Topics in Combinatorial Optimization Orlando Lee Unicamp 10 de - - PowerPoint PPT Presentation

Topics in Combinatorial Optimization

Orlando Lee – Unicamp 10 de abril 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

Basic definitions

We need some definitions and notation related to linear algebra, convexity, polyhedra and linear programming. We assume some familiarity with these topics and present just the minimum needed in order to present the following topics:

integral polyhedra total dual integral (TDI) systems totally unimodular (TU) matrices

Fix Rn as the grounding vector space. For vectors x, y let xy denote the inner product of x and y.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Linear independence

A vector v is a linear combination of x1, . . . , xk if there exist λ1, . . . , λk such that λ1x1 + · · · + λkxk = v. We say that vectors x1, . . . , xk are linearly independent if there do not exist λ1, . . . , λk, not all equal to 0, such that λ1x1 + · · · + λkxk = 0.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Affine independence

A vector v is an affine combination of x1, . . . , xk if there exist λ1, . . . , λk such that λ1x1 + · · · + λkxk = v and λ1 + . . . + λk = 1. We say that vectors x1, . . . , xk are affinely independent if there do not exist λ1, . . . , λk, not all equal to 0, such that λ1x1 + · · · + λkxk = 0 and λ1 + . . . + λk = 0.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Linear and affine independence

So v is an affine combination of x1, . . . , xk if and only if v

• is a

linear combination of x1

1

• , . . . ,

xk

1

• ,
• r

x1, . . . , xk are affinely independent if and only if x1

• , . . . ,

xk

• are linearly independent.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Linear and affine hull

Let X be a subset of Rn. Let lin(X) := {λ1x1 + · · · + λkxk : k 1, x1, . . . , xk ∈ X, λ1, . . . , λk ∈ R} be the linear hull of X. Let X be a subset of Rn. Let aff(X) := {λ1x1 + · · · + λkxk : k 1, x1, . . . , xk ∈ X, λ1, . . . , λk ∈ R, λ1 + . . . + λk = 1} be the affine hull of X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Dimension and (affine) rank

The rank of a subset X ⊆ Rn, denote rank(X), is the size of a maximal subset of linear independent vectors of X. From basic linear algebra, we know that every maximal subset of linearly independent vectors of a set X ⊆ Rn has the same size. So rank is well-defined. The affine rank of a subset X ⊆ Rn, denoted aff-rank(X), is the size of a maximal subset of affine independent vectors of X.

• Exercise. Prove that this concept is well-defined, that is, every

maximal subset of affinely independent vectors of a set X ⊆ Rn has the same size.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Dimension and (affine) rank

Exercise. (a) Prove that if 0 ∈ aff(X) then aff-rank(X) = rank(X) + 1. (b) Prove that if 0 ∈ aff(X) then aff-rank(X) = rank(X). The dimension of a set X ⊆ Rn is dim(X) := aff-rank(X) − 1. We say that X ⊆ Rn is full-dimensional if dim(X) = n.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Dimension and (affine) rank

In linear independence, the vector 0 is privileged. To generate Rn with linear combinations, we need n linear independent vectors. In affine independence, there is no privileged vector. To generate Rn with affine combinations, we need n + 1 affine independent vectors.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Convexity

A subset C of Rn is convex if λx + (1 − λ)y ∈ C for every x, y ∈ C and every λ ∈ R with 0 λ 1. A vector v is a convex combination of x1, . . . , xk if there exist λ1, . . . , λk 0 such that λ1x1 + · · · + λkxk = v and λ1 + . . . + λk = 1. Let X be a subset of Rn. Let conv(X) := {λ1x1 + · · · + λkxk : k 1, x1, . . . , xk ∈ X, λ1, . . . , λk 0, λ1 + . . . + λk = 1} be the convex hull of X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Convexity

• Theorem. (Carath´

eodory, 1911) For any subset X ⊆ Rn and any vector x ∈ conv(X), there exist affinely independent vectors x1, . . . , xk ∈ X such that x ∈ conv({x1, . . . , xk}). Roughly speaking, every element of conv(X) can be written as a convex combinatiorn of k affine independent vectors of X, where k is at most the affine rank of X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cones

