Nontrivial Monotone Graph Property Research Articles

Elusive properties of infinite graphs

AbstractA graph property is said to be elusive (or evasive) if every algorithm testing this property by asking questions of the form “is there an edge between vertices and ?” requires, in the worst case, to ask about all pairs of vertices. The unsettled Aanderaa–Karp–Rosenberg conjecture is that every nontrivial monotone graph property is elusive for any finite vertex set. We show that the situation is completely different for infinite vertex sets: the monotone graph properties “every vertex has degree at least ” and “every connected component has size at least ,” where and are natural numbers, are not elusive for infinite vertex sets, but the monotone graph property “the graph contains a cycle” is elusive for arbitrary vertex set. On the other hand, we also prove that every algorithm testing some natural monotone graph properties, for example, “every vertex has degree at least ” or “connected” on the vertex set should check “lots of edges,” more precisely, all the edges of an infinite complete subgraph.

Shuffles and Circuits (On Lower Bounds for Modern Parallel Computation)

The goal of this article is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in motivating application domains, such as graph connectivity problems. We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the “fan-in” of each machine is limited to s bits, where s is smaller than the input size n , and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems. We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the “unbounded width” version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any nontrivial monotone graph property—such as connectivity—requires a super-constant number of rounds when every machine receives only a subpolynomial (in n ) number of input bits s . Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds would separate P from NC 1 .

Perfectness is an Elusive Graph Property

A graph property is called elusive (or evasive) if every algorithm for testing this property has to read in the worst case $n\choose 2$ entries of the adjacency matrix of the given graph. Several graph properties have been shown to be elusive, e.g., planarity or k-colorability. A famous conjecture of Karp says that every nontrivial monotone graph property is elusive. We prove that a nonmonotone but hereditary graph property is elusive: perfectness.

On the elusiveness of Hamiltonian property

Decision tree complexity is an important measure of computational complexity. Agraph property is a set of graphs such that if some graphG is in the set then each isomorphic graph toG is also in the set. LetP be a graph property onn vertices, if every decision tree algorithm recognizingP must examine at leastk pairs of vertices in the worst case, then it is said that thedecision tree complexity ofP isk. If every decision tree algorithm recognizingP must examine alln(n−1)/2 pairs of vertices in the worst case, thenP is said to beelusive. Karp conjectured that every nontrivial monotone graph property is elusive. This paper concerns the elusiveness of Hamiltonian property. It is proved that ifn=p+1, pq orpq+1, (wherep,q are distinct primes), then Hamiltonian property onn vertices is elusive.

Decision tree complexity of graph properties with dimension at most 5

A graph property is a set of graphs such that if the set contains some graphG then it also contains each isomorphic copy ofG (with the same vertex set). A graph propertyP onn vertices is said to be elusive, if every decision tree algorithm recognizingP must examine alln(n−1)/2, pairs of vertices in the worst case. Karp conjectured that every nontrivial monotone graph property is elusive. In this paper, this conjecture is proved for some cases. Especially, it is shown that if the abstract simplicial complex of a nontrivial monotone graph propertyP has dimension not exceeding 5, thenP is elusive.

Lower bounds to randomized algorithms for graph properties

For any property P on n-vertex graphs, let C(P) be the minimum number of edges needed to be examined by any decision tree algorithm for determining P. In 1975 Rivest and Vuillemin settled the Aanderra-Rosenberg Conjecture, proving that C(P)=Ω(n 2) for every nontrivial monotone graph property P. An intriguing open question is whether the theorem remains true when randomized algorithms are allowed. In this paper we show that Ω(n( log n) 1 12 edges need to be examined by any randomized algorithm for determining any nontrivial monotone graph property.

Further results on the Aanderaa-Rosenberg conjecture

Some results on the “evasiveness” of graph properties are obtained, extending the work of Rivest and Vuillemin. In particular, it is shown that any nontrivial monotone graph property on n vertices is at least n 2 9 - evasive ; particular stronger results are obtained for values of n that are powers of primes less than 14. A new result allowing interpolation among n values is obtained, as are some stronger results on restricted classes of properties.

On recognizing graph properties from adjacency matrices

A conjecture of Aanderaa and Rosenberg [15] motivates this work. We investigate the maximum number C( P) of arguments of P that must be tested in order to compute P, a Boolean function of d Boolean arguments. We present evidence for the general conjecture that C( P) = d whenever P(0 d ) ≠ P(1 d ) and P is invariant under a transitive permutation group acting on the arguments. A non-constructive argument (not based on the construction of an “oracle”) settles this question for d a prime power. We use this result to prove the Aanderaa-Rosenberg conjecture: at least v 2 16 entries of the adjacency matrix of a v-vertex undirected graph G must be examined in the worst case to determine if G has any given non-trivial monotone graph property.