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
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The multiple knapsack problem (MKP) is a natural and well-known generalization of the single knapsack problem and is defined as follows. We are given a set of @math items and @math bins (knapsacks) such that each item @math has a profit @math and a size @math , and each bin @math has a capacity @math . The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the generalized assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation, for it is implicit in the work of Shmoys and Tardos [Math. Program. A, 62 (1993), pp. 461-474], and thus far, this was also the best known approximation for MKP @. The main result of this paper is a polynomial time approximation scheme (PTAS) for MKP @. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a PTAS-preserving reduction from an arbitrary instance of MKP to an instance with @math distinct sizes and profits. We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node @math in a given tree @math . Upon querying a node @math of the tree, the strategy receives as a reply an indication of the connected component of @math containing the target @math . The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target. Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme: for any @math , there exists a @math -approximation strategy with a computation time of @math . Thus, the problem is not APX-hard, unless @math . By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time @math -approximation. This improves previous @math -approximation approaches, where the @math -notation disregards @math -factors. In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node @math upper-bounds the number of edges incident to @math that can be probed. The goal is to maximize the expected profit of the constructed matching. For this problem (Algorithmica 2012) provided a @math -approximation algorithm for bipartite graphs, and a @math -approximation for general graphs. In this work we improve the approximation factors to @math and @math , respectively. We also consider an online version of the bipartite case, where one side of the partition arrives node by node, and each time a node @math arrives we have to decide which edges incident to @math we want to probe, and in which order. Here we present a @math -approximation, improving on the @math -approximation of The main technical ingredient in our result is a novel way of probing edges according to a random but non-uniform permutation. Patching this method with an algorithm that works best for large probability edges (plus some additional ideas) leads to our improved approximation factors. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 .