What is complexity of simplex algorithm for binary integer programming problem? For worst case or average case?
I'm solving assignment problem.
References:
https://en.wikipedia.org/wiki/Integer_programming
https://en.wikipedia.org/wiki/Simplex_algorithm
Since it's for the assignment problem, that changes matters. In that case, as the wiki page notes, the constraint matrix is totally unimodular, which is exactly what you need to make your problem an instance of normal linear programming as well (that is, you can drop the integrality constraint, and the result will still be integral).
So, it can be solved in polynomial time. The Simplex algorithm doesn't guarantee that however.
Of course there are also other polynomial time algorithms to solve the assignment problem.
In a general sense, binary integer programming is one of Karp's 21 NP-complete problems, so assuming P!=NP it's safe to say that Simplex's worst-case running time is lower-bounded by Ω(poly(n)). Again, in general, similar to SAT solvers, the "average" case is going to be heavily dependent upon what you're taking the average across. Until you've got more specific information about the class of problems you're trying to solve with simplex, I don't think there is a good answer.
I'll do some more thinking and update when I have more information.
Can a backtrack and "branch and bound" problem be always solved using dynamic programming?? i.e. given a problem which can be solved using a backtrack method be also solved using dynamic programming
In the general case, whether dynamic programming can be applied, maybe. But whether dynamic programming will definitely lead to an efficient or a pseudo-efficient solution, no.
For example, there can be a number of NP complete Integer Linear Programming problems that need to be solve using branch & bound or through brute-force backtracking since dynamic programming formulation is not possible.
For example this question that I asked some time back, I could not form a DP formulation and I had to resort to finding a solver for my ILP problem. Strange but practical 2D bin packing optimization
There is certainly no such comparison of backtracking and DP because in general DP is used for optimization problems where you need best of many possible solutions whereas backtracking is used to search a single solution to a problem. Whereas you may have a good DP solution to problems which can be solved using Branch and Bound but not always as some problems may not be decomposable to smaller subproblems hence DP solution may not exist.
Recently I've been looking at some greedy algorithm problems. I am confused about locally optimal. As you know, greedy algorithms are composed of locally optimal choices. But combining of locally optimal decisions doesn't necessarily mean globally optimal, right?
Take making change as an example: using the least number of coins to make 15¢, if we have
10¢, 5¢, and 1¢ coins then you can achieve this with one 10¢ and one 5¢. But if we add in a 12¢ coin the greedy algorithm fails as (1×12¢ + 3×1¢) uses more coins than (1×10¢ + 1×5¢).
Consider some classic greedy algorithms, e.g. Huffman, Dijkstra. In my opinion, these algorithms are successful as they have no degenerate cases which means a combination of locally optimal steps always equals global optimal. Do I understand right?
If my understanding is correct, is there a general method for checking if a greedy algorithm is optimal?
I found some discussion of greedy algorithms elsewhere on the site.
However, the problem doesn't go into too much detail.
Generally speaking, a locally optimal solution is always a global optimum whenever the problem is convex. This includes linear programming; quadratic programming with a positive definite objective; and non-linear programming with a convex objective function. (However, NLP problems tend to have a non-convex objective function.)
Heuristic search will give you a global optimum with locally optimum decisions if the heuristic function has certain properties. Consult an AI book for details on this.
In general, though, if the problem is not convex, I don't know of any methods for proving global optimality of a locally optimal solution.
There are some theorems that express problems for which greedy algorithms are optimal in terms of matroids (also:greedoids.) See this Wikipedia section for details: http://en.wikipedia.org/wiki/Matroid#Greedy_algorithms
A greedy algorithm almost never succeeds in finding the optimal solution. In the cases that it does, this is highly dependent on the problem itself. As Ted Hopp explained, with convex curves, the global optimal can be found, assuming you are to find the maximum of the objective function of course (conversely, concave curves also work if you are to minimise). Otherwise, you will almost certainly get stuck in the local optima. This assumes that you already know the objective function.
Another factor which I can think of is the neighbourhood function. Certain neighbourhoods, if large enough, will encompass both the global and local maximas, so that you can avoid the local maxima. However, you can't make the neighbourhood too large or search will be slow.
In other words, whether you find a global optimal or not with greedy algorithms is problem specific, although for most cases, you will not find the globally optimal.
You need to design a witness example where your premise that the algorithm is a global one fails. Design it according to the algorithm and the problem.
Your example of the coin change was not a valid one. Coins are designed purposely to have all the combinations possible, but not to add confusion. Your addition of 12c is not warranted and is extra.
With your addition, the problem is not coin change but a different one (even though the subject are coins, you can change the example to what you want). For this, you yourself gave a witness example to show the greedy algorithm for this problem will get stuck in a local maximum.
What properties should the problem have so that I can decide which method to use dynamic programming or greedy method?
