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21
On kColumn Sparse Packing Programs
, 2009
"... We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k ..."
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We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an ek+o(k)approximation algorithm for kcolumn sparse PIPs, improving on recent results of k2 · 2k [14] and O(k2) [3, 5]. We also show that the integrality gap of our linear programming relaxation is at least 2k − 1; it is known that kcolumn sparse PIPs are Ω(k log k)hard to approximate [8]. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over kcolumn sparse packing constraints.
On Expressing Value Externalities in Position Auctions
"... Externalities are recognized to exist in the sponsored search market, where two colocated ads compete for user attention. Existing work focuses on the effect of another ad on the quantity of clicks received. We focus instead on the negative effect of another ad on the value per click, and propose a ..."
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Cited by 8 (1 self)
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Externalities are recognized to exist in the sponsored search market, where two colocated ads compete for user attention. Existing work focuses on the effect of another ad on the quantity of clicks received. We focus instead on the negative effect of another ad on the value per click, and propose a general model of externalities, in which a bidder has no value for a slot under a set of certain conditions, each on one other bidder’s allocated slot. We provide a generic greedy algorithm for the winner determination problem (WDP) in this model together with a pricing scheme that closely follow the Generalized Second Price (GSP) auction used in practice. For value externalities that satisfy a property of downwardmonotonicity, these mechanisms provide no new opportunities for manipulation beyond the ones already available via untruthful claims about bid value in GSP under the standard slot auction model. Our main instantiation of downwardmonotonic constraints is an identityspecific language, in which a bidder can require that it precedes some subset of other bidders. For this language’s WDP, we establish worstcase complexity and inapproximability results. This motivates the choice of approximations, e.g. via the greedy algorithm. As another way of circumventing the hardness results, we present fixedparameter algorithms for the WDPs of two sublanguages of the identityspecific model. 1.
On LPbased Approximability for Strict CSPs
"... In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes ..."
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Cited by 7 (1 self)
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In a beautiful result, Raghavendra established optimal Unique Games Conjecture (UGC)based inapproximability for a large class of constraint satisfaction problems (CSPs). In the class of CSPs he considers, of which Maximum Cut is a prominent example, the goal is to find an assignment which maximizes a weighted fraction of constraints satisfied. He gave a generic semidefinite program (SDP) for this class of problems and showed how the approximability of each problem is determined by the corresponding SDP (upto an arbitrarily small additive error) assuming the UGC. He noted that his techniques do no apply to CSPs with strict constraints (all of which must be satisfied) such as Vertex Cover. In this paper we address the approximability of these strictCSPs. In the class of CSPs we consider, one is given a set of constraints over a set of variables, and a cost function over the assignments, the goal is to find an assignment to the variables of minimum cost which satisfies all the constraints. We present a generic linear program (LP) for a large class of strictCSPs and give a systematic way to convert integrality gaps for this LP into UGCbased inapproximability results. Some important problems whose approximability our framework captures are Vertex Cover, Hypergraph Vertex Cover, kpartiteHypergraph Vertex Cover, Independent Set and other covering and packing problems over qary alphabets, and a scheduling problem. For the covering and packing problems, which occur quite commonly in practice as well, we provide a matching rounding algorithm, thus settling their approximability upto an arbitrarily small additive error.
Locally stable marriage with strict preferences
 In Proc. 40th Intl. Coll. Automata, Languages and Programming (ICALP
, 2013
"... Abstract. We study twosided matching markets with locality of information and control. Each male (female) agent has an arbitrary strict preference list over all female (male) agents. In addition, each agent is a node in a fixed network. Agents learn about possible partners dynamically based on th ..."
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Cited by 6 (3 self)
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Abstract. We study twosided matching markets with locality of information and control. Each male (female) agent has an arbitrary strict preference list over all female (male) agents. In addition, each agent is a node in a fixed network. Agents learn about possible partners dynamically based on their current network neighborhood. We consider convergence of dynamics to locally stable matchings that are stable with respect to their imposed information structure in the network. While existence of such states is guaranteed, we show that reachability becomes NPhard to decide. This holds even when the network exists only among one side. In contrast, if only one side has no network and agents remember a previous match every round, reachability is guaranteed and random dynamics converge with probability 1. We characterize this positive result in various ways. For instance, it holds for random memory and for memory with the most recent partner, but not for memory with the best partner. Also, it is crucial which partition of the agents has memory. Finally, we conclude with results on approximating maximum locally stable matchings. 1
Optimizing Password Composition Policies
, 2013
"... A password composition policy restricts the space of allowable passwords to eliminate weak passwords that are vulnerable to statistical guessing attacks. Usability studies have demonstrated that existing password composition policies can sometimes result in weaker password distributions; hence a mor ..."
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Cited by 4 (4 self)
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A password composition policy restricts the space of allowable passwords to eliminate weak passwords that are vulnerable to statistical guessing attacks. Usability studies have demonstrated that existing password composition policies can sometimes result in weaker password distributions; hence a more principled approach is needed. We introduce the first theoretical model for optimizing password composition policies. We study the computational and sample complexity of this problem under different assumptions on the structure of policies and on users’ preferences over passwords. Our main positive result is an algorithm that – with high probability — constructs almost optimal policies (which are specified as a union of subsets of allowed passwords), and requires only a small number of samples of users’ preferred passwords. We complement our theoretical results with simulations using a realworld dataset of 32 million passwords.
Placing Regenerators in Optical Networks to Satisfy Multiple Sets of Requests
, 2010
"... Abstract. The placement of regenerators in optical networks has become an active area of research during the last years. Given a set of lightpaths in a network G and a positive integer d, regenerators must be placed in such a way that in any lightpath there are no more than d hops without meeting a ..."
