Topics in Combinatorial OPtimization Orlando Lee Unicamp 19 de mar - - PowerPoint PPT Presentation

Topics in Combinatorial OPtimization

Orlando Lee – Unicamp 19 de mar¸ co de 2014

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Agradecimentos

Este conjunto de slides foram preparados originalmente para o curso T´

• picos de Otimiza¸

c˜ ao Combinat´

• ria no primeiro

semestre de 2014 no Instituto de Computa¸ c˜ ao da Unicamp. Preparei os slides em inglˆ es simplesmente porque me deu vontade, mas as aulas ser˜ ao em portuguˆ es (do Brasil)! Agradecimentos especiais ao Prof. M´ ario Leston Rey. Sem sua ajuda, certamente estes slides nunca ficariam prontos a tempo. Qualquer erro encontrado nestes slide ´ e de minha inteira responsabilidade (Orlando Lee, 2014).

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

Given a digraph D = (V , A), an arc capacity function u : A → R+, a supply function b : V → R in which b(v) > 0 means producer, and b(v) < 0 means consumer, find x : A → R such that: x(δout(v)) − x(δin(v)) = b(v) for every v ∈ V , and 0 ≤ x(a) ≤ u(a) for every a ∈ A.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

4 2 −3 −3 3 3 3 3 2 2 2 4 1 Necessary condition: b(V ) = 0.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

4 2 3 3 3 3 3 3 2 2 2 4 1 s t Set u(s, v) := b(v) se b(v) > 0, Set u(v, t) := −b(v) se b(v) < 0.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

4 2 3 3 3 3 3 3 2 2 2 4 1 s t The system has a feasible solution if and only if the auxiliary network has a maximum st-flow which saturates all arcs leaving s or entering t.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

Conclusion: checking whether there exists x : A → R such that: (a) x(δout(v)) − x(δin(v)) = b(v) for every v ∈ V , and (b) 0 ≤ x(a) ≤ u(a) for every a ∈ A can be done in polynomial time. For brevity, if x satisfies (a), we say that it is a b-flow. If x satisfies both (a) and (b), we say that it is a feasible b-flow. Later we will see a good characterization for the existence of a feasible b-flow. In other words, necessary and sufficient conditions that provide succinct certificates for both possible answers (yes or no).

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

Let M be a real matrix with dimensions p × q such that: the sum of the elements in row i is αi and the sum of the elements in column j is βj. We can round up or down any element r to ⌈r⌉ or ⌊r⌋ anyway we like.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

A consistent rounding of M is obtained by rounding the elements

• f M, the values αi (i = 1, . . . , p) and the values βj (j = 1, . . . , q)

so that: the sum of the rounded elements in row i is equal to the rounding of αi and the sum of the rounded elements in column j is equal to the rounding of βj. Matrix rounding problem. Given a matrix M, find (if exists) a consistent rounding of M.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

3.1 6.8 7.3 17.2 9.6 2.4 0.7 12.7 3.6 1.2 6.5 11.3 16.3 10.4 14.5 Matrix rounding problem.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

3 3 7 7 17 10 1 13 2 2 6 11 11 16 14 Consistent rounding.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

We reduce Matrix Rounding Problem to the following problem. Given a network (D, s, t, l, u) where l, u are arc capacity functions in which for each a ∈ A:

• l(a) is the lower capacity of a and
• u(a) is the upper capacity of a,

find an st-flow f such that l(a) f (a) u(a) for every a ∈ A. In a few minutes, we will show how to solve this problem.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

• Reduction. construct a network as follows:

let s be the source and t be the target, for each row i we have a vertex i, for each column j′ we have a vertex j′, for each pair i e j′ we have an arc (i, j′) with capacities (⌊mi,j⌋, ⌈mi,j⌉), for each vertex i we have an arc (s, i) with capacities (⌊αi⌋, ⌈αi⌉), and for each vertex j′ we have an arc (j′, t) with capacities (⌊βj⌋, ⌈βj⌉).

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

(l, u) i j′ s t 1 1′ 2 2′ 3 3′ (0, 1) (1, 2) (2, 3) (3, 4) (3, 4) (6, 7) (6, 7) (7, 8) (9, 10) (10, 11) (11, 12) (12, 13) (14, 15) (16, 17) (17, 18)

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Example: matrix rounding

There exists a 1-1 correspondence between consistent roundings of M and feasible flows in the network. If ˜ M := ( ˜ mi,j), ˜ α and ˜ β is a consistent rounding, then f (a) =    ˜ mi,j se a = (i, j), ˜ αi se a = (s, i), ˜ βj se a = (j′, t) is a feasible flow in the network. Similarly, if f is a feasible (integral) flow in the network, then we can obtain a consistent rouding.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Generalized networks

A generalized network is a network (D, s, t, l, u) where each arc a is associated to a pair of capacities (l(a), u(a)). Some typical problems: find a feasible flow f , find a feasible flow with maximum value, and find a feasible flow with minimum value.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Generalized networks

Unlike the case in which l = 0, the problem of finding a feasible flow is more difficult. The resolution of an optimization problem in a generalized network is usually divided into two steps: finding a feasible flow, and given a feasible flow, finding a maximum one.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks

In order to prove optimality of a flow in a generalized network (D, s, t, l, u) we need the following concept. Define the (generalized) capacity of an st-cut δout(X) as cap(X) := u(δout(X)) − l(δin(X)).

