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This is just a collection of nodes in which some pairs of nodes are connected by edges. A dimer is something that covers an edge of a graph. The next picture shows three dimer arrangements of the graph that we saw before. Each of these dimer arrangements has two dimers. Since this graph has five edges, there are also five dimer arrangements with exactly one dimer.

There is also a dimer arrangement with no dimers. We have now accounted for all of the dimer arrangements of this graph — there is no way to put more than two dimers on the graph without overlap. Now to get to the point. The computational problem that I have in mind is that of computing the dimer partition function.

Computational complexity theory

We just consider each dimer arrangement, and add up the weights. The problem with this approach is that, if the graph has many nodes, then the number of dimer arrangements is too large to allow considering them one-by-one. In computational complexity theory, a problem refers to the abstract question to be solved.

In contrast, an instance of this problem is a rather concrete utterance, which can serve as the input for a decision problem. For example, consider the problem of primality testing. The instance is a number e. Stated another way, the instance is a particular input to the problem, and the solution is the output corresponding to the given input.

To further highlight the difference between a problem and an instance, consider the following instance of the decision version of the traveling salesman problem : Is there a route of at most kilometres passing through all of Germany's 15 largest cities? For this reason, complexity theory addresses computational problems and not particular problem instances. When considering computational problems, a problem instance is a string over an alphabet.

Usually, the alphabet is taken to be the binary alphabet i. As in a real-world computer , mathematical objects other than bitstrings must be suitably encoded. For example, integers can be represented in binary notation , and graphs can be encoded directly via their adjacency matrices , or by encoding their adjacency lists in binary.

Even though some proofs of complexity-theoretic theorems regularly assume some concrete choice of input encoding, one tries to keep the discussion abstract enough to be independent of the choice of encoding. This can be achieved by ensuring that different representations can be transformed into each other efficiently.


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Decision problems are one of the central objects of study in computational complexity theory. A decision problem is a special type of computational problem whose answer is either yes or no , or alternately either 1 or 0.

Complexity Theory Overview

A decision problem can be viewed as a formal language , where the members of the language are instances whose output is yes, and the non-members are those instances whose output is no. The objective is to decide, with the aid of an algorithm , whether a given input string is a member of the formal language under consideration. If the algorithm deciding this problem returns the answer yes , the algorithm is said to accept the input string, otherwise it is said to reject the input.

An example of a decision problem is the following. The input is an arbitrary graph. The problem consists in deciding whether the given graph is connected or not. The formal language associated with this decision problem is then the set of all connected graphs — to obtain a precise definition of this language, one has to decide how graphs are encoded as binary strings.

A function problem is a computational problem where a single output of a total function is expected for every input, but the output is more complex than that of a decision problem —that is, the output isn't just yes or no. Notable examples include the traveling salesman problem and the integer factorization problem. It is tempting to think that the notion of function problems is much richer than the notion of decision problems. However, this is not really the case, since function problems can be recast as decision problems. Deciding whether a given triple is a member of this set corresponds to solving the problem of multiplying two numbers.

To measure the difficulty of solving a computational problem, one may wish to see how much time the best algorithm requires to solve the problem. However, the running time may, in general, depend on the instance. In particular, larger instances will require more time to solve. Thus the time required to solve a problem or the space required, or any measure of complexity is calculated as a function of the size of the instance. This is usually taken to be the size of the input in bits. Complexity theory is interested in how algorithms scale with an increase in the input size.

For instance, in the problem of finding whether a graph is connected, how much more time does it take to solve a problem for a graph with 2 n vertices compared to the time taken for a graph with n vertices? If the input size is n , the time taken can be expressed as a function of n. Since the time taken on different inputs of the same size can be different, the worst-case time complexity T n is defined to be the maximum time taken over all inputs of size n. If T n is a polynomial in n , then the algorithm is said to be a polynomial time algorithm. Cobham's thesis argues that a problem can be solved with a feasible amount of resources if it admits a polynomial time algorithm.

A Turing machine is a mathematical model of a general computing machine. It is a theoretical device that manipulates symbols contained on a strip of tape. Turing machines are not intended as a practical computing technology, but rather as a general model of a computing machine—anything from an advanced supercomputer to a mathematician with a pencil and paper. It is believed that if a problem can be solved by an algorithm, there exists a Turing machine that solves the problem.

