We introduce the classified stable matching problem, a problem motivated by academic hiring. Suppose that a number of institutes are hiring faculty members from a pool of applicants. Both institutes and applicants have preferences over the other side. An institute classifies the applicants based on their research areas (or any other criterion), and, for each class, it sets a lower bound and an upper bound on the number of applicants it would hire in that class. The objective is to find a stable matching from which no group of participants has reason to deviate. Moreover, the matching should respect the upper lower bounds of the classes. In the first part of the paper, we study classified stable matching problems whose classifications belong to a fixed set of order types.'' We show that if the set consists entirely of downward forests, there is a polynomial-time algorithm; otherwise, it is NP-complete to decide the existence of a stable matching. In the second part, we investigate the problem using a polyhedral approach. Suppose that all classifications are laminar families and there is no lower bound. We propose a set of linear inequalities to describe stable matching polytope and prove that it is integral. This integrality allows us to find various optimal stable matchings using Ellipsoid algorithm. A further ramification of our result is the description of the stable matching polytope for the many-to-many (unclassified) stable matching problem. This answers an open question posed by Sethuraman, Teo and Qian.
Abraham, Irving and Manlove introduced the student-project allocation problem @cite_2 . It can be shown that in , if all classifications are just partitions over the applicants and there is no lower bound, our problem is equivalent to a special case of their problem. They posed the open question whether there is a polynomial time algorithm for their problem if there is lower bound on the projects (classes). Our result in Section 2 gives a partial positive answer.
@cite_2
2012500540 2949121341 1989262026 1923579438 2044502527 2121240598 2126085282 2889292676 2072113937 48624319 2112741516 2037028865 2033898593 2061130579 2900278988 2108444537 2594745741 1964506478 2034600319 2808191361
We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation. We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1- @math allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a @math - @math allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a @math - @math allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of ! to a @math factor in the additive setting. The main idea for our @math - @math allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time. We demonstrate the impossibility of strongly polynomial algorithms for the allocation problem, in the comparison model and in the algebraic tree computation model, that follow from lower bound results. Consequently, there are no strongly polynomial algorithms for nonlinear (concave) separable optimization over a totally unimodular constraint matrix. This is in contrast to the case when the objective is linear. We present scaling-based algorithms that use a greedy algorithm as a subroutine. The algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem, in that the complexity meets the lower bound derived from the comparison model. For other extensions of the allocation problem the scaling-based algorithms presented here are the fastest known. These algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables. The main aim of this note is to show that a polynomial-time algorithm for the scheduling problem 1|r j ; p j = p| ∑ Uj given by Carlier in (1981) is incorrect. In this problem we are given n jobs with release times and deadlines. All jobs have the same processing time p. The objective is to find a non-preemptive schedule that maximizes the number of jobs completed by their deadlines. The feasibility version of this problem, where we ask whether all jobs can meet their deadlines, has been studied thoroughly. Polynomial-time algorithms for this version were first found, independently, by Simons (1978) and Carlier (1981). A faster algorithm, with running time O(n log n), was subsequently given by (1981). The elegant feasibility algorithm of Carlier (1981) is based on dynamic programming and it processes jobs from left to right on the time-axis. For each time t, it constructs a partial schedule with jobs that complete at or before time t. Carlier also attempted to apply the same technique to design a polynomial-time algorithm for the maximization version, 1|r j ; p j = p| ∑ Uj , and claimed an O(n3 log n)-time algorithm. His result is now widely cited in the literature. We show, however, that this algorithm is not correct, by giving an instance on which it produces a sub-optimal schedule. Our counter-example can be, in fact, extended to support a broader claim, namely that even the general approach from Carlier (1981) cannot yield a polynomial-time algorithm. By this general approach we mean a class of algorithms that processes the input from left to right and make decisions based on the deadline ordering, and not their exact values. The question remains as to how efficiently can we solve the scheduling problem 1|r j ; p j = p| ∑ Uj . Baptiste (1999) gave an O(n7)-time algorithm for the more general version of this problem where jobs have weights. We show how to modify his algorithm to obtain a faster, O(n5)-time algorithm for the non-weighted case. These last two results are discussed only briefly in this note. The complete proofs can be found in the full version of this paper, see (2004). We investigate the optimal allocation of effort to a collection of n projects. The projects are 'restless' in that the state of a project evolves in time, whether or not it is allocated effort. The evolution of the state of each project follows a Markov rule, but transitions and rewards depend on whether or not the project receives effort. The objective is to maximize the expected time-average reward under a constraint that exactly m of the n projects receive effort at any one time. We show that as m and n tend to oo with m n fixed, the per-project reward of the optimal policy is asymptotically the same as that achieved by a policy which operates under the relaxed constraint that an average of m projects be active. The relaxed constraint