Azarakhsh Malekian

To fill or not to fill: The gas station problem

In this article we study several routing problems that generalize shortest paths and the traveling salesman problem. We consider a more general model that incorporates the actual cost in terms of gas prices. We have a vehicle with a given tank capacity. We assume that at each vertex gas may be purchased at a certain price. The objective is to find the cheapest route to go from s to t, or the cheapest tour visiting a given set of locations. We show that the problem of finding a cheapest plan to go from s to t can be solved in polynomial time. For most other versions, however, the problem is NP-complete and we develop polynomial-time approximation algorithms for these versions.

Network Security and Contagion

This paper develops a theoretical model of investments in security in a network of interconnected agents. The network connections introduce the possibility of cascading failures depending on exogenous or endogenous attacks and the profile of security investments by the agents. The general presumption in the literature, based on intuitive arguments or analysis of symmetric networks, is that because security investments create positive externalities on other agents, there will be underinvestment in security. We show that this reasoning is incomplete because of a first-order economic force: security investments are also strategic substitutes. In a general (non-symmetric) network, this implies that underinvestment by some agents will encourage overinvestment by others. We demonstrate by means of examples that not only there will be overinvestment by some agents but also aggregate probabilities of infection can be lower in equilibrium than in the social optimum. We then provide sufficient conditions for underinvestment. This requires both sufficiently convex cost functions (just convexity is not enough) and networks that are either symmetric or locally tree-like (i.e., either trees or in the case of stochastic networks, without local cycles with high probability). We also characterize the impact of network structure on equilibrium and optimal investments. Finally, we show that when the attack location is endogenized (by assuming that the attacker chooses a probability distribution over the location of the attack in order to maximize damage), there is another reason for overinvestment: greater investment by an agent shifts the attack to other parts of the network.

On random sampling auctions for digital goods

In the context of auctions for digital goods, an interesting Random Sampling Optimal Price auction (RSOP) has been proposed by Goldberg, Hartline and Wright; this leads to a truthful mechanism. Since random sampling is a popular approach for auctions that aims to maximize the sellers revenue, this method has been analyzed further by Feige, Flaxman, Hartline and Kleinberg, who have shown that it is 15-competitive in the worst case -- which is substantially better than the previously proved bounds but still far from the conjectured competitive ratio of 4. In this paper, we prove that RSOP is indeed 4-competitive for a large class of instances in which the number λ of bidders receiving the item at the optimal uniform price, is at least 6. We also show that it is 4.68 competitive for the small class of remaining instances thus leaving a negligible gap between the lower and upper bound. Furthermore, we develop a robust version of RSOP -- one in which the sellers revenue is, with high probability, not much below its mean -- when the above parameter λ grows large. We employ a mix of probabilistic techniques and dynamic programming to compute these bounds.

Optimizing query rewrites for keyword-based advertising

We consider the problem of query rewrites in the context of pay-per-click search advertising. Given a three-layer graph consisting of queries, query rewrites, and the corresponding ads that can be served for the rewrites, we formulate a family of graph covering problems whose goals are to suggest a subset of ads with the maximum benefit by suggesting rewrites for a given query. We obtain constant-factor approximation algorithms for these covering problems, under two versions of constraints and a realistic notion of ad benefit. We perform experiments on real data and show that our algorithms are capable of outperforming a competitive baseline algorithm in terms of the benefit of the rewrites.

To fill or not to fill: the gas station problem

In this paper we study several routing problems that generalize shortest paths and the Traveling Salesman Problem. We consider a more general model that incorporates the actual cost in terms of gas prices. We have a vehicle with a given tank capacity. We assume that at each vertex gas may be purchased at a certain price. The objective is to find the cheapest route to go from s to t, or the cheapest tour visiting a given set of locations. Surprisingly, the problem of find the cheapest way to go from s to t can be solved in polynomial time and is not NP-complete. For most other versions however, the problem is NP-complete and we develop polynomial time approximation algorithms for these versions.

