An LP-rounding 2√2-approximation for restricted maximum acyclic subgraph

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 interest in RMAS is mostly due to its connections with the Vertex Pricing problem (VP). VP is known to be (2-ε)-hard to approximate via a reduction from RMAS, and the best known approximation factor for both problems is 4 (which is achieved via fairly simple algorithms). In this paper we present a non-trivial LP-rounding algorithm for RMAS with approximation ratio 22≈2.828. 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.