was considered by Whittle (1988) who described how to use a Lagrangian multiplier approach to assign indices to the projects. He conjectured that the policy of allocating effort to the m projects of greatest index is asymptotically optimal as m and n tend to oo. We show that the conjecture is true if the differential equation describing the fluid approximation to the index policy has a globally stable equilibrium point. This need not be the case, and we present an example for which the index policy is not asymptotically optimal. However, numerical work suggests that such counterexamples are extremely rare and that the size of the suboptimality We study the problem of fairly allocating a set of indivisible goods to a set of people from an algorithmic perspective. fair division has been a central topic in the economic literature and several concepts of fairness have been suggested. The criterion that we focus on is envy-freeness. In our model, a monotone utility function is associated with every player specifying the value of each subset of the goods for the player. An allocation is envy-free if every player prefers her own share than the share of any other player. When the goods are divisible, envy-free allocations always exist. In the presence of indivisibilities, we show that there exist allocations in which the envy is bounded by the maximum marginal utility, and present a simple algorithm for computing such allocations. We then look at the optimization problem of finding an allocation with minimum possible envy. In the general case the problem is not solvable or approximable in polynomial time unless P = NP. We consider natural special cases (e.g.additive utilities) which are closely related to a class of job scheduling problems. Approximation algorithms as well as inapproximability results are obtained. Finally we investigate the problem of designing truthful mechanisms for producing allocations with bounded envy. 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 @math 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 2-approximation 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. We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph G. When the items are arranged in a path, we show that EF1 allocations are guaranteed to exist for arbitrary monotonic utility functions over bundles, provided that either there are at most four agents, or there are any number of agents but they all have identical utility functions. Our existence proofs are based on classical arguments from the divisible cake-cutting setting, and involve discrete analogues of cut-and-choose, of Stromquist's moving-knife protocol, and of the Su-Simmons argument based on Sperner's lemma. Sperner's lemma can also be used to show that on a path, an EF2 allocation exists for any number of agents. Except for the results using Sperner's lemma, all of our procedures can be implemented by efficient algorithms. Our positive results for paths imply the existence of connected EF1 or EF2 allocations whenever G is traceable, i.e., contains a Hamiltonian path. For the case of two agents, we completely characterize the class of graphs @math that guarantee the existence of EF1 allocations as the class of graphs whose biconnected components are arranged in a path. This class is strictly larger than the class of traceable graphs; one can be check in linear time whether a graph belongs to this class, and if so return an EF1 allocation. In the traffic assignment problem, first proposed by Wardrop in 1952, commuters select the shortest available path to travel from their origins to their destinations. We study a generalization of this problem in which competitors, who may control a nonnegligible fraction of the total flow, ship goods across a network. This type of games, usually referred to as atomic games, readily applies to situations in which the competing freight companies have market power. Other applications include intelligent transportation systems, competition among telecommunication network service providers, and scheduling with flexible machines. Our goal is to determine to what extent these systems can benefit from some form of coordination or regulation. We measure the quality of the outcome of the game without centralized control by computing the worst-case inefficiency of Nash equilibria. The main conclusion is that although self-interested competitors will not achieve a fully efficient solution from the system's point of view, the loss is not too severe. We show how to compute several bounds for the worst-case inefficiency that depend on the characteristics of cost functions and on the market structure in the game. In addition, building upon the work of Catoni and Pallotino, we show examples in which market aggregation (or collusion) adversely impacts the aggregated competitors, even though their market power increases. For example, Nash equilibria of atomic network games may be less efficient than the corresponding Wardrop equilibria. When competitors are completely symmetric, we provide a characterization of the Nash equilibrium using a potential function, and prove that this counterintuitive phenomenon does not arise. Finally, we study a pricing mechanism that elicits more coordination from the players by reducing the worst-case inefficiency of Nash equilibria. Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f' is an α-sketch of f if for every set S, the value f'(S) lies between f(S) α and f(S), and f' can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1 2. O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n1 2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1 3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−e. