In Step 8, clearly if y ai z Cj for all j then f is dual feasible so it is acceptable to return to Step 2. Consider 1 found in Step 9. Clearly by construction the numerator is positive. Thus, since a convex combination of Chvatal functions is a Chvatal function, pis a Chvatal function.
Now consider the flow of the algorithm. Notice that each time the algorithm leaves Step 2, it either ends up in Step 4, 5, 6, 8 or 9. Steps 5 and 6 are terminal steps as shown above. Further we have shown that when the algorithm leaves Steps 4, 8 or 9 to return to Step 2, it does so with an integral dual feasible function which forces the objective value to decrease by at least 1.
Thus if the subprob- lems can be solved in polynomial or pseudopolynomial time, a pseudopolynomial time algorithm results. We should note here that in Step 7, it is not necessary to solve RMP to optimali- ty. It is merely necessary to find a dual solution of value less thanf b which proves that the value of RMP is less thanf b. Note that the dual of any linear program- ming solution of RMP will have value at mostf b.
This fact results in computa- tional savings when Step 7 is implemented using cutting planes, as discussed in Section 3. The black boxes The major problems with implementing the primal dual algorithm of Section 2 above are in Steps 2, 3,7 and 8. In particular, note that RFP is an integer program- ming problem that could be as difficult as the original problem P.
Here, we discuss two approaches to dealing with solving RFP and implementing the black boxes of Steps 3 and 8 of the algorithm.
In Section 3. Then, in Section 3. Specifi- cally we specialize the results of Chvatal [5] to build dual superadditive functions using Gomory cutting planes in order to implement Steps 2, 3, 7 and 8 of the algo- rithm. Finally, in Section 3. In this case the cardinality matching algorithm of Edmonds [7] is used to solve RFP. Easy special cases Here we briefly discuss two very special cases where implementation of the algo- rithm is easy.
For any Jc 1,. First consider the case when PJ is feasible if and only if its linear programming relaxation is feasible for each J. If so, then there is a linear function y the optimal dual solution to the linear programming relaxation of RFP that will satisfy the conditions of Step 3 of the algorithm.
LleweNyn, J. Ryan function that occurs in Step 4 of the algorithm is vital to the convergence of the algorithm. However in this case, because the linear dual solution can be used, con- vergence is guaranteed without rounding. Since no basis is ever repeated, the algorithm is finite. Thus, with the rounding step omitted, the dual function will remain linear throughout the execu- tion of the algorithm.
As a consequence, complementary linearity will always be satisfied trivially, RMP need never be solved, and our algorithm reduces to the primal dual algorithm for linear programming. Next, define the group relaxation of P , called here G : max cx, s. As our second easy special setting, we now consider the case when for any column set, J, the problem PJ is feasible if and only if both its linear relaxation and group relaxation are feasible. A simple example where this condition holds is if A is a diagonal matrix with nonnegative integers along its diagonal.
Now, to solve RFP one must first check if the linear relaxation is feasible. If not, then as above, the linear programming dual solution will work in Step 3 of the algorithm directly. If the linear relaxation is feasible, then next check the feasibility of the group relaxa- tion. Moreover, y can be found in polynomial time by a unimodular elimination scheme see [6]. This function satisfies the property required by Step 3 of the algorithm.
If both the linear and group relaxations are feasible, then there is a feasible solution to RFP with value 0, and this solution will be optimal for P. In particular, there will be an optimal solution to the linear programming relaxation of RFP that is integral. This integral solution can be found by pivoting among the alternate optima to the linear programming relaxation or by using other standard techniques.
Note that such an integral solution need only be sought in the final iteration of the algorithm. For any integer programming problem, the first instance of RFP to arise can always be easily solved in the absence of dual degeneracy in the linear programming relaxation.
In the absence of dual degeneracy the set J is exactly equal to the indices of the basic variables of the optimal solution of the linear programming relaxation. In [5] Chvatal has shown the following although using different terminology : Theorem 3.
Theorem 3. Consider the restricted feasibility problem RFP. Similarly, consider the restricted maximization problem RMP. For continuity of exposition, the details of the construction of the function guaranteed by Theorem 3. See also [20] for results relating cutting planes and Chvatal functions. Recall RFP , written as an optimization problem with artificial variables. If the linear programming relaxation of RFP has an optimal solution with value less than 0, let y be the optimal dual solution.
