On the Complexity of Mining Itemsets from the Crowd Using Taxonomies

We study the problem of frequent itemset mining in domains where data is not recorded in a conventional database but only exists in human knowledge. We provide examples of such scenarios, and present a crowdsourcing model for them. The model uses the crowd as an oracle to find out whether an itemset is frequent or not, and relies on a known taxonomy of the item domain to guide the search for frequent itemsets. In the spirit of data mining with oracles, we analyze the complexity of this problem in terms of (i) crowd complexity, that measures the number of crowd questions required to identify the frequent itemsets; and (ii) computational complexity, that measures the computational effort required to choose the questions. We provide lower and upper complexity bounds in terms of the size and structure of the input taxonomy, as well as the size of a concise description of the output itemsets. We also provide constructive algorithms that achieve the upper bounds, and consider more efficient variants for practical situations.Comment: 18 pages, 2 figures. To be published to ICDT'13. Added missing acknowledgemen

Grothendieck Rings of Theories of Modules

The model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, $K_0(M_\mathcal R)$, of a right $R$-module $M$, where $\mathcal R$ is any unital ring. As a corollary we prove a conjecture of Prest that $K_0(M)$ is non-trivial, whenever $M$ is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes.Comment: 42 Page

Ramsey numbers for partially-ordered sets

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

Poset Ramsey number R(P,Qn)R(P,Q_n). III. N-shaped poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+\epsilon)n$, for $\epsilon >0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+\Theta(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C,$ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.Comment: 19 pages, 6 figure