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57
Approximation algorithms for combinatorial auctions with complementfree bidders
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC
, 2005
"... We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items m and in the number of bidders n, even though the “input size ” is exponential in m. The first algori ..."
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Cited by 137 (27 self)
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We exhibit three approximation algorithms for the allocation problem in combinatorial auctions with complement free bidders. The running time of these algorithms is polynomial in the number of items m and in the number of bidders n, even though the “input size ” is exponential in m. The first algorithm provides an O(log m) approximation. The second algorithm provides an O ( √ m) approximation in the weaker model of value oracles. This algorithm is also incentive compatible. The third algorithm provides an improved 2approximation for the more restricted case of “XOS bidders”, a class which strictly contains submodular bidders. We also prove lower bounds on the possible approximations achievable for these classes of bidders. These bounds are not tight and we leave the gaps as open problems. 1
Optimal Approximation for the Submodular Welfare Problem in the value oracle model
 STOC'08
, 2008
"... In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility Pn i=1 wi(Si). In this pap ..."
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Cited by 122 (11 self)
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In the Submodular Welfare Problem, m items are to be distributed among n players with utility functions wi: 2 [m] → R+. The utility functions are assumed to be monotone and submodular. Assuming that player i receives a set of items Si, we wish to maximize the total utility Pn i=1 wi(Si). In this paper, we work in the value oracle model where the only access to the utility functions is through a black box returning wi(S) for a given set S. Submodular Welfare is in fact a special case of the more general problem of submodular maximization subject to a matroid constraint: max{f(S) : S ∈ I}, where f is monotone submodular and I is the collection of independent sets in some matroid. For both problems, a greedy algorithm is known to yield a 1/2approximation [21, 16]. In special cases where the matroid is uniform (I = {S: S  ≤ k}) [20] or the submodular function is of a special type [4, 2], a (1 − 1/e)approximation has been achieved and this is optimal for these problems in the value oracle model [22, 6, 15]. A (1 − 1/e)approximation for the general Submodular Welfare Problem has been known only in a stronger demand oracle model [4], where in fact 1 − 1/e can be improved [9]. In this paper, we develop a randomized continuous greedy algorithm which achieves a (1 − 1/e)approximation for the Submodular Welfare Problem in the value oracle model. We also show that the special case of n equal players is approximation resistant, in the sense that the optimal (1 − 1/e)approximation is achieved by a uniformly random solution. Using the pipage rounding technique [1, 2], we obtain a (1 − 1/e)approximation for submodular maximization subject to any matroid constraint. The continuous greedy algorithm has a potential of wider applicability, which we demonstrate on the examples of the Generalized Assignment Problem and the AdWords Assignment Problem.
Maximizing a Submodular Set Function subject to a Matroid Constraint (Extended Abstract)
 PROC. OF 12 TH IPCO
, 2007
"... Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 ..."
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Cited by 114 (12 self)
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Let f: 2 N → R + be a nondecreasing submodular set function, and let (N, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2approximation [9] for this problem. It is also known, via a reduction from the maxkcover problem, that there is no (1 − 1/e + ɛ)approximation for any constant ɛ> 0, unless P = NP [6]. In this paper, we improve the 1/2approximation to a (1−1/e)approximation, when f is a sum of weighted rank functions of matroids. This class of functions captures a number of interesting problems including set coverage type problems. Our main tools are the pipage rounding technique of Ageev and Sviridenko [1] and a probabilistic lemma on monotone submodular functions that might be of independent interest. We show that the generalized assignment problem (GAP) is a special case of our problem; although the reduction requires N  to be exponential in the original problem size, we are able to interpret the recent (1 − 1/e)approximation for GAP by Fleischer et al. [10] in our framework. This enables us to obtain a (1 − 1/e)approximation for variants of GAP with more complex constraints.
Approximation Algorithms for Orienteering and DiscountedReward TSP
, 2003
"... In this paper, we give the first constantfactor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the DiscountedReward TSP, motivated by robot navigation. In both problems, we are given a graph with lengths on edges and prizes (rewards) on nodes, ..."
