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.
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
@cite_10
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We investigate nonparametric multiproduct pricing problems, in which we want to find revenue maximizing prices for products @math based on a set of customer samples @math . We mostly focus on the unit-demand case, in which products constitute strict substitutes and each customer aims to purchase a single product. In this setting a customer sample consists of a number of nonzero values for different products and possibly an additional product ranking. Once prices are fixed, each customer chooses to buy one of the products she can afford based on some predefined selection rule. We distinguish between the min-buying, max-buying, and rank-buying models. Some of our results also extend to single-minded pricing, in which case products are strict complements and every customer seeks to buy a single set of products, which she purchases if the sum of prices is below her valuation for that set. For the min-buying model we show that the revenue maximization problem is not approximable within factor @math for some constant @math , unless @math , thereby almost closing the gap between the known algorithmic results and previous lower bounds. We also prove inapproximability within @math , @math being an upper bound on the number of nonzero values per customer, and @math under slightly stronger assumptions and provide matching upper bounds. Surprisingly, these hardness results hold even if a price ladder constraint, i.e., a predefined order on the prices of all products, is given. Without the price ladder constraint we obtain similar hardness results for the special case of uniform valuations, i.e., the case that every customer has identical values for all the products she is interested in, assuming specific hardness of the balanced bipartite independent set problem in constant degree graphs or hardness of refuting random 3CNF formulas. Introducing a slightly more general problem definition in which customers are given as an explicit probability distribution, we obtain inapproximability within @math assuming @math . These results apply to single-minded pricing as well. For the max-buying model a polynomial-time approximation scheme exists if a price ladder is given. We give a matching lower bound by proving strong NP-hardness. Assuming limited product supply, we analyze a generic local search algorithm and prove that it is 2-approximate. Finally, we discuss implications for the rank-buying model. Revenue maximization in multi-item settings is notoriously elusive. This paper studies a class of two-item auctions which we call a mixed-bundling auction with reserve prices (MBARP). It calls VCG on an enlarged set of agents by adding the seller---who has reserve valuations for each bundle of items---and a fake agent who receives nothing nor has valuations for any item or bundle, but has a valuation for pure bundling allocations, i.e., allocations where the two items are allocated to a single agent. This is a strict subclass of several known classes of auctions, including the affine maximizer auction (AMA), λ-aution, and the virtual valuations combinatorial auction (VVCA). As we show, a striking feature of MBARP is that its revenue can be represented in a simple closed form as a function of the parameters. Thus, we can solve first-order conditions on the parameters and obtain the optimal MBARP. The optimal MBARP yields significantly higher revenue than prior auctions for which the revenue-maximizing parameters could be solved for in closed form: separate Myerson auctions, pure-bundling Myerson auction, VCG, and mixed-bundling auction without reserve prices. Its revenue even exceeds that obtained via simulation within broader classes: VVCA and AMA. We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Node-weighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as Min-CSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovasz relaxation. The Local Distribution LP is generally stronger than the Lovasz relaxation, but applicable only to Min-CSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovasz relaxation gives a (2--2 k)-approximation for Submodular Multiway Partition with k terminals, improving a recent 2-approximation [2]. We prove that this factor is optimal in two senses: (1) A (2--2 k -- e)-approximation for Submodular Multiway Partition with k terminals would require exponentially many value queries (in the oracle model), or imply NP = RP (for certain explicit submodular functions). (2) For Hypergraph Multiway Cut and Node-weighted Multiway Cut with k terminals, both special cases of Submodular Multiway Partition, we prove that a (2--2 k -- e)-approximation is NP-hard, assuming the Unique Games Conjecture. Both our hardness results are more general: (1) We show that the notion of symmetry gap, previously used for submodular maximization problems [19, 6], also implies hardness results for submodular minimization problems. (2) Assuming the Unique Games Conjecture, we show that the Local Distribution LP gives an optimal approximation for every Min-CSP that includes the Not-Equal predicate. Finally, we connect the two hardness techniques by proving that the integrality gap of the Local Distribution LP coincides with the symmetry gap of the multilinear relaxation (for a related instance). This shows that the appearance of the same hardness threshold for a Min-CSP and the related submodular minimization problem is not a coincidence. We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut.