10 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aanderaa–Karp–Rosenberg conjecture | 3/4 | https://en.wikipedia.org/wiki/Aanderaa–Karp–Rosenberg_conjecture | reference | science, encyclopedia | 2026-05-05T11:01:55.807273+00:00 | kb-cron |
== Deterministic query complexity == For deterministic algorithms, Rosenberg (1973) originally conjectured that for all nontrivial graph properties on
n
{\displaystyle n}
vertices, deciding whether a graph possesses this property requires
Ω
(
n
2
)
{\displaystyle \Omega (n^{2})}
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.
The conjecture was disproved by Aanderaa, who exhibited a directed graph property (the property of containing a "sink") which required only
O
(
n
)
{\displaystyle O(n)}
queries to test. A sink, in a directed graph, is a vertex of indegree
n
−
1
{\displaystyle n-1}
and outdegree zero. The existence of a sink can be tested with less than
3
n
{\displaystyle 3n}
queries. An undirected graph property which can also be tested with
O
(
n
)
{\displaystyle O(n)}
queries is the property of being a scorpion graph, first described in 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. 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
2
)
{\displaystyle \Omega (n^{2})}
queries. This conjecture was resolved by Rivest & Vuillemin (1975) by showing that at least
1
16
n
2
{\displaystyle {\tfrac {1}{16}}n^{2}}
queries are needed to test for any nontrivial monotone graph property. Through successive improvements this bound was further increased to
(
1
3
−
ε
)
n
2
{\displaystyle {\bigl (}{\tfrac {1}{3}}-\varepsilon {\bigr )}n^{2}}
. 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
{\displaystyle n}
vertices is evasive." A property is called evasive if determining whether a given graph has this property sometimes requires all
n
(
n
−
1
)
/
2
{\displaystyle n(n-1)/2}
possible queries. 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
{\displaystyle n}
. The conjecture has been resolved for the case where
n
{\displaystyle n}
is a prime power using a topological approach. The conjecture has also been resolved for all non-trivial monotone properties on bipartite graphs. Minor-closed properties have also been shown to be evasive for large
n
{\displaystyle n}
. In Kahn, Saks & Sturtevant (1984) the conjecture was generalized to properties of other (non-graph) functions too, conjecturing that any non-trivial monotone function that is weakly symmetric is evasive. This case is also solved when
n
{\displaystyle n}
is a prime power.
== Randomized query complexity == Richard Karp also conjectured that
Ω
(
n
2
)
{\displaystyle \Omega (n^{2})}
queries are required for testing nontrivial monotone properties even if randomized algorithms are permitted. No nontrivial monotone property is known which requires less than
1
4
n
2
{\displaystyle {\tfrac {1}{4}}n^{2}}
queries to test. A linear lower bound (i.e.,
Ω
(
n
)
{\displaystyle \Omega (n)}
) on all monotone properties follows from a very general relationship between randomized and deterministic query complexities. The first superlinear lower bound for all monotone properties was given by Yao (1991) who showed that
Ω
(
n
(
log
n
)
1
/
12
)
{\displaystyle \Omega {\bigl (}n(\log n)^{1/12}{\bigr )}}
queries are required. This was further improved by King (1991) to
Ω
(
n
5
/
4
)
{\displaystyle \Omega (n^{5/4})}
, and then by Hajnal (1991) to
Ω
(
n
4
/
3
)
{\displaystyle \Omega (n^{4/3})}
. This was subsequently improved to the current best known lower bound (among bounds that hold for all monotone properties) of
Ω
(
n
4
/
3
(
log
n
)
1
/
3
)
{\displaystyle \Omega {\bigl (}n^{4/3}(\log n)^{1/3}{\bigr )}}
by Chakrabarti & Khot (2007). Some recent results give lower bounds which are determined by the critical probability
p
{\displaystyle p}
of the monotone graph property under consideration. The critical probability
p
{\displaystyle p}
is defined as the unique number
p
{\displaystyle p}
in the range
[
0
,
1
]
{\displaystyle [0,1]}
such that a random graph
G
(
n
,
p
)
{\displaystyle G(n,p)}
(obtained by choosing randomly whether each edge exists, independently of the other edges, with probability
p
{\displaystyle p}
per edge) possesses this property with probability equal to
1
2
{\displaystyle {\tfrac {1}{2}}}
. Friedgut, Kahn & Wigderson (2002) showed that any monotone property with critical probability
p
{\displaystyle p}
requires
Ω
(
min
{
n
min
(
p
,
1
−
p
)
,
n
2
log
n
}
)
{\displaystyle \Omega \left(\min \left\{{\frac {n}{\min(p,1-p)}},{\frac {n^{2}}{\log n}}\right\}\right)}
queries. For the same problem, O'Donnell et al. (2005) showed a lower bound of
Ω
(
n
4
/
3
/
p
1
/
3
)
{\displaystyle \Omega (n^{4/3}/p^{1/3})}
. As in the deterministic case, there are many special properties for which an
Ω
(
n
2
)
{\displaystyle \Omega (n^{2})}
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
3
/
2
)
{\displaystyle \Omega (n^{3/2})}
due to Gröger (1992).