text: In the classical Maximum Acyclic Subgraph problem (MAS), given a directed-edge weighted graph, we are required to find an ordering of the nodes that maximizes the total weight of forward-directed edges. MAS admits a 2 approximation, and this approximation is optimal under the Unique Game Conjecture. In this paper we consider a generalization of MAS, the Restricted Maximum Acyclic Subgraph problem (RMAS), where each node is associated with a list of integer labels, and we have to find a labeling of the nodes so as to maximize the weight of edges whose head label is larger than the tail label. The best known (almost trivial) approximation for RMAS is 4. The interest of RMAS is mostly due to its connections with the Vertex Pricing problem (VP). In VP we are given an undirected graph with positive edge budgets. A feasible solution consists of an assignment of non-negative prices to the nodes. The profit for each edge @math is the sum of its endpoints prices if that sum is at most the budget of @math , and zero otherwise. Our goal is to maximize the total profit. The best known approximation for VP, which works analogously to the mentioned approximation algorithm for RMAS, is 4. Improving on that is a challenging open problem. On the other hand, the best known 2 inapproximability result is due to a reduction from a special case of RMAS. In this paper we present an improved LP-rounding @math approximation for RMAS. Our result shows that, in order to prove a 4 hardness of approximation result for VP (if possible), one should consider reductions from harder problems. Alternatively, our approach might suggest a different way to design approximation algorithms for VP.
  @cite_10: We consider the following item pricing problem which has received much attention recently. A seller has an infinite numbers of copies of n items. There are m buyers, each with a budget and an intention to buy a fixed subset of items. Given prices on the items, each buyer buys his subset of items, at the given prices, provided the total price of the subset is at most his budget. The objective of the seller is to determine the prices such that her total profit is maximized. In this paper, we focus on the case where the buyers are interested in subsets of size at most two. This special case is known to be APX-hard ( [1]). The best known approximation algorithm, by Balcan and Blum, gives a 4-approximation [2]. We show that there is indeed a gap of 4 for the combinatorial upper bound used in their analysis. We further show that a natural linear programming relaxation of this problem has an integrality gap of 4, even in this special case. Then we prove that the problem is NP-hard to approximate within a factor of 2 assuming the Unique Games Conjecture; and it is unconditionally NP-hard to approximate within a factor 17 16. Finally, we extend the APX-hardness of the problem to the special case in which the graph formed by items as vertices and buyers as edges is bipartite . We hope that our techniques will be helpful for obtaining stronger hardness of approximation bounds for this problem.
summary: Something more about MAS. How that the 2-apx work? Why does not generalize to RMAS? The intuitive argument is that for MAS the optimum is always at least @math . Our interest in RMAS is motivated by the following problem (VP): we are given an undirected (multi-) graph @math , with positive edge budgets @math . Our goal is to assign a non-negative price @math to each node @math so as to maximize the sum @math over the edges @math such that @math . @cite_10 proved that VP is @math -hard to approximate (under UGC) via a reduction from a special case of RMAS. Their reduction exploits instances of RMAS where labels are non-negative, each @math contains @math , and @math for any distinct @math . Note that such instances still generalize MAS. I meant more details about the reduction to VP. Why do you want to stress the special case of RMAS? The reductions are always special in some sense