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What is the integrality gap?
Integrality gaps essentially represent the inherent limits of a particular linear or convex relaxation in approximating an integer program. Generally, if the integrality gap of a particular relaxation is x, then any approximation algorithm based on that relaxation cannot hope to do better than an x-approximation.
What is greedy set cover algorithm?
Greedy Set Cover: A simple greedy approach to set cover works by at each stage selecting the set that covers the greatest number of uncovered elements. The algorithm is presented in the code block below. We repeatedly select the set of S that covers the greatest number of elements of U and add it to the cover.
Is clique and set cover problem NP complete?
Since VERTEX-COVER can be reduced to CLIQUE in polynomial time, CLIQUE ∈ NP and VERTEX-COVER is NP-Complete, CLIQUE is also NP-Complete.
Is minimum set cover NP complete?
In vertex cover, each vertex (i.e., set) covers the adjacent edges (i.e., elements); in set cover, each set can cover an arbitrary set of elements. Unfortunately, MIN-SET-COVER is NP-hard (i.e., NP-complete as a decision problem).
What is integrality constraint?
The integrality constraints allow MIP models to capture the discrete nature of some decisions. For example, a variable whose values are restricted to 0 or 1, called a binary variable, can be used to decide whether or not some action is taken, such as building a warehouse or purchasing a new machine.
How is integer programming related to LP relaxation?
1 Integer Programming and LP relaxation De nition 10.1.1 An integer program is a linear program in which all variables must be integers. As in a linear program, the constraints in an integer program form a polytope. However, the feasible set is given by the set of all integer-valued points within the polytope, and not the entire polytope.
Is the vertex cover approximation algorithm based on linear programming?
The vertex cover approximation algorithm based on linear programming is very ele- gant and simple, but it requires the solution of a linear program. Our previous vertex cover approximation algorithm, instead, had a very fast linear-time implementation.
Which is the optimal value of the LP?
However, since the feasible region of the LP is larger than the feasible region of the IP, the optimal value of the former is no worse than the optimal value of the latter. This implies that the optimal value to the LP is a lower bound on OPT, the optimal value for the problem we started out with.