algorithm determines an optimal matching in G described by an array
mate.
 For extending the matchings and for changing (u, v), we use an
appropriately labelled alternating tree in H. In the following algorithm, we
keep a variable δj for each j ∈ T which may be viewed as a potential: δj is
the current minimal value of ui+vj−wij. Moreover, p(j) denotes the first
vertex i for which this minimal value is obtained.
 We will use a procedure AUGMENT:
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The Hungarian algorithm
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The Hungarian algorithm
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Time complexity
 Theorem 14.2.4. Hungarian Algorithm determines with complexity O(n3)
an optimal matching for (G,w).
 Let call all the operations executed during one iteration of the while-loop
(4) to (39) a phase.
 Obviously, there are exactly n phases. As updating the feasible nodeweighting (u, v) and calling the procedure AUGMENT both need O(n)
steps, these parts of a phase contribute at most O(n2) steps altogether.
Note that each vertex is inserted into Q and examined in the inner whileloop at most once during each phase. The inner while-loop has complexity
O(n), so that the algorithm consists of n phases of complexity O(n2), which
yields a total complexity of O(n3) as asserted.
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Generalization
 We will restrict our attention to the problem of determining a perfect
matching of maximal weight with respect to a nonnegative weight
function w in a complete bipartite graph Kn,n. We call such a matching an
optimal matching of (Kn,n, w).
 A maximal weighted matching cannot contain any edges of negative
weight. Thus such edges are irrelevant in our context, so that we will
usually assume w to be nonnegative.
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Generalization
 A maximal weighted matching is not necessarily also a matching of
maximal cardinality. Therefore we extend G to a complete bipartite graph
by adding all missing edges e with weight w(e) = 0; then we may assume
that a matching of maximal weight is a complete matching.
 Similarly, we may also assume |S|=|T| by adding an appropriate number
of vertices to the smaller of the two sets (if necessary), and by introducing
further edges of weight 0.
 If we should require a perfect matching of maximal weight in a bipartite
graph containing edges of negative weight, we can add a sufficiently large
constant to all weights first and thus reduce this case to the case of a
nonnegative weight function.
 Hence we may also treat the problem finding a perfect matching of
minimal weight, by replacing w by −w.
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Circulation
 Let G = (V,E) be a digraph; in general, we tacitly assume that G is
connected.
 A mapping f : E → R is called a circulation on G if it satisfies the flow
conservation condition
∑e+=v f(e) = ∑e−=v f(e) for all v∊V .
 Let b:E→R and c:E→R be two further mappings, where b(e) ≤ c(e) for all
e∊E. One calls b(e) and c(e) the lower capacity and the upper capacity of
the edge e, respectively. Then a circulation f is said to be feasible or legal
(with respect to the given capacity constraints b and c) if
(e) ≤ f(e) ≤ c(e)
for all e∊E.
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Circulation
 Let γ:E→R be a further mapping called the cost function. Then the cost of a
circulation f (with respect to γ) is defined as
γ(f) = ∑e∈E γ(e)f(e)
 A feasible circulation f is called optimal or a minimum cost circulation if
γ(f) ≤ γ(g) holds for every feasible circulation g.
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Algorithm of Busacker and Gowen
 Theorem 10.5.3. Let N=(G, c, s, t) be a flow network with integral capacity
function c and nonnegative cost function γ. Then Algorithm OPTFLOW can
be used to determine an optimal flow of value v with complexity O(v|V|2)
or O(v|E|log|V|).
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Circulation & Max Flow
 Max-flow problem: Let N = (G, c, s, t) be a flow network with a flow f of
value w(f), and let G’ be the digraph obtained by adding the edge r = ts
(return arc) to G. We extend the mappings c and f to G’ as follows:
c(r) = ∞ and f(r) = w(f)
 Setting b(e) = 0 for all edges e of G’, f is even feasible.
 Conversely, every feasible circulation f on G’ yields a flow of value f(r) on
G.
 Cost function: γ(r) = −1 and γ(e) = 0 otherwise.
 Then a flow f on N is maximal if and only if the corresponding circulation
has minimal cost with respect to γ: maximal flows on N = (G, c, s, t)
correspond to optimal circulations for (G’, b, c, γ).
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Circulation & Optimal Matching
 Let w : E → R0+ be a weight function for the graph Kn,n. Suppose the
maximal weight of all edges is C.
 Construct a bipartite digraph G’ with vertex set S∪T, where S={1, … , n} and
T={1, ... , n}, and adjoin two additional vertices s and t. The edges of G are
all the (s, i), all the (i, j), and all the (j, t) with i ∈ S and j ∈ T.
 All edges have lower capacity b(e)=0 and upper capacity c(e)=1.
 Cost function: γ(s, i) = γ(j, t) = 0 and γ(i, j) = C-wij
 We just need to find an optimal flow of value n in the flow network (G’, b,
c, γ).