Indeed, this is the statement of the Church—Turing thesis.

Furthermore, it is known that everything that can be computed on other models of computation known to us today, such as a RAM machine , Conway's Game of Life , cellular automata or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used model in complexity theory.

Many types of Turing machines are used to define complexity classes, such as deterministic Turing machines , probabilistic Turing machines , non-deterministic Turing machines , quantum Turing machines , symmetric Turing machines and alternating Turing machines. They are all equally powerful in principle, but when resources such as time or space are bounded, some of these may be more powerful than others.

A deterministic Turing machine is the most basic Turing machine, which uses a fixed set of rules to determine its future actions. A probabilistic Turing machine is a deterministic Turing machine with an extra supply of random bits. The ability to make probabilistic decisions often helps algorithms solve problems more efficiently. Algorithms that use random bits are called randomized algorithms. A non-deterministic Turing machine is a deterministic Turing machine with an added feature of non-determinism, which allows a Turing machine to have multiple possible future actions from a given state.

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Computational Complexity

One way to view non-determinism is that the Turing machine branches into many possible computational paths at each step, and if it solves the problem in any of these branches, it is said to have solved the problem. Clearly, this model is not meant to be a physically realizable model, it is just a theoretically interesting abstract machine that gives rise to particularly interesting complexity classes. For examples, see non-deterministic algorithm. Many machine models different from the standard multi-tape Turing machines have been proposed in the literature, for example random access machines.

Perhaps surprisingly, each of these models can be converted to another without providing any extra computational power. The time and memory consumption of these alternate models may vary. However, some computational problems are easier to analyze in terms of more unusual resources. For example, a non-deterministic Turing machine is a computational model that is allowed to branch out to check many different possibilities at once. The non-deterministic Turing machine has very little to do with how we physically want to compute algorithms, but its branching exactly captures many of the mathematical models we want to analyze, so that non-deterministic time is a very important resource in analyzing computational problems.

For a precise definition of what it means to solve a problem using a given amount of time and space, a computational model such as the deterministic Turing machine is used. The time required by a deterministic Turing machine M on input x is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer "yes" or "no".

Thurner, Hanel, and Klimek write the book on complex systems

A Turing machine M is said to operate within time f n , if the time required by M on each input of length n is at most f n. A decision problem A can be solved in time f n if there exists a Turing machine operating in time f n that solves the problem.


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Since complexity theory is interested in classifying problems based on their difficulty, one defines sets of problems based on some criteria. For instance, the set of problems solvable within time f n on a deterministic Turing machine is then denoted by DTIME f n. Analogous definitions can be made for space requirements. Although time and space are the most well-known complexity resources, any complexity measure can be viewed as a computational resource. Complexity measures are very generally defined by the Blum complexity axioms. Other complexity measures used in complexity theory include communication complexity , circuit complexity , and decision tree complexity.

The complexity of an algorithm is often expressed using big O notation. The best, worst and average case complexity refer to three different ways of measuring the time complexity or any other complexity measure of different inputs of the same size. Since some inputs of size n may be faster to solve than others, we define the following complexities:.

The order from cheap to costly is: Best, average of discrete uniform distribution , amortized, worst. For example, consider the deterministic sorting algorithm quicksort. This solves the problem of sorting a list of integers that is given as the input. The worst-case is when the input is sorted or sorted in reverse order, and the algorithm takes time O n 2 for this case. If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O n log n.

The best case occurs when each pivoting divides the list in half, also needing O n log n time. To classify the computation time or similar resources, such as space consumption , one is interested in proving upper and lower bounds on the maximum amount of time required by the most efficient algorithm solving a given problem.

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The complexity of an algorithm is usually taken to be its worst-case complexity, unless specified otherwise. Analyzing a particular algorithm falls under the field of analysis of algorithms. To show an upper bound T n on the time complexity of a problem, one needs to show only that there is a particular algorithm with running time at most T n.

However, proving lower bounds is much more difficult, since lower bounds make a statement about all possible algorithms that solve a given problem. The phrase "all possible algorithms" includes not just the algorithms known today, but any algorithm that might be discovered in the future. To show a lower bound of T n for a problem requires showing that no algorithm can have time complexity lower than T n. Upper and lower bounds are usually stated using the big O notation , which hides constant factors and smaller terms.