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Cited by 3 (2 self)
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Abstract. The placement of regenerators in optical networks has become an active area of research during the last years. Given a set of lightpaths in a network G and a positive integer d, regenerators must be placed in such a way that in any lightpath there are no more than d hops without meeting a regenerator. While most of the research has focused on heuristics and simulations, the first theoretical study of the problem has been recently provided in [10], where the considered cost function is the number of locations in the network hosting regenerators. Nevertheless, in many situations a more accurate estimation of the real cost of the network is given by the total number of regenerators placed at the nodes, and this is the cost function we consider. Furthermore, in our model we assume that we are given a finite set of p possible traffic patterns (each given by a set of lightpaths), and our objective is to place the minimum number of regenerators at the nodes so that each of the traffic patterns is satisfied. While this problem can be easily solved when d = 1 or p = 1, we prove that for any fixed d, p ≥ 2 it does not admit a PTAS, even if G has maximum degree at most 3 and the lightpaths have length O(d). We complement this hardness result with a constantfactor approximation algorithm with ratio ln(d · p). We then study the case where G is a path, proving that the problem is NPhard for any d, p ≥ 2, even if there are two edges of the path such that any lightpath uses at least one of them. Interestingly, we show that the problem is polynomialtime solvable in paths when all the lightpaths share the first edge of the path, as well as when the number of lightpaths sharing an edge is bounded. Finally, we generalize our model in two natural directions, which allows us to capture the model of [10] as a particular case, and we settle some questions that were left open in [10].
Socially stable matchings
 in the Hospitals / Residents problem. CoRR Technical Report 1303.2041. Available from http://arxiv.org/abs/1303.2041
"... In twosided matching markets, the agents are partitioned into two sets. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their ..."
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Cited by 3 (3 self)
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In twosided matching markets, the agents are partitioned into two sets. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. In this paper we study a generalization of stable matching motivated by the fact that, in most centralized markets, many agents do not have direct communication with each other. Hence even if some blocking pairs exist, the agents involved in those pairs may not be able to coordinate a deviation. We model communication channels with a bipartite graph between the two sets of agents which we call the social graph, and we study socially stable matchings. A matching is socially stable if there are no blocking pairs that are connected by an edge in the social graph. Socially stable matchings vary in size and so we look for a maximum socially stable matching. We prove that this problem is NPhard and, assuming the unique games conjecture, hard to approximate within a factor of 3 2 − ɛ, for any constant ɛ. Weapproximation algorithm. complement the hardness results with a 3 2 1
On the Optimality of a Class of LPbased Algorithms ∗
"... In this paper we will be concerned with a class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, one can write an LP relaxation and then round the solution. For instance, for Vertex Cover one can obtain a 2approximation via this approach. On the other han ..."
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In this paper we will be concerned with a class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, one can write an LP relaxation and then round the solution. For instance, for Vertex Cover one can obtain a 2approximation via this approach. On the other hand, Khot and Regev [KR08] proved that, assuming the Unique Games Conjecture (UGC), it is NPhard to approximate Vertex Cover to within a factor better than 2 − ε for any constant ε> 0. From their, and subsequent proofs of this result, it was not clear why this LP relaxation should be optimal. The situation was akin to Maximum Cut, where a natural SDP relaxation for it was proved by Khot et al. [KKMO07] to be optimal assuming the UGC. A beautiful result of Raghavendra [Rag08] explains why the SDP is optimal (assuming the UGC). Moreover, his result generalizes to a large class of constraint satisfaction problems (CSPs). Unfortunately, we do not know how to extend his framework so that it applies for problems such as Vertex Cover where the constraints are strict. In this paper, we explain why the simple LPbased rounding algorithm for the Vertex Cover problem is optimal assuming the UGC. Complementing Raghavendra’s result, our result generalizes to a class of strict, covering/packing type CSPs. We first write down a natural LP relaxation for this class of problems and present
Vertex Cover in Graphs with Locally Few Colors
, 2011
"... In [14], Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any ∆ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than ∆ dif ..."
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In [14], Erdős et al. defined the local chromatic number of a graph as the minimum number of colors that must appear within distance 1 of a vertex. For any ∆ ≥ 2, there are graphs with arbitrarily large chromatic number that can be colored so that (i) no vertex neighborhood contains more than ∆ different colors (bounded local colorability), and (ii) adjacent vertices from two color classes induce a complete bipartite graph (biclique coloring). We investigate the weighted vertex cover problem in graphs when a locally bounded coloring is given. This generalizes the vertex cover problem in bounded degree graphs to a class of graphs with arbitrarily large chromatic number. Assuming the Unique Game Conjecture (UGC), we provide a tight characterization. We prove that it is UGChard to improve the approximation ratio of 2 − 2/( ∆ + 1) if the given local coloring is not a biclique coloring. A matching upper bound is also provided. Vice versa, when properties (i) and (ii) hold, we present a randomized algorithm with approximation ratio ln ln ∆ ln ∆ of 2 − Ω(1). This matches known inapproximability results for the special case of bounded degree graphs. Moreover, we show that when both the above two properties (i) and (ii) hold, the obtained result finds a natural application in a classical scheduling problem, namely the precedence constrained single machine scheduling problem to minimize the total weighted completion time, denoted as 1prec  ∑ wjCj in standard scheduling notation. In a series of recent papers it was established that this scheduling problem is a special case of the minimum weighted vertex cover in graphs GP of incomparable pairs defined in the dimension theory of partial orders. We show that GP satisfies properties (i) and (ii) where ∆ − 1 is the maximum number of predecessors (or successors) of each job.