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| {{unsolved|computer science|Prove or disprove Aanderaa–Karp–Rosenberg conjecture.}}
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| In [[theoretical computer science]], the '''Aanderaa–Karp–Rosenberg conjecture''' (also known as the '''Aanderaa–Rosenberg conjecture''' or the '''evasiveness conjecture''') is a group of related [[conjecture]]s about the number of questions of the form "Is there an edge between vertex ''u'' and vertex ''v''?" that have to be answered to determine whether or not an [[undirected graph]] has a particular property such as [[planar graph|planarity]] or [[bipartite graph|bipartiteness]]. They are named after [[Stål Aanderaa]], [[Richard M. Karp]], and [[Arnold L. Rosenberg]]. According to the conjecture, for a wide class of properties, no algorithm can guarantee that it will be able to skip any questions: any [[algorithm]] for determining whether the graph has the property, no matter how clever, might need to examine every pair of vertices before it can give its answer. A property satisfying this conjecture is called [[evasive Boolean function|evasive]].
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| More precisely, the Aanderaa–Rosenberg conjecture states that any [[deterministic algorithm]] must test at least a constant fraction of all possible pairs of vertices, in the [[worst case analysis|worst case]], to determine any non-trivial monotone graph property; in this context, a property is monotone if it remains true when edges are added (so planarity is not monotone, but non-planarity is monotone). A stronger version of this conjecture, called the evasiveness conjecture or the Aanderaa–Karp–Rosenberg conjecture, states that exactly {{nowrap|''n''(''n'' − 1)/2}} tests are needed. Versions of the problem for [[randomized algorithm]]s and [[quantum algorithm]]s have also been formulated and studied.
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| The deterministic Aanderaa–Rosenberg conjecture was proven by {{harvtxt|Rivest|Vuillemin|1975}}, but the stronger Aanderaa–Karp–Rosenberg conjecture remains unproven. Additionally, there is a large gap between the conjectured lower bound and the best proven lower bound for randomized and quantum query complexity.
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| ==Example==
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| The property of being non-empty (that is, having at least one edge) is monotone, because adding another edge to a non-empty graph produces another non-empty graph. There is a simple algorithm for testing whether a graph is non-empty: loop through all of the pairs of vertices, testing whether each pair is connected by an edge. If an edge is ever found in this way, break out of the loop, and report that the graph is non-empty, and if the loop completes without finding any edges, then report that the graph is empty. On some graphs (for instance the [[complete graph]]s) this algorithm will terminate quickly, without testing every pair of vertices, but on the [[empty graph]] it tests all possible pairs before terminating. Therefore, the query complexity of this algorithm is ''n''(''n'' − 1)/2: in the worst case, the algorithm performs ''n''(''n'' − 1)/2 tests.
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| The algorithm described above is not the only possible method of testing for non-emptiness, but
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| the Aanderaa–Karp–Rosenberg conjecture implies that every deterministic algorithm for testing non-emptiness has the same query complexity, ''n''(''n'' − 1)/2. That is, the property of being non-empty is ''evasive''. For this property, the result is easy to prove directly: if an algorithm does not perform ''n''(''n'' − 1)/2 tests, it cannot distinguish the empty graph from a graph that has one edge connecting one of the untested pairs of vertices, and must give an incorrect answer on one of these two graphs.
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| ==Definitions==
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| In the context of this article, all [[Graph (mathematics)|graphs]] will be [[simple graph|simple]] and [[undirected graph|undirected]], unless stated otherwise. This means that the edges of the graph form a set (and not a [[multiset]]) and each edge is a pair of distinct vertices. Graphs are assumed to have an [[implicit graph|implicit representation]] in which each vertex has a unique identifier or label and in which it is possible to test the adjacency of any two vertices, but for which adjacency testing is the only allowed primitive operation.