A subset C of Rn is a (convex) cone if C = ∅ and λx + µy ∈ C for every x, y ∈ C and every λ, µ ∈ R+. A vector v is a conic combination of x1, . . . , xk if there exist λ1, . . . , λk 0 such that λ1x1 + · · · + λkxk = v. Let X be a subset of Rn. Let cone(X) := {λ1x1 + · · · + λkxk : x1, . . . , xk ∈ X, λ1, . . . , λk 0} be the cone generated by X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cones

• Theorem. (Carath´

eodory, 1911) For any subset X ⊆ Rn and any vector x ∈ cone(X), there exist affinely independent vectors x1, . . . , xk ∈ X such that x ∈ cone({x1, . . . , xk}). Roughly speaking, every element of cone(X) can be written as a conic combination of k affinely independent vectors of X, where k is at most the affine rank of X. In the statement, if we assume that 0 ∈ X, then we can also write linearly independent vectors instead of affinely independent vectors.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Hyperplanes and half-spaces

A subset H of Rn is a hyperplane if there exist c ∈ Rn \ {0} and δ ∈ R such that H = {x : cx = δ}. We say that the set S := {x : cx δ} is a half-space. If δ = 0 we say that H (S) is a linear hyperplane (half-space).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Polyhedra

A cone C is polyhedral if there is a matrix A such that C = {x : Ax 0}. Equivalently, C is the intersection of finitely many linear half-spaces. Several researchers showed that a convex cone is polyhedral if and only if is finitely generated where C being finitely generated means that there exists X such that C = cone(X).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Polyhedra

A subset P of Rn is a polyhedron if there exists an m × n matrix A and a vector b ∈ Rm such that P = {x : Ax b}. Equivalently, P is the intersection of finitely many half-spaces. We say that the system Ax b determines P. An inequality cx ≤ δ is valid if cx δ holds for every x ∈ P. A polyhedra or cone is rational if the linear system of inequalities that determines them contains only rational inequalities (dx δ with d and δ rational).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Polyhedra

A subset P of Rn is a polytope if there exists X such that P = conv(X). It can be shown that P is a polytope if and only if P is a bounded polyhedron. For subsets A, B of Rn let A + B := {a + b : a ∈ A, b ∈ B}. Minkowski (1936) proved the following fundamental result.

• Theorem. A set P is a polyhedron if and only if P = Q + C for

some polytope A and some polyhedral cone C.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Farka’s lemma

Farka’s lemma has several equivalent statements. It characterizes when a given system of linear (in)equalities Ax b has a feasible solution (or, the polyhedron determined by the system is nonempty).

• Theorem. Ax b is feasible if and only if yb 0 for every y 0

such that yA = 0.

• Corollary. Ax = b has a solution with x 0 if and only if yb 0

for every y such that yA 0.

• Corollary. Ax b has a solution with x 0 if and only if yb 0

for every y 0 such that yA 0.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Linear programming

Linear programming concerns the problem of maximizing or minimizing a linear function over a polyhedron. One of the most important results in the area is the (strong) duality theorem that relates two optimization problems – the primal and the dual – by a minimax equality.

• Theorem. Let A be a matrix and b, c be vectors. Then

max{cx : Ax b} = min{yb : y 0, yA = c}. There exist several equivalent forms of duality theorem, for example: max{cx : Ax b, x 0} = min{yb : y 0, yA c}, max{cx : Ax = b, x 0} = min{yb : yA c}.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Complementary slackness

Consider the linear problem: max{cx : Ax b} = min{yb : y 0, yA = c}. By weak duality we have cx = (yA)x = y(Ax) yb. So a pair x, y of feasible solutions for the primal and the dual, respectively, are both optimum if and only if, the inequality holds with equality. Equivalently, if and only if the following complementary slackness conditions hold: (Ax)i = bi for each i with yi > 0. There exist other forms of complementary slackness, one for each pair of primal and dual problems.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Optimality geometrically

Using Carath´ eodory’s theorem we can prove the following result.