Otherwise, the optimal value of the linear programming relaxation of RFP is 0. That is, the following linear programming problem has value less than 0: s. If the optimal solution to RMP is less than f b , then let y be the optimal dual solution. We now give the details of the construction of the function F corresponding to a cut as in Theorem 3. Although these can be derived from [5] we have included them for completeness.
We will assume that we are working on RFP. The adapta- tion for RMP is straightforward: the same procedure is followed with the omission of the artificial variables. Let the slack variables which have been added to the tableau for the rows corre- sponding to the cuts be s,,. Suppose that we want to cut on row i of the current tableau, and that row i has the form: where 6 is fractional. However adding 2 alone has the same effect, since 1 is an equality and remains part of the problem.
Note that the coefficients of the slacks in 1 will always occur in the ith row of B-I, since there were originally unit vectors in these columns. Since yk- LykJ r0 always, F is a Chvatal function. They will remain valid. Moreover, when solving RMP , it is not necessary to continue to add cuts until the optimal integer solution is found. It suffices to add cuts only until the optimal fractional value falls below f b. Then the dual function f can be constructed as above, and it will have value less than f b as required by Step 8 of the algorithm.
Of course, one must still construct this function from the facet defining inequality. Matching The primal dual algorithm of Section 2 can be specialized to find a maximum weight matching in a weighted graph G.
We have chosen this example because RFP can be solved efficiently. When it becomes necessary to solve RMP , deep cuts can be generated using a separation procedure due to Padberg and Rao [ This approach is not likely to be more efficient than specialized weighted matching algorithms, but provides a nice illustration of how the primal dual algorithm can be tailored to a specific problem.
Without loss of generality suppose that any maximum weight matching is a perfect matching G is easily altered so that this is the case. Let the node set of G be V. Given a graph G, an odd set cover of G is a set of node sets Nr,. Edmonds see [7] showed that the following duality holds: the maximum car- dinality of a matching in a graph is equal to the minimum capacity of an odd set cover.
The algorithm given in [7] returns both a maximum cardinality matching and the corresponding odd set cover. We will use this matching duality result here, and assume that we have available an algorithm that finds a maximum cardinality matching which also returns an odd set cover whose capacity is equal to the car- dinality of the matching.
Problem RFP for the matching problem is efficiently solved. Given J, we must determine if P is feasible using only the variables in J. In the matching setting, we must determine if there is a perfect matching in G using only the edges indexed by J.
We begin by finding the maximum cardinality matching on the graph whose edges consist only of those indexed by J. If this is a perfect matching of G, then we have a feasible solution of P satisfying complementary slackness and we proceed to Step 6 or 7 of the algorithm. Otherwise, we have a matching of size p, where 2p is less than the number of nodes in G. Without loss of generality assume that the vertices ur,.
The function satisfies the condition of Step 3 in the algorithm. Now let e be an edge indexed by je J. Since e is indexed by a member of J, it is covered by the odd set cover given by the cardinality matching algorithm. We now consider RMP. When we reach Step 7 of the algorithm we have found a perfect matching using only edges in J, but the value of that perfect matching is less than f b.
As discussed above, this can be accomplished by adding cutting planes until the value of RMP first drops below f b. Recall that RMP will always have value at most f b.
In the matching setting we have the advantage that we know what the facets of the corresponding polytope are. Let S be a set of nodes in the graph, and let E S denote the set of edges whose both endpoints are in S. Then every facet defining inequality is of the form where S is a node set of odd cardinality. It is easy to see that the function defines the facet derived from an odd set S.
Now suppose that we have solved the linear programming relaxation of RMP , and obtained a basic fractional solution with value equal to f b. The fractional solution will violate one of the above facet describing inequalities. Padberg and Rao [17] showed that the separation problem for the matching polytope can be solved in polynomial time.
That is, a facet defining inequality that is violated by a given infeasible solution can be determined in polynomial time. Thus in polynomial time we can generate a cut as described above that will reduce the value of RMP as desired, or find an integer solution with value equal to f b. Future work Future work related to this algorithm shotrId be directed towards finding special structures which allow efficient solution of the restricted subproblems.