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| Informally, a [[graph property]] is a property of a graph that is independent of labeling. More formally, a graph property is a mapping from the set of all graphs to {0,1} such that isomorphic graphs are mapped to the same value. For example, the property of containing at least 1 vertex of degree 2 is a graph property, but the property that the first vertex has degree 2 is not, because it depends on the labeling of the graph (in particular, it depends on which vertex is the "first" vertex). A graph property is called non-trivial if it doesn't assign the same value to all graphs. For instance, the property of being a graph is a trivial property, since all graphs possess this property. On the other hand, the property of being empty is non-trivial, because the [[empty graph]] possesses this property, but non-empty graphs do not. A graph property is said to be [[Hereditary property#Monotone property|monotone]] if the addition of edges does not destroy the property. Alternately, if a graph possesses a monotone property, then every [[supergraph]] of this graph on the same vertex set also possesses it. For instance, the property of being [[planar graph|nonplanar]] is monotone: a supergraph of a nonplanar graph is itself nonplanar. However, the property of being [[regular graph|regular]] is not monotone.
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| The [[big Oh notation]] is often used for query complexity. In short, ''f''(''n'') is O(''g''(''n'')) if for large enough ''n'', ''f''(''n'') ≤ ''c g''(''n'') for some positive constant ''c''. Similarly, ''f''(''n'') is Ω(''g''(''n'')) if for large enough ''n'', ''f''(''n'') ≥ ''c g''(''n'') for some positive constant ''c''. Finally, ''f''(''n'') is Θ(''g''(''n'')) if it is both O(''g''(''n'')) and Ω(''g''(''n'')).
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| ==Query complexity==
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| The deterministic query complexity of evaluating a function on ''n'' bits (''x''<sub>1</sub>, ''x''<sub>2</sub>, ..., ''x''<sub>''n''</sub>) is the number of bits ''x''<sub>i</sub> that have to be read in the worst case by a deterministic algorithm to determine the value of the function. For instance, if the function takes value 0 when all bits are 0 and takes value 1 otherwise (this is the [[Logical disjunction|OR]] function), then the deterministic query complexity is exactly ''n''. In the worst case, the first {{nowrap|''n'' − 1}} bits read could all be 0, and the value of the function now depends on the last bit. If an algorithm doesn't read this bit, it might output an incorrect answer. (Such arguments are known as adversary arguments.) The number of bits read are also called the number of queries made to the input. One can imagine that the algorithm asks (or queries) the input for a particular bit and the input responds to this query.
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| The randomized query complexity of evaluating a function is defined similarly, except the algorithm is allowed to be randomized, i.e., it can flip coins and use the outcome of these coin flips to decide which bits to query. However, the randomized algorithm must still output the correct answer for all inputs: it is not allowed to make errors. Such algorithms are called [[Las Vegas algorithm]]s, which distinguishes them from [[Monte Carlo algorithm]]s which are allowed to make some error. Randomized query complexity can also be defined in the Monte Carlo sense, but the Aanderaa–Karp–Rosenberg conjecture is about the Las Vegas query complexity of graph properties.
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| Quantum query complexity is the natural generalization of randomized query complexity, of course allowing quantum queries and responses. Quantum query complexity can also be defined with respect to Monte Carlo algorithms or Las Vegas algorithms, but it is usually taken to mean Monte Carlo quantum algorithms.
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| In the context of this conjecture, the function to be evaluated is the graph property, and the input is a string of size ''n''(''n'' − 1)/2, which gives the locations of the edges on an ''n'' vertex graph, since a graph can have at most ''n''(''n'' − 1)/2 possible edges. The query complexity of any function is upper bounded by ''n''(''n'' − 1)/2, since the whole input is read after making ''n''(''n'' − 1)/2 queries, thus determining the input graph completely.
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| ==Deterministic query complexity==
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| For deterministic algorithms, {{Harvtxt|Rosenberg|1973}} originally conjectured that for all nontrivial graph properties on ''n'' vertices, deciding whether a graph possesses this property requires Ω(''n''<sup>2</sup>) queries. The non-triviality condition is clearly required because there are trivial properties like "is this a graph?" which can be answered with no queries at all.
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| The conjecture was disproved by Aanderaa, who exhibited a directed graph property (the property of containing a "sink") which required only O(''n'') queries to test. A ''sink'', in a directed graph, is a vertex of indegree ''n''-1 and outdegree 0. This property can be tested with less than 3''n'' queries {{harv|Best|van Emde Boas|Lenstra|1974}}. An undirected graph property which can also be tested with O(''n'') queries is the property of being a scorpion graph, first described in {{harvtxt|Best|van Emde Boas|Lenstra|1974}}. A scorpion graph is a graph containing a three-vertex path, such that one endpoint of the path is connected to all remaining vertices, while the other two path vertices have no incident edges other than the ones in the path.