• Theorem. If the optimum value of

max{cx : Ax b} = min{yb : y 0, yA = c} is finite, then the minimum is attained by a vector y such that the rows of A corresponding to the positive components of y are linearly independent. Geometrically, this says that c belongs to the cone generated by the rows of A.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Optimality geometrically

Sketch of proof. Let x∗ be an optimum solution of the primal and let δ := cx∗. Let y 0 be a optimum dual solution; so δ = yb. Then (c δ) = y[A b], that is, the vector (c δ) belongs to the cone generated by the rows

• f the [A b].

By Carath´ eodory’s theorem, c belongs to the cone generated by a linearly independent subset of rows of this matrix. So we may assume that the positive entries of y correspond to a submatrix [A′ b′] of full row rank. By complementary slackness, A′x∗ = b′ and hence, A′ has full row rank.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Faces, facets and vertices

Let P := {x : Ax b} be a polyhedron in Rn. Let c a nonzero vector and let δ := max{cx : Ax b}. The hyperplane H := {x : cx = δ} is called a supporting hyperplane of P. A subset F of P is a face of P if F = P or F = P ∩ H for some supporting hyperplane H of P. F is a face of P ⇐ ⇒ F is the set of optimum solutions of max{cx : Ax b} for some c ∈ Rn. An inequality cx ≤ δ determines or induces a face F if F = {x ∈ P : cx = δ}.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Faces, facets and vertices

Equivalently, F is a face of P if and only if F = {x ∈ P : A′x = b′} for some subsystem A′x b′ of Ax b. So a face itself is a polyhedron. An inequality ax β from Ax b is tight or active in a face F if ax = β holds for every x ∈ F. An inequality ax β from Ax b is an implict equality if Ax b implies ax = β.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Faces, facets and vertices

• Theorem. Let P := {x : Ax b} be a polyhedron of Rn. Let

A′x b′ be a subsystem of implicit equalities in Ax b. Then dimP = n − rank(A′). A facet is an inclusionwise maximal face F of P with F = P. A system Ax b is minimal or irredundant if each proper subsystem A′x b′ has a solution x not satisfying Ax b. If a system Ax b is irredundant and P is full-dimensional, then each of its inequalities determines a facet.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Faces, facets and vertices

A face F of P is minimal if it is an inclusionwise minimal face. Any minimal face is an affine subspace of Rn and all minimal faces are translates of each other. They all have dimension n − rank(A). If each minimal face has dimension 0 then P is called pointed. A vertex or extremal point of P is an element x such that {x} is a minimal face. A polytope is the convex hull of its vertices. For any element z of P = {x : Ax b}, let Azx ≤ bz the subsystem of Ax b consisting of those inequalities that are satisfied by z with equality.

• Theorem. Let P = {x : Ax b} be a polyhedron in Rn and let

z ∈ P. Then z is a vertex of P if and only if rank(Az) = n.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Linear programming

In the 60’s Dantzig introduced the Simplex algorithm for solving linear programming problems. The algorithm was very efficient in practice but it was known that there were (rare) cases in which it could take an exponential number of iterations to converge. In late 70’s, Kaichyan showed that linear programming problems could be solved in polynomial time using the ellipsoid method, which was a method used previously to solve nonlinear problems. In practice, the algorithm was very slow and numerically unstable (it uses square root). In the 80’s, interior points methods were proposed to solve linear programming problems. The algorithms were competitive with the Simplex method.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Optimization versus separation

The separation problem with respect to a convex subset P of Rn consists in given a vector y (a) either answers correctly that y ∈ P, (b) or returns a (separating) hyperplane cx = δ such that P ⊆ {x : dx δ} and cy > δ. This problem is trivial if P is a polyhedron given by a linear system

• f inequalities Ax b.

