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Black-Box Reductions for Cost-Sharing Mechanism Design
"... We consider the design of strategyproof cost-sharing mechanisms. We give two simple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a minimum-cost solution for a set of players. Our first reduc ..."
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We consider the design of strategyproof cost-sharing mechanisms. We give two simple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a minimum-cost solution for a set of players. Our first reduction shows that any truthful, α-approximation mechanism for the socialcost minimization (SCM) problem satisfying a technical no-bossiness condition can be morphed into a truthful mechanism that achieves an O(α log n)-approximation where the prices recover the cost incurred. Thus, we decouple the task of truthfully computing an outcome with near-optimal social cost from the cost-sharing problem. This is fruitful since truthful mechanism-design, especially for single-dimensional problems, is a relatively well-understood and manageable task. Our second reduction nicely complements the first one by showing that any LP-based ρ-approximation for the problem of finding a min-cost solution for a set of players yields a truthful, no-bossy, (ρ + 1)-approximation for the SCM problem (and hence, a truthful (ρ + 1) log n-approximation cost-sharing mechanism). These reductions find a slew of applications, yielding, as corollaries, the first or improved polytime costsharing mechanisms for a variety of problems. For example, our first reduction coupled with the celebrated VCG mechanism shows that for any costsharing problem (with a monotone cost function) one can obtain a truthful mechanism that achieves an O(log n)-approximation where the prices recover the cost incurred. Other applications include O(log n)approximation mechanisms for: survivable network design problems, facility location (FL) problems including capacitated and connected FL problems, and minimummakespan scheduling on unrelated machines. Our results demonstrate that in contrast with our current understanding of group-strategyproof and acyclic mechanisms, strategyproofness allows for ample flexibility in cost-sharing mechanism design enabling one to effectively leverage various algorithmic results.
Primal-dual algorithms for combinatorial optimization problems
, 2007
"... Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a ..."
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Combinatorial optimization problems such as routing, scheduling, covering and packing problems abound in everyday life. At a very high level, a combinatorial optimization problem amounts to finding a solution with minimum or maximum cost among a large number of feasible solutions. An algorithm for a given optimization problem is said to be exact if it always returns an optimal solution and is said to be efficient if it runs in time polynomial on the size of its input. The theory of NP-completeness suggests that exact and efficient algorithms are unlikely to exist for the class of NP-hard problems. Unfortunately, a large number of natural and interesting combinatorial optimization problems are NP-hard. One way to cope with NP-hardness is to relax the optimality requirement and instead look for solutions that are provably close to the optimum. This is the main idea behind approximation algorithms. An algorithm is said to be a ρ-approximation if it always returns a solution whose cost is at most a ρ factor away from the optimal cost. Arguably, one of the most important techniques in the design of combinatorial algorithms is the primal-dual schema in which the cost of the primal solution is compared to the cost of a dual solution. In this dissertation we study the primal-dual schema in the design of approximation algorithms for a number of covering and scheduling problems.
LAGRANGIAN RELAXATION AND PARTIAL COVER
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LM ..."
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Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) α-approximation as a black box can yield an approximation factor better than 4/3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρ-Blocks.
Partial Interval Set Cover -- Trade-offs Between Scalability and Optimality
, 2013
"... Given an interval I = {1, 2,..., n} of points, a collection I of subintervals of I and a fraction 0 ≤ r ≤ 1, we consider the following variation of partial set cover. We wish to find an optimal subset of I covering at least an r-fraction of I. While this problem is easily solved exactly in quadratic ..."
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Given an interval I = {1, 2,..., n} of points, a collection I of subintervals of I and a fraction 0 ≤ r ≤ 1, we consider the following variation of partial set cover. We wish to find an optimal subset of I covering at least an r-fraction of I. While this problem is easily solved exactly in quadratic time using classical methods, we focus on developing scalable algorithms which return near-optimal solutions and run in near-linear time. We give a (1 + ɛ)-approximation algorithm running in O ( 1 ɛ · min{n + |I|, |I | log |I|}}) time. We also prove a tight approximation ratio of 2 for a simple greedy algorithm for this problem, improving on the bound of 9 given in [10].
Partial Multicovering and the d-consecutive Ones Property
, 2011
"... A d-interval is the union of d disjoint intervals on the real line. In the d-interval stabbing problem (d-is) we are given a set of d-intervals and a set of points, each d-interval I has a stabbing requirement r(I) and each point has a weight, and the goal is to find a minimum weight multiset of poi ..."