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| Then Aanderaa and Rosenberg formulated a new conjecture (the '''Aanderaa–Rosenberg conjecture''') which says that deciding whether a graph possesses a non-trivial monotone graph property requires Ω(''n''<sup>2</sup>) queries.<ref>{{Harvtxt|Triesch|1996}}</ref> This conjecture was resolved by {{harvtxt|Rivest|Vuillemin|1975}} by showing that at least ''n''<sup>2</sup>/16 queries are needed to test for any nontrivial monotone graph property. The bound was further improved to ''n''<sup>2</sup>/9 by {{Harvtxt|Kleitman|Kwiatkowski|1980}}, then to ''n''<sup>2</sup>/4 - o(''n''<sup>2</sup>) by {{Harvtxt|Kahn|Saks|Sturtevant|1983}}, then to (8/25)''n''<sup>2</sup> - o(''n''<sup>2</sup>) by {{Harvtxt|Korneffel|Triesch|2010}}, and then to ''n''<sup>2</sup>/3 - o(''n''<sup>2</sup>) by {{Harvtxt|Scheidweiler|Triesch|2013}}.
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| [[Richard Karp]] conjectured the stronger statement (which is now called the '''evasiveness conjecture''' or the '''Aanderaa–Karp–Rosenberg conjecture''') that "every nontrivial monotone graph property for graphs on ''n'' vertices is evasive."<ref>{{Harvtxt|Lutz|2001}}</ref> A property is called ''evasive'' if determining whether a given graph has this property sometimes requires all ''n''(''n'' − 1)/2 queries.<ref>{{Harvtxt|Kozlov|2008|pp=226–228}}</ref> This conjecture says that the best algorithm for testing any nontrivial monotone property must (in the worst case) query all possible edges. This conjecture is still open, although several special graph properties have shown to be evasive for all ''n''. The conjecture has been resolved for the case where ''n'' is a [[prime power]] by {{Harvtxt|Kahn|Saks|Sturtevant|1983}} using a [[Topological combinatorics|topological]] approach. The conjecture has also been resolved for all non-trivial monotone properties on bipartite graphs by {{harvtxt|Yao|1988}}. [[Minor (graph theory)|Minor]]-closed properties have also been shown to be evasive for large ''n'' {{harv|Chakrabarti|Khot|Shi|2001}}.
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| ==Randomized query complexity==
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| Richard Karp also conjectured that Ω(''n''<sup>2</sup>) queries are required for testing nontrivial monotone properties even if randomized algorithms are permitted. No nontrivial monotone property is known which requires less than ''n''<sup>2</sup>/4 queries to test. A linear lower bound (i.e., Ω(''n'')) follows from a very general [[Query_complexity#Relationship_between_different_models|relationship between randomized and deterministic query complexities]]. The first superlinear lower bound for this problem was given by {{harvtxt|Yao|1991}} who showed that Ω(''n'' log<sup>1/12</sup> ''n'') queries are required. This was further improved by {{harvtxt|King|1988}} to Ω(''n''<sup>5/4</sup>), and then by {{Harvtxt|Hajnal|1991}} to Ω(''n''<sup>4/3</sup>). This was subsequently improved to the current best known lower bound of Ω(''n''<sup>4/3</sup> log<sup>1/3</sup> ''n'') by {{harvtxt|Chakrabarti|Khot|2001}}.
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| Some recent results give lower bounds which are determined by the critical probability ''p'' of the monotone graph property under consideration. The critical probability ''p'' is defined as the unique ''p'' such that a [[Erdős–Rényi model|random graph]] ''G''(''n'', ''p'') possesses this property with probability equal to 1/2. A random graph ''G''(''n'', ''p'') is a graph on ''n'' vertices where each edge is chosen to be present with probability ''p'' independent of all the other edges. {{Harvtxt|Friedgut|Kahn|Wigderson|2002}} showed that any monotone property with critical probability ''p'' requires <math>\Omega\left(\min\left\{\frac{n}{\min(p,1-p)},\frac{n^2}{\log n}\right\}\right)</math> queries. For the same problem, {{Harvtxt|O'Donnell|Saks|Schramm|Servedio|2005}} showed a lower bound of Ω(''n''<sup>4/3</sup>/''p''<sup>1/3</sup>).
