4 research outputs found
Implicit representation conjecture for semi-algebraic graphs
The implicit representation conjecture concerns hereditary families of
graphs. Given a graph in such a family, we want to assign some string of bits
to each vertex in such a way that we can recover the information about whether
2 vertices are connected or not using only the 2 strings of bits associated
with those two vertices. We then want to minimise the length of this string.
The conjecture states that if the family is hereditary and small enough (it
only has graphs of size ), then bits per vertex
should be sufficient. The trivial bounds on this problem are that: (1) some
families require at least bits per vertex ; (2)
bits per vertex are sufficient for all families.
In this paper, we will be talking about a special case of the implicit
representation conjecture, where the family is semi-algebraic (which roughly
means that the vertices are points in some euclidean space, and the edges are
defined geometrically, or according to some polynomials). We will first prove
that the `obvious' way of storing the information, where we store an
approximation of the coordinates of each vertex, doesn't work. Then we will
come up with a way of storing the information that requires
bits per vertex, where is some small constant depending only on the
family. This is a slight improvement over the trivial bound, but is still a
long way from proving the conjecture.Comment: 17 page
Randomized Communication and Implicit Representations for Matrices and Graphs of Small Sign-Rank
We prove a characterization of the structural conditions on matrices of
sign-rank 3 and unit disk graphs (UDGs) which permit constant-cost public-coin
randomized communication protocols. Therefore, under these conditions, these
graphs also admit implicit representations.
The sign-rank of a matrix is the smallest rank
of a matrix such that for all ; equivalently, it is the smallest dimension in which can be
represented as a point-halfspace incidence matrix with halfspaces through the
origin, and it is essentially equivalent to the unbounded-error communication
complexity. Matrices of sign-rank 3 can achieve the maximum possible
bounded-error randomized communication complexity , and
meanwhile the existence of implicit representations for graphs of bounded
sign-rank (including UDGs, which have sign-rank 4) has been open since at least
2003. We prove that matrices of sign-rank 3, and UDGs, have constant randomized
communication complexity if and only if they do not encode arbitrarily large
instances of the Greater-Than communication problem, or, equivalently, if they
do not contain arbitrarily large half-graphs as semi-induced subgraphs. This
also establishes the existence of implicit representations for these graphs
under the same conditions.Comment: 28 page
Applications of entropy to extremal problems
The Sidorenko conjecture gives a lower bound on the number of homomorphisms from a bipartite graph to another graph. Szegedy [28] used entropy methods to prove the conjecture in some cases. We will refine these methods to also give lower bounds for the number of injective homomorphisms from a bipartite graph to another bipartite graph, and a lower bound for the number of homomorphisms from a k-partite hypergraph to another k-partite hypergraph, as well as a few other similar problems.
Next is a generalisation of the Kruskal Katona Theorem [19, 17]. We are given integers k 4 we will make a lot of progress towards finding a solution.
The next chapter is to do with Turán-type problems. Given a family of k-hypergraphs F, ex(n;F) is the maximum number of edges an F-free n-vertex k-hypergraph can have. We prove that for a rational r, there exists some finite family F of k-hypergraphs for which ex(n;F) = Ɵ(nk-r) if and only if 0 < r < k - 1 or r = k.
The final chapter will deal with the implicit representation conjecture, in the special case of semi-algebraic graphs. Given a graph in such a family, we want to assign a name to each vertex in such a way that we can recover each edge based only on the names of the two incident vertices. We will first prove that one `obvious' way of storing the information doesn't work. Then we will come up with a way of storing the information that requires O(n1-E) bits per vertex, where E is some small constant depending only on the family