Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP

Given a metric space on n points, an {\alpha}-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an {\alpha} factor of the optimal Steiner tree cost for X. An {\alpha}-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an {\alpha} factor of the optimal as before, and satisfies the additional property that for any set X' that differs in a single vertex from X, the tree distributions for X and X' are "close" to each other. Universal and differentially private algorithms for TSP are defined similarly. An {\alpha}-approximate universal algorithm for the Steiner tree problem or TSP is also an {\alpha}-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(log n)-approximate differentially private algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root.Comment: 14 page

Computing Equilibrium in Matching Markets

Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items

Optimization under uncertainty

Uncertainty exists everywhere; from the future price of a stock, to the routing time of a network packet and playing a slot machine. In presence of such uncertainty, one often has to take a decision without knowing the entire information. While one approach is to optimize for the worst possible scenario (i.e. a {\em robust} approach), this is undesirable as it leads to large scale inefficiencies. Instead, we focus on designing optimization techniques to handle such uncertainty by assuming the presence of statistical information about the input. In this regard, we study two important classes of allocation problems: (a) allocation in a general stochastic packing framework, and (b) allocation to selfish buyers in a Bayesian setting (i.e. Bayesian mechanism design). We first consider a generic stochastic packing framework where the item sizes or profits come from item-specific distributions; this framework encapsulates a wide variety of natural scenarios that include (a) deadline scheduling, (b) bandwidth allocation to bursty connections, and (c) asset allocation for a risk averse investor. We design new techniques to construct a PTAS for any stochastic packing problem in this framework, assuming a slight relaxation in the packing constraints. Our techniques can handle adaptivity, and extend to a wide variety of risk-averse objectives. Our techniques also yield a new approximation result when the packing constraints are strict. We next focus on Bayesian mechanism design. In a mechanism, buyers\u27 valuations are private and the seller has to optimize only based on his priors. This leads to a unique set of stochastic challenges; we focus on three specific aspects: As the outcome of a mechanism is random, there is risk associated with it; thus, the buyers\u27 as well as the seller\u27s risk-aversion play a strong role in the design of the mechanism. We design approximately optimal DSIC and BIC mechanisms for a risk-averse seller in presence of potentially risk-averse buyers. This uses our result on utility equivalence of correlated random variables for a concave objective function. In addition to being a private information, a buyer\u27s valuation may as well depend upon the entire allocation. In particular, we consider the positive network externality effect on a buyer\u27s valuation and design approximately optimal mechanisms for various single-parameter and multi-parameter settings. Finally, we study the problem of designing mechanisms with desirable payment properties, such as (a) quitting rights for buyers, (b) ex post individual rationality in presence of budget constraints, and (c) cost to borrow money beyond budget; we give a generic technique to design optimal mechanisms in presence of such desirable but complex payment constraints

Optimization under uncertainty

Uncertainty exists everywhere; from the future price of a stock, to the routing time of a network packet and playing a slot machine. In presence of such uncertainty, one often has to take a decision without knowing the entire information. While one approach is to optimize for the worst possible scenario (i.e. a {\em robust} approach), this is undesirable as it leads to large scale inefficiencies. Instead, we focus on designing optimization techniques to handle such uncertainty by assuming the presence of statistical information about the input. In this regard, we study two important classes of allocation problems: (a) allocation in a general stochastic packing framework, and (b) allocation to selfish buyers in a Bayesian setting (i.e. Bayesian mechanism design). We first consider a generic stochastic packing framework where the item sizes or profits come from item-specific distributions; this framework encapsulates a wide variety of natural scenarios that include (a) deadline scheduling, (b) bandwidth allocation to bursty connections, and (c) asset allocation for a risk averse investor. We design new techniques to construct a PTAS for any stochastic packing problem in this framework, assuming a slight relaxation in the packing constraints. Our techniques can handle adaptivity, and extend to a wide variety of risk-averse objectives. Our techniques also yield a new approximation result when the packing constraints are strict. We next focus on Bayesian mechanism design. In a mechanism, buyers\u27 valuations are private and the seller has to optimize only based on his priors. This leads to a unique set of stochastic challenges; we focus on three specific aspects: As the outcome of a mechanism is random, there is risk associated with it; thus, the buyers\u27 as well as the seller\u27s risk-aversion play a strong role in the design of the mechanism. We design approximately optimal DSIC and BIC mechanisms for a risk-averse seller in presence of potentially risk-averse buyers. This uses our result on utility equivalence of correlated random variables for a concave objective function. In addition to being a private information, a buyer\u27s valuation may as well depend upon the entire allocation. In particular, we consider the positive network externality effect on a buyer\u27s valuation and design approximately optimal mechanisms for various single-parameter and multi-parameter settings. Finally, we study the problem of designing mechanisms with desirable payment properties, such as (a) quitting rights for buyers, (b) ex post individual rationality in presence of budget constraints, and (c) cost to borrow money beyond budget; we give a generic technique to design optimal mechanisms in presence of such desirable but complex payment constraints

Mechanisms and Allocations with Positive Network Externalities

With the advent of social networks such as Facebook and LinkedIn, and online offers/deals web sites, network externalties raise the possibility of marketing and advertising to users based on influence they derive from their neighbors in such networks. Indeed, a user’s knowledge of which of his neighbors “liked ” the product, changes his valuation for the product. Much of the work on the mechanism design under network externalities has addressed the setting when there is only one product. We consider a more natural setting when there are multiple competing products, and each node in the network is a unit-demand agent. We first consider the problem of welfare maximization under various different types of externality functions. Specifically we get a O(lognlog(nm)) approximation for concave externality functions, 2 O(d)-approximation for convex externality functions that are bounded above by a polynomial of degree d, and we give a O(log 3 n)-approximation when the externality function is submodular. Our techniques involve formulating non-trivial linear relaxations in each case, and developing novel rounding schemes that yield bounds vastly superior to those obtainable by directly applying results from combinatorial welfare maximization. We then consider the problem of Nash equilibrium where each node in the network is a player whose strategy space corresponds to selecting an item. We develop tight characterization of the conditions under which a Nash equilibrium exists in this game. Lastly, we consider the question of pricing and revenue optimizatio

Improved Approximation Results for Stochastic Knapsack Problems

In the stochastic knapsack problem, we are given a set of items each associated with a probability distribution on sizes and a profit, and a knapsack of unit capacity. The size of an item is revealed as soon as it is inserted into the knapsack, and the goal is to design a policy that maximizes the expected profit of items that are successfully inserted into the knapsack. The stochastic knapsack problem is a natural generalization of the classical knapsack problem, and arises in many applications, including bandwidth allocation, budgeted learning, and scheduling. An adaptive policy for stochastic knapsack specifies the next item to be inserted based on observed sizes of the items inserted thus far. The adaptive policy can have an exponentially large explicit description and is known to b