• Lemma. Let f be a feasible flow and δout(X) be an st-cut. Then

val(f ) ≤ cap(X) with equality only if f (a) = u(a) for every a ∈ δout(X) and f (a) = 0 for every a ∈ δin(X). Furthermore, if equality holds, then f is a maximum flow and δout(X) is a minimum generalized capacity st-cut.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks

Assume we know a feasible flow f in (D, s, t, l, u). Define a residual network D′

f = (V , A′ f ) with residual capacity

r as follows: (a) if f (a) < u(a) then a ∈ A′

f and r(a) = u(a) − f (a),

(b) if l(a) < f (a) then a−1 ∈ A′

f and r(a−1) = f (a) − l(a).

Note that the definiton of residual capacity is similar to case l = 0.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks (2,3)

(l, u) s s s t t t (1, 2) (2, 4) (1, 4) (0, 3) f 1 1 1 3 2 2 2 2 2 2 r

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks

Solve the MaxFlow problem on the network (D′

f , s, t, r). Let

f ′ be the maximum flow returned by the algorithm. Intuitively, f ′ is how much we must deviate from f to reach a maximum flow in the original network. Define a flow f ∗ of the original network as follows: (a) if a ∈ AD′

f ∩ A then f ∗(a) := f (a) + f ′(a), and

(b) if a−1 ∈ AD′

f ∩ A−1 then f ∗(a) := f (a) − f ′(a).

Clearly, f ∗ is a feasible flow in the original network. (Why?)

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks

We will show that f ∗ is a (feasible) maximum flow in the generalized network. Let δout(X) be a minimum st-cut of (D′

f , s, t, r). Then

f ′(e) = r(e) for every e ∈ δout

D′

f (X) and

f ′(e) = 0 for every e ∈ δin

D′

f (X).

Recall that (a) for each e ∈ δout

D′

f (X) either e = a for some arc a ∈ δout(X) or

e = a−1 for some arc a ∈ δin(X) and (b) for each e ∈ δin

D′

f (X) either e = a for some arc a ∈ δin(X) or

e = a−1 for some arc a ∈ δout(X).

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

MaxFlow in generalized networks

Analyzing these four possibilities we conclude that in D we have that (a) if a ∈ δout(X) then f ∗(a) = u(a) and (b) if a ∈ δin(X) then f ∗(a) = l(a). Hence, f ∗ is a maximum flow and δout(X) is a minimum generalized capacity cut in the generalized network.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Generalized MaxFlow MinCut theorem

• Theorem. (Generalized MaxFlow MinCut)

Let (D, s, t, l, u) be a generalized network. Then the value of a maximum flow is equal to the generalized capacity of a minimum st-cut. Note that any MaxFlow algorithm can be used for finding a maximum flow in a generalized network, as long as we have an initial feasible solution.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible flows

Let (D, s, t, l, u) be a generalized network. Two basic questions: How does one find a feasible flow, if it exists? If it does not exist, how does one exhibit a simple certificate

• f this fact?

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible circulations

(l, u) (l, u) s s t t (1, 2) (1, 2) (2, 4) (2, 4) (1, 3) (1, 3) (1, 4) (1, 4) (0, 3) (0, 3) (0, ∞) Add an arc (t, s) to the network with capacities (0, ∞). The generalized network has a feasible st-flow if and only if the new network has a feasible circulation.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible circulation problem

So the problem consists in finding x : A → R such that: x(δout(v)) − x(δin(v)) = 0 for every v ∈ V , l(a) ≤ x(a) ≤ u(a) for every a ∈ A. We say that x is a feasble circulation.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Feasible circulation problem

Replacing x(a) by x′(a) + l(a) we obtain: x′(δout(v)) − x′(δin(v)) = b(v) for every v ∈ V , 0 ≤ x′(a) ≤ u(a) − l(a) for every a ∈ A. where b(v) := l(δin(v)) − l(δout(v)). Therefore it suffices to find x′ satisfying conditions above. We have seen how this can be solved by a reduction to a MaxFlow problem!