Dynamic programming problems exhibit optimal substructure. This means that the solution to the problem can be expressed as a function of solutions to subproblems that are strictly smaller.
One example of such a problem is matrix chain multiplication.
Greedy algorithms can be used only when a locally optimal choice leads to a totally optimal solution. This can be harder to see right away, but generally easier to implement because you only have one thing to consider (the greedy choice) instead of multiple (the solutions to all smaller subproblems).
One famous greedy algorithm is Kruskal's algorithm for finding a minimum spanning tree.
The second edition of Cormen, Leiserson, Rivest and Stein's Algorithms book has a section (16.4) titled "Theoretical foundations for greedy methods" that discusses when the greedy methods yields an optimum solution. It covers many cases of practical interest, but not all greedy algorithms that yield optimum results can be understood in terms of this theory.
I also came across a paper titled "From Dynamic Programming To Greedy Algorithms" linked here that talks about certain greedy algorithms can be seen as refinements of dynamic programming. From a quick scan, it may be of interest to you.
There's really strict rule to know it. As someone already said, there are some things that should turn the red light on, but at the end, only experience will be able to tell you.
We apply greedy method when a decision can be made on the local information available at each stage.We are sure that following the set of decisions at each stage,we will find the optimal solution.
However, in dynamic approach we may not be sure about making a decision at one stage, so we carry a set of probable decisions , one of the probable elements may take to a solution.
What is the difference between a heuristic and an algorithm?
An algorithm is the description of an automated solution to a problem. What the algorithm does is precisely defined. The solution could or could not be the best possible one but you know from the start what kind of result you will get. You implement the algorithm using some programming language to get (a part of) a program.
Now, some problems are hard and you may not be able to get an acceptable solution in an acceptable time. In such cases you often can get a not too bad solution much faster, by applying some arbitrary choices (educated guesses): that's a heuristic.
A heuristic is still a kind of an algorithm, but one that will not explore all possible states of the problem, or will begin by exploring the most likely ones.
Typical examples are from games. When writing a chess game program you could imagine trying every possible move at some depth level and applying some evaluation function to the board. A heuristic would exclude full branches that begin with obviously bad moves.
In some cases you're not searching for the best solution, but for any solution fitting some constraint. A good heuristic would help to find a solution in a short time, but may also fail to find any if the only solutions are in the states it chose not to try.
An algorithm is typically deterministic and proven to yield an optimal result
A heuristic has no proof of correctness, often involves random elements, and may not yield optimal results.
Many problems for which no efficient algorithm to find an optimal solution is known have heuristic approaches that yield near-optimal results very quickly.
There are some overlaps: "genetic algorithms" is an accepted term, but strictly speaking, those are heuristics, not algorithms.
Heuristic, in a nutshell is an "Educated guess". Wikipedia explains it nicely. At the end, a "general acceptance" method is taken as an optimal solution to the specified problem.
Heuristic is an adjective for
experience-based techniques that help
in problem solving, learning and
discovery. A heuristic method is used
to rapidly come to a solution that is
hoped to be close to the best possible
answer, or 'optimal solution'.
Heuristics are "rules of thumb",
educated guesses, intuitive judgments
or simply common sense. A heuristic is
a general way of solving a problem.
Heuristics as a noun is another name
for heuristic methods.
In more precise terms, heuristics
stand for strategies using readily
accessible, though loosely applicable,
information to control problem solving
in human beings and machines.
While an algorithm is a method containing finite set of instructions used to solving a problem. The method has been proven mathematically or scientifically to work for the problem. There are formal methods and proofs.
Heuristic algorithm is an algorithm that is able to produce an
acceptable solution to a problem in
many practical scenarios, in the
fashion of a general heuristic, but
for which there is no formal proof of
its correctness.
An algorithm is a self-contained step-by-step set of operations to be performed 4, typically interpreted as a finite sequence of (computer or human) instructions to determine a solution to a problem such as: is there a path from A to B, or what is the smallest path between A and B. In the latter case, you could also be satisfied with a 'reasonably close' alternative solution.
There are certain categories of algorithms, of which the heuristic algorithm is one. Depending on the (proven) properties of the algorithm in this case, it falls into one of these three categories (note 1):
Exact: the solution is proven to be an optimal (or exact solution) to the input problem
Approximation: the deviation of the solution value is proven to be never further away from the optimal value than some pre-defined bound (for example, never more than 50% larger than the optimal value)
Heuristic: the algorithm has not been proven to be optimal, nor within a pre-defined bound of the optimal solution
Notice that an approximation algorithm is also a heuristic, but with the stronger property that there is a proven bound to the solution (value) it outputs.
For some problems, noone has ever found an 'efficient' algorithm to compute the optimal solutions (note 2). One of those problems is the well-known Traveling Salesman Problem. Christophides' algorithm for the Traveling Salesman Problem, for example, used to be called a heuristic, as it was not proven that it was within 50% of the optimal solution. Since it has been proven, however, Christophides' algorithm is more accurately referred to as an approximation algorithm.
