1.3 Modeling with exponentially many constr.

1.3 Modeling with exponentially many constr.

Some strong formulations (or even formulation itself) may involve exponen-tially many constraints (cutting plane method is used to solve the LP relax-ations of them)

The minimum spanning tree problem undirected graph ( ). Every edge has cost .Find a spanning tree (acyclic connected subgraph of ) of minimum cost.

Let define one end node of is in

Integer Programming 2013 1

Subtour elimination formulation min

Cutset formulationmin

Thm 1.1 :

(a) , and there exist examples for which the inclusion is strict.(b) can have fractional extreme points.


It can be shown that LP relaxation of subtour elimination formulation gives integer optimal solutions. (polymatroid)

Why consider IP formulation although there exist good algorithms (Kruskal, Prim)?

Algorithms may fail if problem structure changed a little bit: degree constrained spanning tree problem, Shortest total path length spanning tree problem, Steiner tree problem, capacitated spanning tree problem, …

Formulation of a basic problem may be used as part of a formulation for a larger complicated problem.

Theoretical analysis, e.g. strength of 1-tree relaxation of TSP.


The traveling salesman problem

undirected graph. Every edge has cost .Find a tour (a cycle that visits all nodes exactly once) of minimum cost.

Cutset formulationminimizesubject to

. Subtour elimination formulation

minimizesubject to

. LP relaxations of both formulations give the same solution set.


Remarks

For directed version of the problem, the following formulation is possible, which is smaller in size. But it is a bad formulation. (refer exercise 1.21 in text page 32)

Note that, are continuous variables in the above formulation. Undirected TSP is a special case of directed case, we may replace each edge

by two directed arcs with opposite direction and having the same costs as the edge.


Is the formulation correct?The formulation has variables. If feasible, we only read values ( projection of to space)We need to show that (1) any tour solution satisfies the constraints and (2) any non-tour solution does not satisfy the constraints.(1) For any tour , if node is th node in the tour, assign .(2) If is 0,1 and satisfies degree constraints, it is either a tour or consists of subtours. If subtours exist, there is one that does not include node 1. Add the constraints along the arcs in the subtour.


Comparing the LP relaxation of the cutset formulation (A) (in directed case version) and the LP relaxation of the previous formulation (B): It can be shown that the projection of the polyhedron B onto y space gives a polyhe-dron which completely contains A (the inclusion is strict), hence cutset formu-lation (or subtour elimination formulation) is stronger.

Although the previous formulation is not strong, it can be an alternative to use if you only have a generic IP software to use, not the sophisticated one to handle the cutset constraints.


How to Solve the LP relaxation of the Cut-Set Formulation? (many constr.)


Solve LP relaxation (w/o cut-set constraints)

If y* tour, stop.O/w find violated cut-set

violated cut-set?

Solve LP after adding theCut-set constraint.

Y

N

Stop

If the obtained solution is not a tour, branch and apply the same procedure again. Choose the best solution

Branching : If , solve two subproblems after setting and .

Branch-and-cut approach ( cutting plane alg.) Ideas for TSP formulation can be used for various routing, sequencing prob-

lems. Branch-and-cut Ideas useful to solve many difficult IP problems. What can we do for the LP with many variables? For the LP with many vars.

and constraints? TSP site: http://www.tsp.gatech.edu/


http://www.tsp.gatech.edu/

The perfect matching problem

Match persons into pairs perfectly. Cost if person is matched with person . minimize

subject to .

(see Fig 1.7)

Add oddor odd

Both have .


Cut covering problems

General problem class that includes many problems on network and graph , undirected graph costs for

Cut covering problems minimize

subject to .

There exists an optimal solution which is minimal w.r.t. inclusion. ( )


minimize

subject to .

The minimum spanning tree for all

The traveling salesman problem for all

The perfect matching problem for all with odd

The Steiner tree problem needs to be connected by a tree possibly using nodes in .

for all with otherwise


The survivable network design problemCosts for all requirements for every pair of nodes Select a set of edges from at minimum cost, so that between every pair of nodes and there are at least paths that do not share any edges ( edge-dis-joint paths)

The vehicle routing problem


Dircted vs. undirected formulations

Steiner tree problem minimize

subject to (1.8) .

(a) (b) (c)Let be the set of edges, whose endpoints lie in different .

minimize

subject to satisfying (a)-(c) . (1.9)


Directed version

Find a minimum cost directed subtree that contains a directed path between some given root vertex 1 (), and every other terminal in .

minimize subject to (1.10)

.

can recover by setting for all (for linear relaxations)


(1.8) is a special case of (1.9) with .

Assume that the root vertex and consider

Add above together with for and

setting we get feasible for the linear relaxation of (1.9)

There are examples such that

For TSP, directed formulation has the same strength


1.4 Modeling with exponentially many variables

Column generation methodEnumerate partial feasible solutions and represent their interactions in the master model. (Decomposition)Important modeling tool in applications

The cutting stock problemLarge rolls of paper of width (raw). Customer demand rolls of width (fi-nal), . ( )Minimize the number of large rolls used while satisfying customer demand.

Cutting pattern , : produce rolls of width in th cutting pattern (number of possible cutting patterns can be enormous)A feasible cutting pattern must satisfy and is nonnegative integer. (inte-ger knapsack constraint)


Formulationminimize subject to

is the number of rolls of width (raw) cut by cutting pattern . LP relaxation can be solved by column generation. Fractional optimal solu-

tion may be rounded down and a few more raws may be used to produce addi-tional finals. (close to optimal)


Combinatorial auctions: set of bidders, : set of items being auctioned : bid that bidder is willing to pay for Assume that if Bidders are allowed to bid on combinations of different items.Let

maximize subject to


The vehicle routing problemtransportation network: undirected, cost Node 0 is central depot. Node represents customers with demand .Company has vehicles with capacities Assume demand of each customer cannot be divided into several vehicles.

Let if partial tour is used, and zero, otherwise () : equals one if node is visited in partial solution . : cost of tour

minimize subject to


1.3 Modeling with exponentially many constr.

Documents

remarksinteger programming

ynstop8integer programming

variablesinteger programming

cutset formulation

ip formulation

integer optimal solutions

previous formulation

steiner tree problem