Abstract We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph $$G = (V, E)$$ , where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges $$T \subseteq E$$ such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. This is a natural generalization of the maximum coverage problem. Our interest in this problem stems from its application to bid optimization in sponsored search auctions. It is easily seen that this problem is at least as hard as budgeted maximum coverage (where the costs are associated with the selected hyperedges instead of the covered vertices). This implies $$(1-1/e+\epsilon )$$ -inapproximability for any $$\epsilon> 0$$ . Furthermore, standard greedy approaches do not yield constant factor approximations for our variant of the problem. In fact, through a reduction from Densest k -Subgraph, it can be established that our problem is inapproximable up to a constant factor, conditional on the exponential time hypothesis. Our main results are as follows: (i.) We obtain a $$(1 - 1/\sqrt{e})/2$$ -approximation algorithm for graphs. (ii.) We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic ). We extend this result to incidence graphs with a fixed-size feedback hyperedge node set. (iii.) We give a $$(1-\varepsilon )/(2d^2)$$ -approximation algorithm for all $$\varepsilon> 0$$ , where d is the maximum vertex degree.

This paper proposes an algorithm that decomposes the Periodic Event Scheduling Problem (PESP) into trees that can efficiently be solved. By identifying at an early stage which partial solutions can lead to a feasible solution, the decomposed components can be integrated back while maintaining feasibility if possible. If not, the modifications required to regain feasibility can be found efficiently. These techniques integrate dynamic programming into standard search methods. The performance of these heuristics are very satisfying, as the problem using publicly available benchmarks can be solved within a reasonable amount of time, in an alternative way than the currently accepted leading-edge techniques. Furthermore, these heuristics do not necessarily rely on linearity of the objective function, which facilitates the research of timetabling under nonlinear circumstances.

We study the following natural variant of the budgeted maximum coverage problem: We are given a budget B and a hypergraph G = (V, E), where each vertex has a non-negative cost and a non-negative profit. The goal is to select a set of hyperedges T subseteq E such that the total cost of the vertices covered by T is at most B and the total profit of all covered vertices is maximized. Besides being a natural generalization of the well-studied maximum coverage problem, our motivation for investigating this problem originates from its application in the context of bid optimization in sponsored search auctions, such as Google AdWords. It is easily seen that this problem is strictly harder than budgeted max coverage, which means that the problem is (1-1/e)-inapproximable. The difference of our problem to the budgeted maximum coverage problem is that the costs are associated with the covered vertices instead of the selected hyperedges. As it turns out, this difference refutes the applicability of standard greedy approaches which are used to obtain constant factor approximation algorithms for several other variants of the maximum coverage problem. Our main results are as follows: - We obtain a (1 - 1/sqrt(e))/2-approximation algorithm for graphs. - We derive a fully polynomial-time approximation scheme (FPTAS) if the incidence graph of the hypergraph is a forest (i.e., the hypergraph is Berge-acyclic). We also extend this result to incidence graphs with a fixed-size feedback hyperedge node set. - We give a (1-epsilon)/(2d^2)-approximation algorithm for every epsilon > 0, where d is the maximum degree of a vertex in the hypergraph.