It is perhaps surprising that this can be solved in some cases even if the linear system of inequalities has exponential size (on the dimension).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Optimization versus separation

A classical result of Gr¨

• tschel, Lov´

asz and Schrijver (1982) roughly says that we can solve an optimization problem over max{cx : x ∈ P} in polynomial time as long as there is a separation oracle for P, that is, an efficient algorithm that solves the separation problem. So P can be given/described implicitly by this oracle. The exact description of the result is very technical. Moreover, its proof is very complex and relies on the ellipsoid method. Informally, it establishes that the optimization problem and its separation problem are polynomially equivalent. The result itself has several implications in combinatorial

• ptimization problems.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Primal-dual method

The primal-dual method is a method for solving LP problems. Many efficient combinatorial algorithms are based on this idea. Suppose we want to solve the following LP problem: max{cx : Ax = b, x 0}. The dual problem is min{yb : yA ≥ c}. Complementary slackness xj > 0 ⇒ yaj = cj where aj denotes the vector corresponding to the column j of A.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Primal-dual method

Idea: let y be a feasible dual solution. Let A= be the submatrix of A consisting of those columns aj of A for which yaj = cj holds. To find a feasible primal solution we solve the restricted linear program: A=x′ = b, x′ 0. If such x′ exists, by adding components 0, we obtain a vector x 0 such that Ax = 0 satisfying complementary slackness. Thus, x and y are optimum solutions of the primal and the dual problems.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Primal-dual method

On the other hand, if no solution exists, by Farka’s lemma, there exists y ′ such that y′A= 0 and y′b < 0. Let ǫ be the largest real number such that (y + ǫy ′)A c. If ǫ = ∞, then the problem is unbounded and the LP is infeasible. Otherwise, note that ǫ > 0. So (y + ǫy ′)b < yb. Start a new iteration with y ← y + ǫy ′. The algorithm for the Maximum Weight Perfect Maching Problem we described uses the primal-dual method.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Integral polyhedra

A vector x is said integral if each of its components is integer. A polyhedron P is integral if it is rational and each face of P contains an integral vector. In particular, every vertex is integral. If P = {x : Ax b} is integral then the LP problem max{cx : Ax b} has an integral optimum solution, if it is finite. Hence, in this case max{cx : Ax b, x ∈ Zn} = max{cx : Ax b}.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Integral polyhedra

• Theorem. Let P be a rational polyhedron in Rn. Then P is

integral if and only if for each rational vector c ∈ Rn, the LP max{cx : Ax b} has an integral optimum solution, if it is finite.

• Theorem. (Edmonds and Giles, 1977) A rational polyhedron P in

Rn is integral if and only if for each integral vector c ∈ Rn the value of max{cx : Ax b} is integer if it is finite.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Totally unimodular matrices

Totally unimodular matrices play an important role in integer programming and combinatorial optimization. A matrix A is totally unimodular (TU) if each square submatrix of A has determinant in {0, +1, −1}. In particular, each component

• f a TU matrix is in {0, +1, −1}.
• Theorem. Let A be a TU m × n matrix and let b ∈ Zm. Then the

polyhedron P := {x : Ax b}. is integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

TU matrices

Sketch of proof. We prove only the case in which P is pointed. Let x be a vertex of P. So there exists a submatrix A′ of A consisting of linearly independent columns of A with rank n such that xj = 0 only if the column aj of A is in A′. Let x′ be the vector obtained from x by deleting the (zero) entries which do no correspond to columns of A′. Hence, A′x′ = b and therefore, x′ = (A′)−1b. (Jump of the cat) From matrix theory each entry of the inverse (A′)−1 is equal to a determinant of a cofactor of A divided by the determinant of A. Since A is TU, this implies that (A′)−1 is integral and since b is integral, it follows that x is integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

TU matrices

• Corollary. Let A be a TU m × n matrix and let b ∈ Zm. Then the

polyhedron P := {x : Ax = b, x 0} is integral.

• Proof. Note that P can be described as

P =   x :   A −A −I   x   b −b      . Since A is TU, the constraint matrix above is also TU and the result follows from the previous theorem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

TU matrices

• Corollary. Let A be a TU m × n matrix, b ∈ Zm and l, u ∈ Zn

+.

