We study bounded degree graph problems, mainly the problem of k-Dimensional Matching (k-DM), namely, the problem of finding a maximal matching in a k-partite k-uniform balanced hyper-graph. We prove that k-DM cannot be efficiently approximated to within a factor of O( k ln k ) unless P = NP . This improves the previous factor of k 2O( √ ln ) by Trevisan [Tre01]. For low k values we prove NP-har...

Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems. We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem. These results follow from a new inapproximability result for...

We consider the following singularly perturbed Neumann problem ε∆u− u+ u = 0 in Ω, u > 0 in Ω, ∂u ∂ν = 0 on ∂Ω, where p is subcritical and Ω is a smooth and bounded domain in R with its unit outward normal ν. Lin-Ni-Wei [18] proved that there exists ε0 such that for 0 < ε < ε0 and for each integer k bounded by 1 ≤ k ≤ δ(Ω, n, p) (ε| log ε|)n (0.1) where δ(Ω, n, p) is a constant depending only o...

The first point here is that convex sets can be separated by linear functionals. Second, continuous linear functionals on subspaces of a locally convex topological vectorspace have continuous extensions to the whole space. Proofs are for real vectorspaces. The complex versions are corollaries. A crucial corollary is that on locally convex topological vectorspaces continuous linear functionals s...

We prove that for every k and every ε > 0, there exists g such that every graph with tree-width at most k and odd-girth at least g has circular chromatic number at most 2 + ε.

Szemerédi’s Regularity Lemma proved to be a powerful tool in the area of extremal graph theory. Many of its applications are based on its accompanying Counting Lemma: If G is an `-partite graph with V (G) = V1 ∪ · · · ∪ V` and |Vi| = n for all i ∈ [`], and all pairs (Vi, Vj) are ε-regular of density d for 1 ≤ i < j ≤ ` and ε d, then G contains (1± f`(ε))d “ ` 2 ” × n` cliques K`, where f`(ε) → ...

Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weigh...

Let’s quickly review some results from previous lectures. Definition 1 Let (Enc,Dec) denote an encryption scheme where Enc : {0, 1}m×{0, 1}n → {0, 1}λ and Dec : {0, 1}m × {0, 1}λ → {0, 1}n are functions. A correct encryption scheme satisfies ∀r, x Decr(Encr(x)) = x. Let R and X denote distributions on r and x respectively. (Enc,Dec) is said to be (k, ε)-secure if it is (R, ε)-secure for every k...

In this work, we study the k-means cost function. The (Euclidean) k-means problem can be described as follows: given a dataset X ⊆ R and a positive integer k, find a set of k centers C ⊆ R such that Φ(C,X) def = ∑ x∈X minc∈C ||x− c|| 2 is minimized. Let ∆k(X) def = minC⊆Rd Φ(C,X) denote the cost of the optimal k-means solution. It is simple to observe that for any dataset X, ∆k(X) decreases as ...