Due to restrictions on what computers can do, it is not always possible to efficiently find the best solution possible. If there is enough structure in a problem, there may be an efficient way to traverse the solution space, even though the solution space is huge (i.e. in the shortest path problem).
Heuristics are typically applied to improve the running time of algorithms, by adding 'expert information' or 'educated guesses' to guide the search direction. In practice, a heuristic may also be a sub-routine for an optimal algorithm, to determine where to look first.
(note 1): Additionally, algorithms are characterised by whether they include random or non-deterministic elements. An algorithm that always executes the same way and produces the same answer, is called deterministic.
(note 2): This is called the P vs NP problem, and problems that are classified as NP-complete and NP-hard are unlikely to have an 'efficient' algorithm. Note; as #Kriss mentioned in the comments, there are even 'worse' types of problems, which may need exponential time or space to compute.
There are several answers that answer part of the question. I deemed them less complete and not accurate enough, and decided not to edit the accepted answer made by #Kriss
Actually I don't think that there is a lot in common between them. Some algorithm use heuristics in their logic (often to make fewer calculations or get faster results). Usually heuristics are used in the so called greedy algorithms.
Heuristics is some "knowledge" that we assume is good to use in order to get the best choice in our algorithm (when a choice should be taken). For example ... a heuristics in chess could be (always take the opponents' queen if you can, since you know this is the stronger figure). Heuristics do not guarantee you that will lead you to the correct answer, but (if the assumptions is correct) often get answer which are close to the best in much shorter time.
An Algorithm is a clearly defined set of instructions to solve a problem, Heuristics involve utilising an approach of learning and discovery to reach a solution.
So, if you know how to solve a problem then use an algorithm. If you need to develop a solution then it's heuristics.
Heuristics are algorithms, so in that sense there is none, however, heuristics take a 'guess' approach to problem solving, yielding a 'good enough' answer, rather than finding a 'best possible' solution.
A good example is where you have a very hard (read NP-complete) problem you want a solution for but don't have the time to arrive to it, so have to use a good enough solution based on a heuristic algorithm, such as finding a solution to a travelling salesman problem using a genetic algorithm.
Algorithm is a sequence of some operations that given an input computes something (a function) and outputs a result.
Algorithm may yield an exact or approximate values.
It also may compute a random value that is with high probability close to the exact value.
A heuristic algorithm uses some insight on input values and computes not exact value (but may be close to optimal).
In some special cases, heuristic can find exact solution.
A heuristic is usually an optimization or a strategy that usually provides a good enough answer, but not always and rarely the best answer. For example, if you were to solve the traveling salesman problem with brute force, discarding a partial solution once its cost exceeds that of the current best solution is a heuristic: sometimes it helps, other times it doesn't, and it definitely doesn't improve the theoretical (big-oh notation) run time of the algorithm
I think Heuristic is more of a constraint used in Learning Based Model in Artificial Intelligent since the future solution states are difficult to predict.
But then my doubt after reading above answers is
"How would Heuristic can be successfully applied using Stochastic Optimization Techniques? or can they function as full fledged algorithms when used with Stochastic Optimization?"
http://en.wikipedia.org/wiki/Stochastic_optimization
One of the best explanations I have read comes from the great book Code Complete, which I now quote:
A heuristic is a technique that helps you look for an answer. Its
results are subject to chance because a heuristic tells you only how
to look, not what to find. It doesn’t tell you how to get directly
from point A to point B; it might not even know where point A and
point B are. In effect, a heuristic is an algorithm in a clown suit.
It’s less predict- able, it’s more fun, and it comes without a 30-day,
money-back guarantee.
Here is an algorithm for driving to someone’s house: Take Highway 167
south to Puy-allup. Take the South Hill Mall exit and drive 4.5 miles
up the hill. Turn right at the light by the grocery store, and then
take the first left. Turn into the driveway of the large tan house on
the left, at 714 North Cedar.
Here’s a heuristic for getting to someone’s house: Find the last
letter we mailed you. Drive to the town in the return address. When
you get to town, ask someone where our house is. Everyone knows
us—someone will be glad to help you. If you can’t find anyone, call us
from a public phone, and we’ll come get you.
The difference between an algorithm and a heuristic is subtle, and the
two terms over-lap somewhat. For the purposes of this book, the main
difference between the two is the level of indirection from the
solution. An algorithm gives you the instructions directly. A
heuristic tells you how to discover the instructions for yourself, or
at least where to look for them.
They find a solution suboptimally without any guarantee as to the quality of solution found, it is obvious that it makes sense to the development of heuristics only polynomial. The application of these methods is suitable to solve real world problems or large problems so awkward from the computational point of view that for them there is not even an algorithm capable of finding an approximate solution in polynomial time.