Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

In the \emph{tollbooth problem}, we are given a tree \bT=(V,E) with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of \bT such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when \bT is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of \bT. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard

Pricing Network Edges to Cross a River.

We consider a Stackelberg pricing problem in directed networks:Tariffs (prices) have to be defined by an operator, the leader, for a subset of the arcs. Clients, the followers, choose paths to route their demand through the network selfishly and independently of each other, on the basis of minimal total cost. The problem is to find tariffs such as to maximize the operator''s revenue. We consider the case where each client takes at most one tariff arc to route the demand.The problem, which we refer to as the river tarification problem, is a special case of problems studied previously in the literature.We prove that the problem is strongly NP-hard.Moreover, we show that the polynomially solvable case of uniform tarification yields an m--approximation algorithm, and this is tight. We suggest a new type of analysis that allows to improve the result to \bigO{\log m}, whenever the input data is polynomially bounded. We furthermore derive an \bigO{m^{1-\varepsilon}}--inapproximability result for problems where the operator must serve all clients, and we discuss some polynomial special cases. Finally, a computational study with instances from France Telecom suggests that uniform pricing performs better in practice than theory would suggest.operations research and management science;

Approximation Algorithms for the Highway Problem under the Coupon Model

When a store sells items to customers, the store wishes to determine the prices of the items to maximize its profit. Intuitively, if the store sells the items with low (resp. high) prices, the customers buy more (resp. less) items, which provides less profit to the store. So it would be hard for the store to decide the prices of items. Assume that the store has a set V of n items and there is a set E of m customers who wish to buy those items, and also assume that each item i \in V has the production cost d_i and each customer e_j \in E has the valuation v_j on the bundle e_j \subseteq V of items. When the store sells an item i \in V at the price r_i, the profit for the item i is p_i=r_i-d_i. The goal of the store is to decide the price of each item to maximize its total profit. In most of the previous works, the item pricing problem was considered under the assumption that p_i \geq 0 for each i \in V, however, Balcan, et al. [In Proc. of WINE, LNCS 4858, 2007] introduced the notion of loss-leader, and showed that the seller can get more total profit in the case that p_i < 0 is allowed than in the case that p_i < 0 is not allowed. In this paper, we consider the line and the cycle highway problem, and show approximation algorithms for the line and/or cycle highway problem for which the smallest valuation is s and the largest valuation is \ell or all valuations are identical.Comment: 13 pages, 5 figure

Beyond The Technical Dimensions of Water: Community-Based Efforts for Water Justice in the High Desert

Water injustice is a result of the co-constitution of water and power from the global to the local level. In the small High Desert town of Adelanto, CA, concerns over the water quality and its association with overall quality of life incentivized a collaborative project to investigate dealings of water. A multidimensional analysis of the water system uncovers an instance of water injustice that is complex and linked to water issues regionally in the Inland Southern California area, as well as globally. Globalized discourses, climate change, global capitalism and flows of immigration are some of the global processes linked to water in Adelanto. A dominating narrative of water scarcity and a technocratic approach to water has left little room for local knowledge and community participation while also producing a problematic water system that has ultimately fostered a sense of mistrust in the safety of water among residents. Further, comparison of different sources of information on the water quality reveal misalignment between the reporting of contaminant levels. Testimonies from residents express concerns over the odd taste, smell and appearance of water, and indicate that the issue is likely connected to overall disinvestment from immigrant, low-income communities and other marginalized communities. Findings stress the importance of local knowledge in the face of a technocratic water system that has produced intersecting forms of injustice. Equitable, safe and trustworthy access to drinkable water depends upon developing community water monitoring systems, empowering community members to make political claims, and fostering intimate connection with water

The Feasibility of Pulp and Paper Production in Utah

The objectives of this study are: l. To determine the adequacy of markets for paper products. 2. To identify and quantify the costs of the factors affecting pulp and paper production. The available information indicates that adequate markets exist for paper produced in Utah. Chemicals, water, labor, and power are available at costs low enough not to prohibit pulp and paper production. An adequate, low priced supply of wood could be purchased from the United States Forest Service. Intermountain and national paper markets would enable a paper producer in Utah to take advantage of lower comparative transportation costs

Social, Economic and Health Impact of Industrial Pollution in Dindigul District, Tamil Nadu

Background: Pollution of air, water and soil poisons the environment and cause diseases. The illness in a social context needs further exploration in this particular context. That also led to a reduction in agricultural production, drinking water contamination, and cause illness to both human and animal. That is how it is contributing to alterations in social structure specifically the social institutions. By studying the problem of pollution, we address the impact of pollution on health, society and the environment as a whole. Methods/Design: For the mentioned objectives, the explanatory design was adapted. The study will have three components. A quantitative survey using an interview schedule conducted among the members of the selected villages of Dindigul, district. Specific questions on Social, Economic and Health impacts of industrial pollution will be administered. There will be two different locations, one closer to the Tanneries and villages located 10 Kilometers away from the Tanneries. Even the later villages there are other industries such as handlooms. This will help to understand the difference in the impact of industrial pollution with specific reference to tanneries. Qualitative data aims to conduct in-depth interviews among the village leaders and Focus Group Discussions among the community members. The basic water and soil tests form the villages near tanneries will give additional evidence to understand the physical effects of pollution on the rural environment. The study has received ethical approval from the Institutional Ethics committee. Discussion: Due to the industrialization in the city of Dindigul and the surrounding area, the pollution from industrial outlets not only affect the environment, but it also affects the social structure, interactions, and employment