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general. Motivated by the allocation problem facing publishers in display advertising we formulate the online assignment with forecast problem, a version of the online allocation problem where the algorithm has access to random samples from the future set of arriving vertices. We provide a solution that allows us to serve Internet users in an online manner that is provably nearly optimal. Our technique applies to the forecast version of a large class of online assignment problems, such as online bipartite matching, allocation, and budgeted bidders, in which we wish to minimize the value of some convex objective function subject to a set of linear supply and demand constraints. Our solution utilizes a particular subspace of the dual space, allowing us to describe the optimal primal solution implicitly in space proportional to the demand side of the input graph. More importantly, it allows us to prove that representing the primal solution using such a compact allocation plan yields a robust online algorithm which makes near-optimal online decisions. Furthermore, unlike the primal solution, we show that the compact allocation plan produced by considering only a sampled version of the original problem generalizes to produce a near optimal solution on the full problem instance. Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields. SummaryGale and Shapley have an algorithm for assigning students to universities which gives each student the best university available in a stable system of assignments. The object here is to prove that students cannot improve their fate by lying about their preferences. Indeed, no coalition of students can simultaneously improve the lot of all its members if those outside the coalition state their true preferences. In a traditional classification problem, we wish to assign one of k labels (or classes) to each of n objects, in a way that is consistent with some observed data that we have about the problem. An active line of research in this area is concerned with classification when one has information about pairwise relationships among the objects to be classified; this issue is one of the principal motivations for the framework of Markov random fields, and it arises in areas such as image processing, biometry, and document analysis. In its most basic form, this style of analysis seeks to find a classification that optimizes a combinatorial function consisting of assignment costs---based on the individual choice of label we make for each object---and separation costs---based on the pair of choices we make for two "related" objects.We formulate a general classification problem of this type, the metric labeling problem; we show that it contains as special cases a number of standard classification frameworks, including several arising from the theory of Markov random fields. From the perspective of combinatorial optimization, our problem can be viewed as a substantial generalization of the multiway cut problem, and equivalent to a type of uncapacitated quadratic assignment problem.We provide the first nontrivial polynomial-time approximation algorithms for a general family of classification problems of this type. Our main result is an O(log k log log k)-approximation algorithm for the metric labeling problem, with respect to an arbitrary metric on a set of k labels, and an arbitrary weighted graph of relationships on a set of objects. For the special case in which the labels are endowed with the uniform metric---all distances are the same---our methods provide a 2-approximation algorithm. We study the problem of allocating indivisible items to agents with additive valuations, under the additional constraint that bundles must be connected in an underlying item graph. Previous work has considered the existence and complexity of fair allocations. We study the problem of finding an allocation that is Pareto-optimal. While it is easy to find an efficient allocation when the underlying graph is a path or a star, the problem is NP-hard for many other graph topologies, even for trees of bounded pathwidth or of maximum degree 3. We show that on a path, there are instances where no Pareto-optimal allocation satisfies envy-freeness up to one good, and that it is NP-hard to decide whether such an allocation exists, even for binary valuations. We also show that, for a path, it is NP-hard to find a Pareto-optimal allocation that satisfies maximin share, but show that a moving-knife algorithm can find such an allocation when agents have binary valuations that have a non-nested interval structure. We consider structural and algorithmic questions related to the Nash dynamics of weighted congestion games. In weighted congestion games with linear latency functions, the existence of pure Nash equilibria is guaranteed by a potential function argument. Unfortunately, this proof of existence is inefficient and computing pure Nash equilibria in such games is a PLS-hard problem even when all players have unit weights. The situation gets worse when superlinear (e.g., quadratic) latency functions come into play; in this case, the Nash dynamics of the game may contain cycles and pure Nash equilibria may not even exist. Given these obstacles, we consider approximate pure Nash equilibria as alternative solution concepts. A ρ--approximate pure Nash equilibrium is a state of a (weighted congestion) game from which no player has any incentive to deviate in order to improve her cost by a multiplicative factor higher than ρ. Do such equilibria exist for small values of ρ? And if so, can we compute them efficiently? We provide positive answers to both questions for weighted congestion games with polynomial latency functions by exploiting an “approximation” of such games by a new class of potential games that we call Ψ-games. This allows us to show that these games have d!-approximate pure Nash equilibria, where d is the maximum degree of the latency functions. Our main technical contribution is an efficient algorithm for computing O(1)-approximate pure Nash equilibria when d is a constant. For games with linear latency functions, the approximation guarantee is 3+√5 2 + Oγ for arbitrarily small γ > 0; for latency functions with maximum degree d≥ 2, it is d2d+o(d). The running time is polynomial in the number of bits in the representation of the game and 1 γ. As a byproduct of our techniques, we also show the following interesting structural statement for weighted congestion games with polynomial latency functions of maximum degree d ≥ 2: polynomially-long sequences of best-response moves from any initial state to a dO(d2)-approximate pure Nash equilibrium exist and can be efficiently identified in such games as long as d is a constant. To the best of our knowledge, these are the first positive algorithmic results for approximate pure Nash equilibria in weighted congestion games. Our techniques significantly extend our recent work on unweighted congestion games through the use of Ψ-games. The