Then the polyhedron P := {x : Ax = b, l x u} is integral.

• Proof. Similar to the previous one.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Two important examples of TU matrices

Let G = (V , E) be a graph. The incidence matrix of G is the {0, 1}-matrix M with rows indexed by V and columns indexed by E such that: M[v, e] = 1 if e ∈ δ(v),

• therwise.

Let D = (V , A) be a digraph. The incidence matrix of D is the {0, +1, −1}-matrix M with rows indexed by V and columns indexed by A such that: M[v, a] =    1 if a ∈ δout(v), −1 if a ∈ δin(v),

• therwise.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrices of bipartite graphs are TU

• Theorem. Let G be a bipartite graph. Then the incidence matrix

M of G is TU.

• Proof. Suppose G = (V , E) is (X, Y )-bipartite. Then there exist

matrices MX, MY with rows indexed by X and Y respectively, and both with columns indexed by E such that: M = MX MY

• and each MX, MY has exacty one 1 in each column. So if we

add rows in MX by 1 and subtract rows in MY we obtain the zero vector. So rank(M) |V | − 1.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrices of bipartite graphs are TU

Let B a submatrix of M. We prove by induction that det(B) ∈ {0, +1, −1}. If B has order 1 then det(B) ∈ {0, +1}. So suppose that B has order at least two. Then B satisfies one of the following: (a) every column has exactly two 1’s, or (b) there exists a column of zeroes, or (c) there exists a column with exactly one 1. If (a) or (b) holds then det(B) = 0. If (c) holds, then assume that B[v, e] = 1. Let B′ the matrix

• btained from B by deleting row v and column e. Then

det(B) = (−1)v+e det(B′). By induction det(B′) ∈ {0, +1, −1} and therefore det(B) ∈ {0, +1, −1}.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrices of bipartite graphs are TU

• Corollary. Let G be a bipartite graph containing a perfect
• matching. Then every extreme point of the polytope

x(δ(v)) = 1 for every v ∈ V , x(e) 0 for every e ∈ E is integral.

• Corollary. Let G = (V , E) be a bipartite graph and w ∈ ZE

+. Then

every extreme point of the polyhedron y(u) + y(v) w(e) for every e = uv ∈ E, is integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Incidence matrices of digraphs are TU

• Theorem. Let D be a digraph.Then the incidence matrix M of D

is TU.

• Proof. Let B be a submatrix of M. We prove by induction that

det(B) ∈ {0, +1, −1}. If B has order 1 then det(B) ∈ {0, +1, −1}. So suppose that B has order at least two. Then B satisfies one of the following: (a) every column has exactly one +1 and one −1, or (b) there exists a column of zeroes, or (c) there exists a column with exactly one +1 or one −1. If (a) or (b) holds then det(B) = 0. If (c) holds, then assume that B[v, e] = ±1. Let B′ the matrix

• btained from B by deleting row v and column e. Then

det(B) = ±(−1)v+e det(B′). By induction det(B′) ∈ {0, +1, −1} and therefore det(B) ∈ {0, +1, −1}.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Minimum cost flow/circulation

Let D = (V , A) be a digraph, let b ∈ ZV be a demand/supply function, let c : A → R be a cost function and let l, u : A → Z+ be capacity functions. Consider the following problem: max

• a∈A c(a)x(a)

s.t. x(δout(v)) − x(δin(v)) = b(v) for every v ∈ V , l(a) x(a) u(a) for every a ∈ A. This is known as Minimum Cost Flow/Circulation problem. This can be rewritten as min{cx : Mx = b, l x u}. From the total unimodularity of M we have that P := {x : Mx = b, l ≤ x ≤ u} is integral.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Minimum cost flow/circulation

min{cx : Mx = b, l x u}. This can be used to model (and solve) several combinatorial problems: Shortest Path problem, MaxFlow problem, Directed Chinese Postman problem etc. There also exist (several) combinatorial polynomial algorithms for the Minimum Cost Flow problem. Many real applications can be modeled as a problem of this type. We will not discuss them here. Consult the references. Let us show how to model two well-known problems in this framework.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Typical problems in this framework

Minimum/shortest st-path problem: take l = 0, u = 1 and b(v) :=    +1 if v = s −1 if v = t

• therwise.