Online allocation of display advertisements subject to advanced sales contracts

In this paper we propose a utility model that accounts for both sales and branding advertisers. We first study the computational complexity of optimization problems related to both online and offline allocation of display advertisements. Next, we focus on a particular instance of the online allocation problem, and design a simple online algorithm with provable approximation guarantees. Our algorithm is near optimal as is shown by a matching lower bound. Finally, we report on experiments to establish actual case behavior on some real datasets, with encouraging results.

Network security and contagion

We develop a theoretical model of security investments in a network of interconnected agents. Network connections introduce the possibility of cascading failures due to an exogenous or endogenous attack depending on the profile of security investments by the agents. We provide a tractable decomposition of individual payoffs into an own effect and an externality, which also enables us to characterize individual investment incentives recursively (by considering the network with one agent removed at a time). Using this decomposition, we provide characterization of equilibrium and socially optimal investment levels as a function of the structure of the network, highlighting the role of a new set of network centrality measures in shaping the levels of equilibrium and optimal investments. When the attack location is endogenized (by assuming that the attacker chooses a probability distribution over the location of the attack in order to maximize damage), similar forces still operate, but now because greater investment by an agent shifts the attack to other parts of the network, the equilibrium may involve too much investment relative to the social optimum.

A novel approach to propagating distrust

Trust propagation is a fundamental topic of study in the theory and practice of ranking and recommendation systems on networks. The Page Rank [9] algorithm ranks web pages by propagating trust throughout a network, and similar algorithms have been designed for recommendation systems. How might one analogously propagate distrust as well? This is a question of practical importance and mathematical intrigue (see, e.g., [2]). However, it has proven challenging to model distrust propagation in a manner which is both logically consistent and psychologically plausible. We propose a novel and simple extension of the Page Rank equations, and argue that it naturally captures most types of distrust that are expressed in such networks. We give an efficient algorithm for implementing the system and prove desirable properties of the system.

Competitive equilibrium in two sided matching markets with general utility functions

Two sided matching markets are among the most studied models in market design. There is a vast literature on the structure of competitive equilibria in these markets, yet most of it is focused on quasilinear settings. General (non-quasilinear) utilities can, for instance, model smooth budget constraints as a special case. Due to the difficulty of dealing with arbitrary non-quasilinear utilities, most of the existing work on non-quasilinear utilities is limited to the special case of hard budget constraints in which the utility of each agent is quasilinear as long as her payment is within her budget limit and is negative infinity otherwise. Most of the work on competitive equilibria with hard budget constraints rely on some form of ascending auction. For general non-quasilinear utilities, such ascending auctions may not even converge in finite time. As such, almost all of the existing work on general non-quasilinear utilities have resorted to non-constructive proofs based on fixed point theorems or discretization. We present the first direct characterization of competitive equilibria in such markets. Our approach is constructive and solely based on induction. Our characterization reveals striking similarities between the payments at the lowest competitive equilibrium for general utilities and VCG payments for quasilinear utilities. We also show that lowest competitive equilibrium is group strategyproof for the agents on one side of the market (e.g., for buyers).

Improved algorithms for data migration

Our work is motivated by the need to manage data on a collection of storage devices to handle dynamically changing demand. As demand for data changes, the system needs to automatically respond to changes in demand for different data items. The problem of computing a migration plan among the storage devices is called the data migration problem. This problem was shown to be NP-hard, and an approximation algorithm achieving an approximation factor of 9.5 was presented for the half-duplex communication model in [Khuller, Kim and Wan: Algorithms for Data Migration with Cloning, SIAM J. on Computing, Vol. 33(2):448–461 (2004)]. In this paper we develop an improved approximation algorithm that gives a bound of 6.5+o(1) using various new ideas. In addition, we develop better algorithms using external disks and get an approximation factor of 4.5. We also consider the full duplex communication model and develop an improved bound of 4 +o(1) for this model, with no external disks.