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A d-interval is the union of d disjoint intervals on the real line. In the d-interval stabbing problem (d-is) we are given a set of d-intervals and a set of points, each d-interval I has a stabbing requirement r(I) and each point has a weight, and the goal is to find a minimum weight multiset of points that stabs each d-interval I at least r(I) times. In practice there is a trade-off between fulfilling requirements and cost, and therefore it is interesting to study problems in which one is required to fulfill only a subset of the requirements. In this paper we study variants of d-is in which a feasible solution is a multiset of points that may satisfy only a subset of the stabbing requirements. In partial d-is we are given an integer t, and the sum of requirements satisfied by the computed solution must be at least t. In prize collecting d-is each d-interval has a penalty that must be paid for every unit of unsatisfied requirement. We also consider a maximization version of prize collecting d-is in which each d-interval has a prize that is gained for every time, up to r(I), it is stabbed. Our study is motivated by several resource allocation and geometric facility location problems. We present a (ρ+d−1 ρ)-approximation algorithm for prize collecting d-is, where ρ = minI r(I), and an O(d)-approximation algorithm for partial d-is. We obtain the latter result by designing a general framework for approximating partial multicovering problems that extends the framework for approximating partial covering problems from [21]. We also show that maximum prize collecting d-is is at least as hard to approximate as maximum independent set, even for d = 2, and present a d-approximation algorithm for maximum prize collecting d-dimensional rectangle stabbing.
Classification by Set Cover: The Prototype Vector Machine
, 2009
"... We introduce a new nearest-prototype classifier, the prototype vector machine (PVM). It arises from a combinatorial optimization problem which we cast as a variant of the set cover problem. We propose two algorithms for approximating its solution. The PVM selects a relatively small number of represe ..."
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We introduce a new nearest-prototype classifier, the prototype vector machine (PVM). It arises from a combinatorial optimization problem which we cast as a variant of the set cover problem. We propose two algorithms for approximating its solution. The PVM selects a relatively small number of representative points which can then be used for classification. It contains 1-NN as a special case. The method is compatible with any dissimilarity measure, making it amenable to situations in which the data are not embedded in an underlying feature space or in which using a non-Euclidean metric is desirable. Indeed, we demonstrate on the much studied ZIP code data how the PVM can reap the benefits of a problem-specific metric. In this example, the PVM outperforms the highly successful 1-NN with tangent distance, and does so retaining fewer than half of the data points. This example highlights the strengths of the PVM in yielding a low-error, highly interpretable model. Additionally, we apply the PVM to a protein classification problem in which a kernel-based distance is used. 1
FPTAS’s for Trimming Weighted Trees
"... Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost o ..."
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Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at most k and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an O(nl/ε)-time O(l 2 /ε + n)-space algorithm for MaxMstCut, and O(nl(1/ε+log n))-time O(l 2 /ε+n)-space algorithms for the other two problems, MinLstCut and MaxLstCut, where n is the number of vertices in the tree, l the number of leaves, and ε> 0 the prescribed error bound. Key words.
Black-Box Reductions for Cost-Sharing Mechanism
"... We consider the design of strategyproof cost-sharing mechanisms. We give two sim-ple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a min-cost solution for a set of players. Our first reductio ..."
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We consider the design of strategyproof cost-sharing mechanisms. We give two sim-ple, but extremely versatile, black-box reductions, that in combination reduce the cost-sharing mechanism-design problem to the algorithmic problem of finding a min-cost solution for a set of players. Our first reduction shows that any truthful, α-approximation mechanism for the social-cost minimization (SCM) problem satisfying a technical no-bossiness condition can be morphed into a truthful mechanism that achieves an O(α log n)-approximation where the prices recover the cost incurred. Thus, we de-couple (modulo no-bossiness) the task of truthfully computing an outcome with near-optimal social cost from the cost-sharing problem. This is fruitful since truthful mechanism-design, especially for single-dimensional problems, is a relatively well-understood and manageable task. Our second reduction nicely complements the first one by showing that any LP-relative ρ-approximation for the problem of finding a min-cost solution for a set of players yields a truthful, no-bossy, (ρ+ 1)-approximation for the SCM problem (and hence, a truthful (ρ + 1) log n-approximation cost-sharing mechanism).