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Linear Programming
 A linear programming problem (or, for short, an LP) is an optimization
problem of the following kind: we want to maximize (or minimize) a linear
objective function with respect to some given constraints which have the
form of linear equalities or inequalities.
 Formally:
(LP)
maximize x1c1 + . . . + xncn
subject to ai1x1 + . . . + ainxn ≤ bi (i=1, … , m),
xj ≥ 0 (j = 1, ... , n)
 Concisely; A=(aij) (m×n), x=(x1, … , xn) and c=(c1, … , cn) :
(LP)
maximize cxT subject to AxT ≤ bT and x ≥ 0
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Linear Programming
 (LP)
maximize cxT subject to AxT ≤ bT and x ≥ 0
 Adding conditions:
x should be integral: integer linear programming problem (ILP)
xi∈{0, 1} (for i=1, ... , n): zero-one linear program (ZOLP)
 LP is a polynomial problem (e.g. ellipsoid algorithm of Khachiyan)
 ILP and ZOLP are NP-Hard.
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Matchings & Linear Programs
 Let G=(V,E) be a complete bipartite graph with a nonnegative weight
function w. Then the optimal matchings of G are precisely the solutions of
the following ZOLP:
maximize ∑e∊E w(e)xe
subject to xe∊{0, 1} for all e∊E and ∑e∈δ(v) xe =1 for all v∊V
( δ(v) denotes the set of edges incident with v)
 An edge e is contained in the corresponding perfect matching iff xe=1.
 xe∊{0, 1} can be replaced by xe∊ℤ and xe≥0.
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Matchings & Linear Programs
 Using the incidence matrix A of G, we can write the ILP:
maximize wxT subject to AxT=1T and x ≥ 0
(14.3)
where x=(xe)e∊E ∊ ℤE
v1
e1
0,1
v3
v2
e2
1, 0
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Matchings & Linear Programs
 If the set of all admissible vectors x∊ℝn for a given LP is bounded and
nonempty, then all these vectors form a polytope. optimal solutions for
the LP can always be found among the vectors corresponding to vertices
of the polytope (though there may exist further optimal solutions)
 Vertices of the polytope can be defined as those points at which an
appropriate objective function achieves its
unique maximum over the polytope.
 Incidence vectors of perfect matchings M
of G are vertices of the polytope in ℝE
defined by the constraints given in (14.3)
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Matchings & Linear Programs
 Hoffman and Kruskal [HoKr56]:
Let A be an integer matrix. If A is totally unimodular, then the vertices of
the polytope { x: AxT=bT , x ≥ 0} are integral whenever b is integral.
 Wiki:
A totally unimodular matrix is a matrix for which every square non-singular
submatrix is unimodular.
A unimodular matrix M is a square integer matrix with determinant +1 or
−1.
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Matchings & Linear Programs
 Then the following four conditions together are sufficientfor A to be totally
unimodular:
- Every column of A contains at most two non-zero entries;
- Every entry in A is 0, +1, or −1;
- If two non-zero entries in a column of A have the same sign, then the
row of one is in B, and the other in C;
- If two non-zero entries in a column of A have opposite signs, then the
rows of both are in B, or both in C.
 For every bipartite graph G, the incidence matrix A of G is totally
unimodular.
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Matchings & Linear Programs
 Using the incidence matrix A of G,
we can write the ILP:
maximize wxT subject to AxT=1T and
x ≥ 0 where x=(xe)e∊E ∊ ℤE (14.3)
 Hoffman and Kruskal: If A is totally
unimodular, then the vertices of
the polytope { x: AxT=bT , x ≥ 0} are
integral whenever b is integral.
 Incidence vectors of perfect
matchings M of G are vertices of
the polytope in ℝE defined by
the constraints given in (14.3)
 For every bipartite graph G, the
incidence matrix A of G is
totally unimodular.
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Matchings & Linear Programs
 Theorem 14.3.4. Let A be the incidence matrix of a complete bipartite
graph G=(V,E). Then the vertices of the polytope
P={ x∊ℝE : AxT=1T , x ≥ 0}
coincide with the incidence vectors of perfect matchings of G. Hence the
optimal matchings are precisely those solutions of the LP (14.3) which
correspond to vertices of P (for a given weight function).
v1
e1
0,1
v3
v2
e2
1, 0
 ILP (14.3) would is equivalent to the corresponding LP and could be solved
with one of the known algorithms for linear programs.
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Dual Linear Program
 For any linear program
(LP) maximize cxT subject to AxT ≤ bT and x ≥ 0
the dual LP is the linear program
(DP) minimize byT subject to AT yT ≥ cT and y ≥ 0
where y = (y1, … , ym)
 Strong duality theorem: Let x and y be admissible vectors for (LP) and
(DP), respectively. Then one has cxT ≤ byT
with equality if and only if x and y are actually optimal solutions for their
respective linear programs.
 (LP) Maximize wxT subject to AxT ≤ 1T and x ≥ 0
 (DP) Minimize 1yT subject to ATyT ≥ wT and y ≥ 0
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Thanks
for attention
Questions?