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Hoffman’s condition

Given a digraph D = (V , A) and capacities l, u : A → R with l u, we want to find out if there exists a feasible circulation. A necessary condition is the following: l(δin(X)) ≤ u(δout(X)) for every X ⊆ V . We will show that this condition is also sufficient. This can be done using the reduction to a MaxFlow on the previous slide (see also the first three slides). If the algorithm stops and returns a flow saturating all arcs leaving s then we obtain the desired flow. Otherwise, find a minimum cut δout(X). This s¯ t-set X is a set violating Hoffman’s cut condition. (Exercise)

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Hoffman’s circulation theorem

• Theorem. (Hoffman, 1960) Let D = (V , A) be a digraph with

capacities l, u : A → R such that l u. Then there exists a circulation x such that l x u if and only if l(δin(X)) ≤ u(δout(X)) for every X ⊆ V . Furthermore, if l, u are integral then x can be chosen integral. We present here another proof of Hoffman’s theorem. The proof does not imply a polynomial algorithm for this problem, but illustrates some finer points.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Proof of Hoffman’s theorem

We can assume that l 0 by modifying the network if

• necessary. (Why?)

Initialize x := 0. Thus, x u but possibly l x. We say that an arc a is unfeasible if x(a) < l(a) and feasible if l(a) x(a). Idea of the algorithm/proof: at each iteration we choose an unfeasible arc a and try to increase x(a). If this is not possible, then we find a set X violating Hoffman’s cut condition.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Proof of Hoffman’s theorem

Define a residual graph D(x) as follows. For a feasible arc a (1) if x(a) < u(a) then put a in D(x) with residual capacity u(a) − x(a), and (2) if l(a) < x(a) then put a−1 in D(x) with residual capacity x(a) − l(a). For an unfeasible arc a (3) put a in D(x) with residual capacity u(a) − x(a).

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Proof of Hoffman’s theorem

At each iteration the algorithm selects an unfeasible arc a = (v, w). Suppose first that there exists a wv-path P in D(x). Since the residual capacities are positive, the algorithm increases the flow along the cycle P + a, reducing the “unfeasibility”of a. Note that this operation does not affect the feasiblity of the remaining arcs.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Proof of Hoffman’s theorem

Suppose then that there exists no wv-path in D(x). Let X be the set of vertices reachable from w in D(x). Then x(a) = u(a) for every a ∈ δout(X) and x(a) ≤ l(a) for every a ∈ δin(X). Note that (v, w) belongs to δin(X) e x(v, w) < l(v, w). Therefore, l(δin(X)) > u(δ+(X)), and X violates Hoffman’s cut condition.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Some consequences

• Theorem. Let D = (V , A) be a digraph, b : V → R be a

function with b(V ) = 0 and l, u : A → R be arc capacity functions with l u. Then the system x(δout(v)) − x(δin(v)) = b(v) for every v ∈ V , l(a) ≤ x(a) ≤ u(a) for every a ∈ A, has a a solution if and only if u(δout(X)) − l(δin(X)) ≥ b(X) for every X ⊆ V . Furthermore, if b, l, u are integral, we can choose x integral.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Some consequences

Sketch of proof. Add a new vertex r and for each v ∈ V add an arc (v, r) with l(u, r) := u(v, r) := b(v). Then there exists a solution x if and only if there exists a circulation x′ in the new network with l ≤ x′ ≤ u. Hoffman’s cut condition is equivalent to the cut condition of the theorem.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Some consequences

• Theorem. (Gale, 1957) Let D = (V , A) be a digraph,

b : V → R be a function with b(V ) = 0 and u : A → R an arc capacity function. Then the system x(δout(v)) − x(δin(v)) = b(v) para todo v ∈ V , 0 ≤ x(a) ≤ u(a) para todo a ∈ A, has a solution if and only if u(δout(X)) ≥ b(X) for every X ⊆ V . Furthermore, if b, u are integral, we can choose x integral.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Some consequences

• Exercise. Derive from Hoffman’s theorem a necessary and

sufficient cut condition that a generalized network (D, s, t, l, u) must satisfy to have a feasible flow.

• Theorem. (MinFlow MaxCut) Let (D, s, t, l, u) be a

generalized network which has a feasible st-flow. Then the minimum value of an st-flow is equal to the maximum of l(δout(X)) − u(δin(X)) for every s¯ t-set X. Furthermore, if l, u are integral then there exists an integral minimum flow.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

Some consequences

Sketch of proof. (very very sketchy. . . ) Let α be a real parameter. Add a new arc (t, s) with l(t, s) := 0 and u(t, s) := α. Then there exists a feasible flow f with val(f ) = α if and only if there exists a feasible circulation x′ in the new network with x′(t, s) = α. By Hoffman’s cut condition this holds if and only if l(δout(X)) − u(δin(X)) α for every s¯ t-set X. Thus the minimum value of α for which this holds corresponds to the maximum of the statement of the theorem.

Orlando Lee – Unicamp Topics in Combinatorial OPtimization

References

• A. Schrijver, Combinatorial Optimization, Vol. A, Springer.

W.J. Cook, W.H. Cunningham, W.R. Pulleyblank and A. Schrijver, Combinatorial Optimization, Wiley. R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice-Hall. Some old slides of mine. . .

Orlando Lee – Unicamp Topics in Combinatorial OPtimization