If (D, c) contains no negative cycle, then min{cx : Mx = b, 0 x 1} is equivalent to the Shortest Path problem. If (D, c) contains a negative cycle, the integral solution may not have any combinatorial interpretation for the Shortest st-Path problem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Typical problems in this framework

Maxflow problem: add a new arc (t, s) with cost c(t, s) = −1, take as objective function min x(t, s)c(t, s) = max x(t, s); all the remaining arcs have zero cost. Take l = 0 and b = 0. So max{x(t, s) : Mx = 0, 0 x u} is equivalent to the Maxflow problem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Integral polyhedra

Edmonds (1965) in a seminal paper characterized the perfect matching polytope of a graph G = (V , E). More precisely, let MG := {χM : M is a perfect matching of G}. The matching polytope of G is Ppm(G) := conv(MG).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Perfect matching polytope

We have seen that when G = (V , E) is bipartite then Ppm(G) is determined by x(δ(v)) = 1 for every v ∈ V , x(e) 0 for every e ∈ E. If G is not bipartite, then the previously result does not necessarily hold. However, Edmonds showed a nice characterization of Ppm(G), albeit with an exponential number of constraints.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Perfect matching polytope

Let G = (V , E) a graph with a perfect matching (so |V | is even). We say that a cut δ(X) is odd if |X| is odd. It is easy to see that if δ(X) is an odd cut and M is a perfect matching then |M ∩ δ(X)| 1 (in fact, it must be odd).

• Theorem. (Edmonds, 1965) Let G = (V , E) be a graph. Then

Ppm(G) is determined by: x(δ(v)) = 1 for every v ∈ V , x(δ(X)) 1 for every X ⊆ V odd, x(e) 0 for every e ∈ E.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Integral polyhedra

How can we show that such description is actually complete, that is, those linear inequalities do determine Ppm(G)? There are some possibilities: using standard arguments from polyhedral combinatorics, or showing that for any objective function, the corresponding linear programming problem has always an integral optimum solution. We will not use the first method, but a more refined approach of the second one.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

TDI systems

A system Ax b is totally dual integral (TDI) if A and b are rational and for every integral vector c ∈ Zn, the dual problem in: min{cx : Ax b} = max{yb : yA = c, y 0} has an integral optimum solution y, if it exists. The importance of this concept comes from the following result of Edmonds and Giles (1977).

• Theorem. If Ax b is TDI and b is integral, then Ax b

determines an integral polyhedron.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

TDI systems

We list some useful results.

• Theorem. Suppose Ax 1, x 0 is a TDI system. Then the

system Ax 1, 0 x 1 is also TDI.

• Theorem. A system Ax b, x ≥ 0 is TDI if and only if for every

c ∈ Zn the dual in: min{cx : Ax b, x 0} = max{yb : yA c, y 0} has an integral optimum solution y, if it exists.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

The perfect matching polytope

• Theorem. Let G = (V , E) be a graph. Then the system

x(δ(v)) = 1 for every v ∈ V , x(δ(X)) 1 for every X ⊆ V odd, x(e) 0 for every e ∈ E. is TDI. Furthermore, it determines Ppm(G). A typical way to prove this is to design an (primal-dual) algorithm that finds an integral optimal dual solution (and an integral

• ptimum primal solution) for each integral vector c. In the result

above, this is complicated, but we will show some other examples.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Path polyhedra

Let D = (V , A) be a digraph and let s, t ∈ V with s = t. Let Pst-path(D) := conv{χP : P is an st-path of D}. Since the problem of finding a longest st-path is NP-hard, there is little hope of finding a complete characterization of Pst-path(D). However, we can consider the dominant P↑

st-path(D) of Pst-path(D).

