Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems

In this paper we study the (in)approximability of two distance-based relaxed variants of the maximum clique problem (Max Clique), named Max d-Clique and Max d-Club: A d-clique in a graph $$G = (V, E)$$G=(V,E) is a subset $$S\subseteq V$$S⊆V of vertices ...

A d-clique in a graph $$G = V, E$$G=V,E is a subset $$S\subseteq V$$S⊆V of vertices such that for pairs of vertices $$u, v\in S$$u,v∈S, the distance between u and v is at most d in G. A d-club in a graph $$G = V, E$$G=V,E is a subset $$S'\subseteq V$$S'⊆...

We discuss results obtained jointly with Van Vu on the length of arithmetic progressions in $$\ell $$ℓ-fold sumsets of the form $$\begin{aligned} \ell \mathcal {A}=\{a_1+\dots +a_\ell ~|~a_i\in \mathcal {A}\} \end{aligned}$$ℓA={a1+ź+aℓ|aiźA}and $$\begin{...