concept of approximating nonpotential games by potential ones is interesting in itself and might have further applications. We analyze the problem of learning a single user's preferences in an active learning setting, sequentially and adaptively querying the user over a finite time horizon. Learning is conducted via choice-based queries, where the user selects her preferred option among a small subset of offered alternatives. These queries have been shown to be a robust and efficient way to learn an individual's preferences. We take a parametric approach and model the user's preferences through a linear classifier, using a Bayesian prior to encode our current knowledge of this classifier. The rate at which we learn depends on the alternatives offered at every time epoch. Under certain noise assumptions, we show that the Bayes-optimal policy for maximally reducing entropy of the posterior distribution of this linear classifier is a greedy policy, and that this policy achieves a linear lower bound when alternatives can be constructed from the continuum. Further, we analyze a different metric called misclassification error, proving that the performance of the optimal policy that minimizes misclassification error is bounded below by a linear function of differential entropy. Lastly, we numerically compare the greedy entropy reduction policy with a knowledge gradient policy under a number of scenarios, examining their performance under both differential entropy and misclassification error. We consider the classical problem of minimizing the total weighted flow-time for unrelated machines in the online non-clairvoyant setting. In this problem, a set of jobs J arrive over time to be scheduled on a set of M machines. Each job J has processing length pj, weight wj, and is processed at a rate of lij when scheduled on machine i. The online scheduler knows the values of wj and lij upon arrival of the job, but is not aware of the quantity pj. We present the first online algorithm that is scalable ((1+e)-speed O(1 2)-competitive for any constant e > 0) for the total weighted flow-time objective. No non-trivial results were known for this setting, except for the most basic case of identical machines. Our result resolves a major open problem in online scheduling theory. Moreover, we also show that no job needs more than a logarithmic number of migrations. We further extend our result and give a scalable algorithm for the objective of minimizing total weighted flow-time plus energy cost for the case of unrelated machines. In this problem, each machine can be sped up by a factor of f-1i(P) when consuming power P, where fi is an arbitrary strictly convex power function. In particular, we get an O(γ2)-competitive algorithm when all power functions are of form sγ. These are the first non-trivial non-clairvoyant results in any setting with heterogeneous machines. The key algorithmic idea is to let jobs migrate selfishly until they converge to an equilibrium. Towards this end, we define a game where each job's utility which is closely tied to the instantaneous increase in the objective the job is responsible for, and each machine declares a policy that assigns priorities to jobs based on when they migrate to it, and the execution speeds. This has a spirit similar to coordination mechanisms that attempt to achieve near optimum welfare in the presence of selfish agents (jobs). To the best our knowledge, this is the first work that demonstrates the usefulness of ideas from coordination mechanisms and Nash equilibria for designing and analyzing online algorithms. In citepapa, Papadimitriou formalized the notion of routing stability in BGP as the following coalitional game theoretic problem: Given a network with a multicommodity flow satisfying node capacity and demand constraints, the payoff of a node is the total flow originated or terminated at it. A payoff allocation is in the core if and only if there is no subset of nodes that can increase their payoff by seceding from the network. We answer one of the open problems in citepapa by proving that for any network, the core is non-empty in both the transferable (where the nodes can compensate each other with side payments) and the non-transferable case. In the transferable case we show that such an allocation can be computed in polynomial time. We also generalize this result to the case where a strictly concave utility function is associated with each commodity. We consider the over-time version of the Max-Min Fair Allocation problem. Given a time horizon @math , with at each time t a set of demands and a set of available resources that may change over the time defining instance @math , we seek a sequence of solutions @math that (1) are near-optimal at each time t , and (2) as stable as possible (inducing small modification costs). We focus on the impact of the knowledge of the future on the quality and the stability of the returned solutions by distinguishing three settings: the off-line setting where the whole set of instances through the time horizon is known in advance, the on-line setting where no future instance is known, and the k -lookahead setting where at time t , the instances at times @math are known. We first consider the case without restrictions where the set of resources and the set of agents are the same for all instances and where every resource can be allocated to any agent. For the off-line setting, we show that the over-time version of the problem is much harder than the static one, since it becomes @math -hard even for families of instances for which the static problem is trivial. Then, we provide a @math -approximation algorithm for the off-line setting using as subroutine a ρ-approximation algorithm for the static version. We also give a @math -competitive algorithm for the online setting using also as subroutine a ρ-approximation algorithm for the static version. Furthermore, for the case with restrictions, we show that in the off-line setting it is possible to get a polynomial-time algorithm with the same approximation ratio as in the case without restrictions. For the online setting, we prove that it is not possible to find an online algorithm with bounded competitive ratio. For the 1-lookahead setting however, we give a @math -approximation algorithm using as subroutine a ρ-approximation algorithm for the static version.