Formally, the dominant of a polyhedron P is the polyhedron P↑ consisting of vectors x such that x z for some z ∈ P. Equivalently, P↑ = P + RA

+.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Path polyhedra

Consider the LP problem: min

• a∈A c(a)x(a)

s.a x(δout(X)) 1 for every s¯ t-set X, x(a) 0 for every a ∈ A. and its dual: max

• X y X

s.a

• X:a∈δ(X) yX c(a)

for every a ∈ A, y X 0 for every s¯ t-set X.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Packing st-cuts

When c is integral, then an integral dual solution corresponds to a collection C of st-cuts such that each arc a ∈ A belongs to at most c(a) st-cuts in C. We have seen the following result (Exercise).

• Theorem. Let D = (V , A) be a digraph, let s, t ∈ V with s = t

and let c : A → Z+ a capacity function. Then the length of a shortest (weighted) st-path is equal to maximum size of a collection C of st-cuts such that each arc a ∈ A belongs to at most c(a) st-cuts in C.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Path polyhedra

• Theorem. Let D = (V , A) be a digraph and let s, t ∈ V with

s = t. Then the system x(δ(X)) 1 for every s¯ t-set X, x(a) 0 for every a ∈ A. is TDI. Furthermore, it determines P↑

st-path(D).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cut polyhedra

Let D = (V , A) be a digraph and let s, t ∈ V with s = t. Let Pst-cut(D) := conv{χC : C is an st-cut of D}. Again, there is little hope of finding a system describing this polytope, because MaxCut is NP-hard. So we consider the dominant P↑

st-cut(D) rather than Pst-cut(D).

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cut polyhedra

Consider the LP problem: min

• a∈A c(a)x(a)

s.a x(P) 1 for every st-path P, x(a) 0 for every a ∈ A. and its dual: max

• P y P

s.a

• P:a∈A(P) yP c(a)

for every a ∈ A, y P 0 for every st-path P.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Packing st-paths

When c is integral, then an integral dual solution corresponds to a collection P of st-paths such that each arc a ∈ A belongs to at most c(a) st-paths in P. We have seen the following result.

• Theorem. (Integral MaxFlow MinCut Theorem) Let D = (V , A) be

a digraph, let s, t ∈ V with s = t and let c : A → Z+ a capacity

• function. Then the weight of a minimum st-cut is equal to

maximum size of a collection P of st-paths such that each arc a ∈ A belongs to at most c(a) st-paths in P.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Cut polyhedra

• Theorem. Let D = (V , A) be a digraph and let s, t ∈ V with

s = t. Then the system x(P) 1 for every st-path P, x(a) 0 for every a ∈ A. is TDI. Furthermore, it determines P↑

st-cut(D).

Note that the dual problem consists in finding a maximum (fractional) packing of st-paths. So, it is equivalent to the MaxFlow problem (recall the Flow Decomposition Theorem). It provides another polyhedral formulation of the MaxFlow problem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Chain polytope

Let (S, ) be a poset. Let Pchain(S) := conv{χC : C is a chain of S}.

• Theorem. Let (S, ) be a poset. Then the system

x(A) 1 for every antichain A, x(s) 0 for every s ∈ S. is TDI. Furthermore, it determines Pchain(D). This follows from the weighted Mirsky’s theorem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

Antichain polytope

Let (S, ) be a poset. Let Pantichain(S) := conv{χA : A is an antichain of S}.

• Theorem. Let (S, ) be a poset. Then the system

x(C) 1 for every chain C, x(s) 0 for every s ∈ S. is TDI. Furthermore, it determines Pantichain(D). This follows from the weighted Dilworth’s theorem.

Orlando Lee – Unicamp Topics in Combinatorial Optimization

References

• A. Schrijver, Theory of Linear and Integer Programming,
• Wiley. (a classic book)

C.E. Ferreira and Y. Wakabayashi, Combinat´

• ria Poli´

edrica e Planos-de-Corte Faciais. (Escola de Computa¸ c˜ ao)

• A. Schrijver, Combinatorial Optimization, Vol. A, Springer.
• A. Frank, Connections in Combinatorial Optimization, Oxford.

Orlando Lee – Unicamp Topics in Combinatorial Optimization