Integer Linear Programming Problem

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From the course by University of California San Diego

Advanced Algorithms and Complexity

202 ratings

University of California San Diego

Advanced Algorithms and Complexity

202 ratings

Course 5 of 6 in the Specialization Data 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.

From the lesson

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

Meet the Instructors

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 next hard stage problem deals with integers and linear inequalities.

Namely, the problem is called integer linear programming.

The input to this problem is a set, or a collection, or

a system of linear inequalities, which we present here in metrics form.

And our goal is to find integer values for

all the variables that satisfy all the inequalities.

To give it our example, consider the following three inequalities.

The first one says that x1 should be at least one-half.

The second one says that minus x1 plus 8x2 should be non negative.

And the last one says that minus x1 minus 8 times x2 should be at least minus 8.

As usual, we can represent a set of all solutions, all feasible points

to this system of linear inequalities as a convex polygon as follows.

We first draw a half space that contains all points

which satisfy the first inequality, that's shown here in green.

So once again, in the green half space, all the pairs of points (x1,

x2) satisfies the inequality x1 is at least half.

The second, the blue one, half space,

contains all the points that satisfy the second inequality.

Finally, the red half space shown here,

contains all the points that satisfies the last inequality.

In particular, the intersection of these three half spaces,

this triangle, contains all the points that satisfies our three inequalities.

Recall, however, that what we need to find is an integer solution.

That is, we would like x1 and x2 to have integer values.

And so this intersection is non empty, it contains no integer points.

So the integer points are here, the closest ones.

But none of them is inside this region.

Right?

So it turns out that this additional restriction,

namely the restriction that the solution should be integer,

gives us a very hard problem.

In particular, if we just have a system of linear inequalities and we would like

to check whether there is a point that satisfies whether there is a solution to

them, then we can use, for example, simplex method to solve it in practice.

The running time of simplex method is not bounded by polynomial, so

on some pathological cases, it can have exponential running time.

But there are other methods like ellipsoid method or

interior point method that have polynomial upper bounds in the running time.

So in any case,

we can solve systems of linear inequalities efficiently in practice.

But if we additionally require that we need the solution to be integer,

then we get a very difficult problem for

which we have no polynomial algorithm at the moment.

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