More specifi- cally, one aim is to identify cases where the problem can be solved without resorting to the use of cutting planes in the solution of RFP. These problems can be of two types-those where the type of data is known to be such that the subproblems can be solved easily like those cases discussed in Section 3.
With respect to this last type of problem, the richest potential lies with integer pro- gramming problems with known pseudopolynomial algorithms.
One other direction for future work is in the area of column generation. A primal dual integer programming algorithm References [l] C. Blair and R. Jeroslow, The value function of a mixed integer program 1, Discrete Math.
Jeroslow, The value function of an integer program, Math. Programming 23 Jeroslow, Computational complexity of some problems in parametric discrete programming I, Math. Burdet and E. Johnson, A subadditive approach to solve linear integer programs, in: Annals of Discrete Mathematics 1 North-Holland, Amsterdam, Chvatal, Edmonds polytopes and a hierarchy of combinatorial problems, Discrete Math.
Domich, R. Kannan and L. Trotter Jr, Hermite normal form computation using modulo determinant arithmetic, Math. Edmonds, Paths, trees and flowers, Canad. Edmonds and F. Giles, A min-max relation for submodular functions on graphs, in: Annals of Discrete Mathematics 1 North-Holland, Amsterdam, Gomory, Outline of an algorithm for integer solutions to linear programs, Bull. Gomory, Solving linear programs in integers, in: R. Log In Sign Up. Download Free PDF.
Sandjon Landry. Download PDF. A short summary of this paper. To make a computer do anything i. In a computer program you tell a computer, step by step, exactly what you want it to do. The computer then executes the program, following each step mechanically, to accomplish the end goal.
In mathematics, computer science, and related subjects, an algorithm is a finite sequence of steps expressed for solving a problem. Algorithms are used for calculation, data processing, and many other fields. In computing, algorithms are essential because they serve as the systematic procedures that computers require.
A good algorithm is like using the right tool in the workshop. It does the job with the right amount of effort. Using the wrong algorithm or one that is not clearly defined is like trying to cut a piece of plywood with a pair of scissors: although the job may get done, you have to wonder how effective you were in completing it. This description or procedure takes some value or set of values as input, and produces some value or set of values as output.
A set of instructions is not an algorithm if it does not have a definite stopping place, or if the instructions are too vague to be followed clearly. The stopping place may be at variable points in the general procedure, but something in the procedure must determine precisely where the stopping place is for a particular case.
Let us follow an example to help us understand the concept of algorithm in a better way. Here are three different ways algorithms that you might give your friend for getting to your home.
All these three algorithms accomplish the same goal, but each algorithm does it in a different way. Each algorithm also has a different cost and a different travel time.
Taking a taxi, for example, is the fastest way, but also the most expensive. Taking the bus is definitely less expensive, but a whole lot slower. You choose the algorithm based on the circumstances. In computer programming, there are often many different algorithms to accomplish any given task. Each algorithm has advantages and disadvantages in different situations. Sorting is one place where a lot of research has been done, because computers spend a lot of time sorting lists.
The language used does not obey the rules of a particular programming language. One should feel free to write in a non-ambiguous way every instruction of the algorithm regardless of constrains imposed by the syntax rules of a particular programming language.
Some of the ways in which algorithms can be represented are, pseudo code and flow charts. Writing pseudo code is one of the best ways to plan a computer program. Each step in the flowchart is followed by an arrow that indicates which step to do next. It helps the programmer to put efforts more efficiently on that part.
In that case, flowchart becomes complex and clumsy. It also assists in finding the key elements of a process, while drawing clear lines between where one process ends and the next one starts. Flowcharts also uncover steps that are redundant or misplaced. Elongated circle: indicate the beginning start or end stop of the algorithm b.
Rectangle: indicates instructions or actions c. Diamond: indicates a decision that has to be made d. Arrow: indicates the direction of flow e. Parallelogram: indicates data input and output f. Delay: used to indicate a delay or wait in the process for input from some other process. Connector : connector symbol is used to connect different flow lines. There should not be any room for ambiguity in understanding the flowchart. Variable: an object in a program whose value can be modified during the execution of the program.
In the above flow chart, x, y and z are variables.
0コメント