Independent Set Problem

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Del curso dictado por University of California San Diego

Advanced Algorithms and Complexity

291 calificaciones

University of California San Diego

Advanced Algorithms and Complexity

291 calificaciones

Curso 5 de 6 en SpecializationData Structures and Algorithms

You've learned the basic algorithms now and are ready to step into the area of more complex problems and algorithms to solve them. Advanced algorithms build upon basic ones and use new ideas. We will start with networks flows which are used in more typical applications such as optimal matchings, finding disjoint paths and flight scheduling as well as more surprising ones like image segmentation in computer vision. We then proceed to linear programming with applications in optimizing budget allocation, portfolio optimization, finding the cheapest diet satisfying all requirements and many others. Next we discuss inherently hard problems for which no exact good solutions are known (and not likely to be found) and how to solve them in practice. We finish with a soft introduction to streaming algorithms that are heavily used in Big Data processing. Such algorithms are usually designed to be able to process huge datasets without being able even to store a dataset.

De la lección

NP-complete Problems

Although many of the algorithms you've learned so far are applied in practice a lot, it turns out that the world is dominated by real-world problems without a known provably efficient algorithm. Many of these problems can be reduced to one of the classical problems called NP-complete problems which either cannot be solved by a polynomial algorithm or solving any one of them would win you a million dollars (see Millenium Prize Problems) and eternal worldwide fame for solving the main problem of computer science called P vs NP. It's good to know this before trying to solve a problem before the tomorrow's deadline :) Although these problems are very unlikely to be solvable efficiently in the nearest future, people always come up with various workarounds. In this module you will study the classical NP-complete problems and the reductions between them. You will also practice solving large instances of some of these problems despite their hardness using very efficient specialized software based on tons of research in the area of NP-complete problems.

Brute Force Search5:42

Search Problems9:44

Traveling Salesman Problem7:57

Hamiltonian Cycle Problem8:09

Longest Path Problem1:42

Integer Linear Programming Problem3:08

Independent Set Problem3:07

Conoce a los instructores

Alexander S. Kulikov
Visiting Professor
Department of Computer Science and Engineering
Michael Levin
Lecturer
Computer Science
Daniel M Kane
Assistant Professor
Department of Computer Science and Engineering / Department of Mathematics
Neil Rhodes
Adjunct Faculty
Computer Science and Engineering

Our last hard set problem that we mentioned here, deals with graphs again.

It is called an independent set problem.

Here, we're given a graph and the budget b.

And our goal is to select at least b vertices such that

there is no edge between any pair of selected vertices.

For example, in this graph that we've seen before in this lecture,

is there is an independent set of size seven?

So the selected vertices are shown here in red.

And it is not difficult to check that there is no edge between a pair

of red vertices.

And the particularization implies that an independent set is indeed

a search problem.

It is easy to check whether a given set of vertices is an independent set, and

that it has size at least b.

It is interesting to note,

that the problem can be easily solved if the given graph is a three.

Namely, it can be solved by the following simple greedy strategy,

given a tree if you want to find even just the independence side of maximum

size we can do the following.

The first thing to do is let's just take all the leaves into a solution.

Then, lets remove all the leaves from the three together with all its parents.

And then, lets just continue this process.

To prove that this algorithm produces and optimal solution,

we need to show the take in all the leaves and our solution is a safe move,

that it is consistent with an optimal solution.

This is usually done as follows.

Assume that there is some optimal solution in which not all the leafs are taken.

Assume that, just for concreteness, assume that this is suboptimal solution.

Not all the leaves are taken here because, well, we have this leaf, this leaf, and

this leaf, which is not in the solution.

Then we show that it can be transformed into another solution which,

without decreasing its size, such that it contains all the leaves.

Indeed, let's just take all these market leaves into a solution.

This will probably require us to discard from a solution all it's parents, but

it will not decrease the size of the solution.

So what we get is another solution whose size is, in this case, actually the same.

But it contains all the leaves.

Which proves that there always exists an optimal solution

which contains all the leaves.

And this in turn means that it is safe to take all the leaves.

We will see the details of this algorithm later in this class.

But in general, once again, if we are given a tree then we can find

an independent set of maximum size in this tree.

Very efficiently in linear time.

But at the moment, we have no polynomial algorithm that finds,

that even checks where there's a reason and

independent set of size b in the given graph in polynomial time.

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