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| As in the deterministic case, there are many special properties for which an Ω(''n''<sup>2</sup>) lower bound is known. Moreover, better lower bounds are known for several classes of graph properties. For instance, for testing whether the graph has a subgraph isomorphic to any given graph (the so-called [[subgraph isomorphism]] problem), the best known lower bound is Ω(''n''<sup>3/2</sup>) due to {{harvtxt|Gröger|1992}}.
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| ==Quantum query complexity==
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| For bounded-error [[quantum query complexity]], the best known lower bound is Ω(''n''<sup>2/3</sup> log<sup>1/6</sup> ''n'') as observed by Andrew Yao.<ref>The result is unpublished, but mentioned in {{Harvtxt|Magniez|Santha|Szegedy|2005}}.</ref> It is obtained by combining the randomized lower bound with the quantum adversary method. The best possible lower bound one could hope to achieve is Ω(''n''), unlike the classical case, due to [[Grover's algorithm]] which gives an O(''n'') query algorithm for testing the monotone property of non-emptiness. Similar to the deterministic and randomized case, there are some properties which are known to have an Ω(''n'') lower bound, for example non-emptiness (which follows from the optimality of Grover's algorithm) and [[triangle finding problem|the property of containing a triangle]]. More interestingly, there are some graph properties which are known to have an Ω(''n''<sup>3/2</sup>) lower bound, and even some properties with an Ω(''n''<sup>2</sup>) lower bound. For example, the monotone property of nonplanarity requires Θ(''n''<sup>3/2</sup>) queries {{harv|Ambainis|Iwama|Nakanishi|Nishimura|2008}} and the monotone property of containing more than half the possible number of edges (also called the majority function) requires Θ(''n''<sup>2</sup>) queries {{Harv|Beals|Buhrman|Cleve|Mosca|2001}}.
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| ==Notes==
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| | journal=SIAM Journal on Discrete Mathematics
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| | number=1
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| | url=http://epubs.siam.org/doi/abs/10.1137/120888703}}
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| *{{Citation
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| | first = Eberhard
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| | title = On the recognition complexity of some graph properties
| |
| | journal = [[Combinatorica]]
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| | year = 1996
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| }}.
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| *{{Citation
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| | volume = 17
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| | issue = 3
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| | pages = 517–520
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| | last = Yao
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| | first = Andrew Chi-Chih | authorlink = Andrew Yao
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| | title = Monotone bipartite graph properties are evasive
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| | journal = [[SIAM Journal on Computing]]
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| | year = 1988
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| | doi = 10.1137/0217031
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| }}.
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| | first = Andrew Chi-Chih
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| | authorlink = Andrew Yao
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| | title = Lower bounds to randomized algorithms for graph properties
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| | doi = 10.1016/0022-0000(91)90003-N
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| }}.
| |
| {{refend}}
| |
| | |
| ==Further reading==
| |
| *{{Citation | last1=Bollobás | first1=Béla | author1-link=Béla Bollobás | title=Extremal Graph Theory | publisher=[[Dover Publications]] | location=New York | isbn=978-0-486-43596-1 | year=2004 | contribution = Chapter VIII. Complexity and packing | pages = 401–437}}.
| |
| *{{cite arxiv|eprint=cs/0205031v1|author1=[[Laszlo Lovasz]]|last2=Young | first2=Neal E.|title=Lecture Notes on Evasiveness of Graph Properties|class=cs.CC|year=2002}}
| |
| *{{Citation
| |
| | last = Chronaki
| |
| | first = Catherine E
| |
| | title = A survey of Evasiveness: Lower Bounds on the Decision-Tree Complexity of Boolean Functions
| |
| | year = 1990
| |
| | id = {{citeseerx|10.1.1.37.1041}}
| |
| | postscript = .
| |
| }}
| |
| *{{cite web|author=Michael Saks|authorlink=Michael Saks (mathematician)|title=Decision Trees: Problems and Results, Old and New|url=http://www.math.rutgers.edu/~saks/PUBS/dtsurvey.pdf}}
| |
| | |
| {{DEFAULTSORT:Aanderaa-Karp-Rosenberg conjecture}}
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| [[Category:Conjectures]]
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| [[Category:Graph theory]]
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| [[Category:Combinatorics]]
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| [[Category:Unsolved problems in computer science]]
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| [[Category:Computational complexity theory]]
| |