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Cited by 81 (1 self)
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In this paper, we give the first constantfactor approximation algorithm for the rooted Orienteering problem, as well as a new problem that we call the DiscountedReward TSP, motivated by robot navigation. In both problems, we are given a graph with lengths on edges and prizes (rewards) on nodes, and a start node s. In the Orienteering Problem, the goal is to find a path that maximizes the reward collected, subject to a hard limit on the total length of the path. In the DiscountedReward TSP, instead of a length limit we are given a discount factor fl, and the goal is to maximize total discounted reward collected, where reward for a node reached at time t is discounted by fl . This is similar to the objective considered in Markov Decision Processes (MDPs) except we only receive a reward the first time a node is visited. We also consider tree and multiplepath variants of these problems and provide approximations for those as well. Although the unrooted orienteering problem, where there is no fixed start node s, has been known to be approximable using algorithms for related problems such as kTSP (in which the amount of reward to be collected is fixed and the total length is approximately minimized), ours is the first to approximate the rooted question, solving an open problem of [3, 1].
Online budgeted matching in random input models with applications to adwords
 In SODA 2008
"... We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a r ..."
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Cited by 72 (10 self)
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We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 − 1/e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1/2, and that the optimal algorithm presented there has a ratio of 1 − 1/e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auctionstyle process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking algorithm of [KVV90] has a ratio of 1 − 1/e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. 1
Approximation Algorithms for DeadlineTSP and Vehicle Routing with TimeWindows
 STOC'04
, 2004
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Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)app ..."
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Cited by 63 (0 self)
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Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires X  to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
A Recursive Greedy Algorithm for Walks in Directed Graphs
 PROC. OF IEEE FOCS
, 2005
"... Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the ori ..."
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Cited by 52 (3 self)
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Given an arcweighted directed graph G = (V, A, ℓ) and a pair of nodes s, t, we seek to find an st walk of length at most B that maximizes some given function f of the set of nodes visited by the walk. The simplest case is when we seek to maximize the number of nodes visited: this is called the orienteering problem. Our main result is a quasipolynomial time algorithm that yields an O(log OPT) approximation for this problem when f is a given submodular set function. We then extend it to the case when a node v is counted as visited only if the walk reaches v in its time window [R(v), D(v)]. We apply the algorithm to obtain several new results. First, we obtain an O(log OPT) approximation for a generalization of the orienteering problem in which the profit for visiting each node may vary arbitrarily with time. This captures the time window problem considered earlier for which, even in undirected graphs, the best approximation ratio known [4] is O(log 2 OPT). The second application is an O(log 2 k) approximation for the kTSP problem in directed graphs (satisfying asymmetric triangle inequality). This is the first nontrivial approximation algorithm for this problem. The third application is an O(log 2 k) approximation (in quasipoly time) for the group Steiner problem in undirected graphs where k is the number of groups. This improves earlier ratios [15, 19, 8] by a logarithmic factor and almost matches the inapproximability threshold on trees [20]. This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Ω(log 1−ɛ OPT), even in undirected graphs. Even though our algorithm runs in quasipoly time, we believe that the implications for the approximability of several basic optimization problems are interesting.
Maximizing Submodular Set Functions Subject to Multiple Linear Constraints
, 2009
"... The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we presen ..."
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Cited by 51 (1 self)
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The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we present the first known approximation algorithms for the problem of maximizing a nondecreasing submodular set function subject to multiple linear constraints. Given a ddimensional budget vector ¯ L, for some d ≥ 1, and an oracle for a nondecreasing submodular set function f over a universe U, where each element e ∈ U is associated with a ddimensional cost vector, we seek a subset of elements S ⊆ U whose total cost is at most ¯ L, such that f(S) is maximized. We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 − ε)(1 − e−1)approximation to the optimum for any ε> 0, where d> 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 − e−1 is (almost) matched for both of these problems.
Improved Algorithms for Orienteering and Related Problems
, 2007
"... In this paper we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to pointorienteeringproblem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an st ..."
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Cited by 50 (6 self)
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In this paper we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to pointorienteeringproblem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an st walk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as kMST. Our main results are the following. • A 2 + ɛ approximation in undirected graphs, improving upon the 3approximation from [5]. • An O(log 2 OPT) approximation in directed graphs. Previously, only a quasipolynomial time algorithm achieved a polylogarithmic approximation [12] (a ratio of O(log OPT)). The above results are based on, or lead to, improved algorithms for several other related problems.