We present new algebraic approaches for several well-known combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite matching, we obtain a simple, purely algebraic algorithm with running time O(n) where n is the number of vertices and ω is the m...

An (r, w; d) cover-free family (CFF ) is a family of subsets of a finite set such that the intersection of any r members of the family contains at least d elements that are not in the union of any other w members. The minimum number of elements for which there exists an (r, w; d) − CFF with t blocks is denoted by N((r, w; d), t). In this paper, we show that the value of N((r, w; d), t) is equal...

It is well known that imprimitive P -polynomial association schemes X = (X, {Ri}0≤i≤d) with k1 > 2 are either bipartite or antipodal, i.e., intersection numbers satisfy either ai = 0 for all i, or bi = cd−i for all i 6= [d/2]. In this paper, we show that imprimitiveQ-polynomial association schemesX = (X, {Ri}0≤i≤d) with d > 6 and k∗ 1 > 2 are either dual bipartite or dual antipodal, i.e., dual ...

Every lattice is isomorphic to a lattice whose elements are sets of sets and whose operations are intersection and the operation ∨∗ defined by A ∨∗ B = A ∪ B ∪ {Z : (∃X ∈ A)(∃Y ∈ B)X ∩ Y ⊆ Z}. This representation spells out precisely Birkhoff’s and Frink’s representation of arbitrary lattices, which is related to Stone’s set-theoretic representation of distributive lattices. (AMS Subject Classi...

let $g$ be a finite group with the identity $e$‎. ‎the subgroup intersection graph $gamma_{si}(g)$ of $g$ is the graph with vertex set $v(gamma_{si}(g)) = g-e$ and two distinct vertices $x$ and $y$ are adjacent in $gamma_{si}(g)$ if and only if $|leftlangle xrightrangle capleftlangle yrightrangle|>1$‎, ‎where $leftlangle xrightrangle $ is the cyclic subgroup of $g$ generated by $xin g$‎. ‎in th...

Threshold private set intersection (TPSI) allows a receiver to obtain the when cardinality of is greater or equal threshold, which has wide range applications such as fingerprint matching, online dating and ridesharing. Existing TPSI protocols are inefficient because almost all them rely on lots expensive public-key techniques require an exponential number possible combinations among shares. In...

A private set intersection (PSI) protocol allows players to obtain the intersection of their inputs. While in its unilateral version only the client can obtain the intersection, the mutual PSI protocol enables all players to get the desired result. In this work, we construct a mutual PSI protocol that is significantly more efficient than the stateof-the-art in the computation overhead. To the b...

This thesis focuses on three fundamental problems in combinatorial optimization: non-bipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems. For the matching problem, we give an algorithm for constructing perfect or maximum cardinality mat...