In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of $n$ items, each of which must be allocated to one of $m$ agents. Each agent $\ell$ has a valuation function $v_\ell$, where $v_\ell(S)$ denotes the welfare obtained by this agent if she receives the set of items $S$. The functions $v_\ell$ are all submodular; as is standard, we assume that they are monotone and $v_\ell(\emptyset) = 0$. The goal is to partition the items into $m$ disjoint subsets $S_1, S_2, \ldots S_m$ in order to maximize the social welfare, defined as $\sum_{\ell = 1}^m v_\ell(S_\ell)$. 
In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. Obtaining a competitive ratio of $1/2 + \Omega(1)$ for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least $0.505$ for the Online Submodular Welfare Maximization problem in the random order model. For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of "second-order supermodular" functions, we provide a different analysis that gives a competitive ratio of $0.51$.

Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order

The proliferation of mobile devices that support the acceleration of data services (especially smartphones) has resulted in a dramatic increase in mobile traffic. Mobile data also increased exponentially, already exceeding the throughput of the backhaul. To improve spectrum utilization and increase mobile network traffic, in combination with content caching, we study the cooperation between primary and secondary networks via content caching. We consider that the secondary base station assists the primary user by pre-caching some popular primary contents. Thus, the secondary base station can obtain more licensed bandwidth to serve its own user. We mainly focus on the time delay from the backhaul link to the secondary base station. First, in terms of the content caching and the transmission strategies, we provide a cooperation scheme to maximize the secondary user’s effective data transmission rates under the constraint of the primary users target rate. Then, we investigate the impact of the caching allocation and prove that the formulated problem is a concave problem with regard to the caching capacity allocation for any given power allocation. Furthermore, we obtain the joint caching and power allocation by an effective bisection search algorithm. Finally, our results show that the content caching cooperation scheme can achieve significant performance gain for the primary and secondary systems over the traditional two-hop relay cooperation without caching.

Cache-Enabled in Cooperative Cognitive Radio Networks for Transmission Performance

Principal-agent relationships between the supervisory board and the management of bidding firms in auctions are widespread in high-stakes auctions. Often only the agent has information about the value of the objects being sold. The board wants to maximize the profit, but the management wants to win the package with the highest value. In environments where it is efficient for firms to coordinate, we show that the principals would coordinate, while the agents would not. We analyze environments with decreasing levels of information that the principal has about the valuations. Sometimes it can also be impossible to set budget constraints which align the agents’ strategies in equilibrium. The analysis helps explain price wars in high-stakes auctions.

A Principal-Agent Model of Bidding Firms in Multi-Unit Auctions.

Online data caching in edge computing

We consider a classical principal–agent model in the contract theory A principal designs the payment $$\mathbf {w}=\left\{ w_{0},w_{1},\ldots ,w_{n}\right\} $$
 to incentivize the agent to enter into the contract Given the payment $$\mathbf {w}$$
 , the agent will take hidden actions from her strategy set $$\mathbf {S_{t}^{n}}$$
 to finish it and from the perspective of the agent, she will select the best strategy to maximize her expected utility Due to the hidden strategy set, the principal obtains the expected revenue $$R(S_{t}^{n})$$
 from the agent Furthermore, the principal has a non-decreasing revenue function r(k), which is common information, where k is the number of successful tasks in the total n independent tasks The objective of the problem is to maximize the principal’s expected profit, ie, $$\max _{S,\mathbf {w}}\left\{ R(S_{t}^{n})-P(S_{t}^{n},\mathbf {w})\right\} $$
 , where $$P(S_{t}^{n},\mathbf {w})$$
 is the agent’s expected payment The difficulty of this problem is due to the asymmetric information If the principal knows all the information about the agent, then the optimal contract can be solved by linear programming Based on Dutting et al (in: Proceedings of the EC, pp 369–387, 2019), we consider the more general model When information is asymmetric, we further analyze that the approximation ratio of the linear contract can reach $$(1-\alpha _{N})/(1-\alpha _{N}^{N})$$
 , which improves the results of Dutting et al (in: Proceedings of the EC, pp 369–387, 2019), where $$\alpha _{N}\in [0,1)$$
 is a given constant and the coefficient of the linear contract

Principal–agent problem under the linear contract

We consider the contract design problem where a principal designs a contract to incentivize an agent to undertake n independent tasks. In the contract, the principal decides the payment $\textit{\textbf{w}}=\{w_0, w_1, ..., w_n\}$. The principal is associated with a non-decreasing revenue function f(k), where k is the number of successful tasks. The objective of the contract design problem is to maximize the principal’s expected profit, i.e., $\max _{\textit{\textbf{s}},\textit{\textbf{w}}}\{F(\textit{\textbf{s}})-W(\textit{\textbf{s}},\textit{\textbf{w}})\}$, where $\textit{\textbf{s}}$ is the agent’s strategy set, $F(\textit{\textbf{s}})$ is the expected revenue of the principal and $W(\textit{\textbf{s}},\textit{\textbf{w}})$ is the expected payment to the agent. Given the payment $\textit{\textbf{w}}$, the agent will choose her strategy from $\textit{\textbf{s}}$ to maximize her expected utility. For each task, the agent may work hard or shirk and her strategy is to decide the number of tasks to work hard on. If the principal knows the strategy set $\textit{\textbf{s}}$, the optimum contract can be found by linear programming. In [7], a more complicated model where the strategy set $\textit{\textbf{s}}$ is unknown to the principal is considered, they showed that the linear contract is robust and presented an 1/N-approximation algorithm given $N\le n$ dominating strategies. In this paper, we further analyze this model and prove the approximation factor can be upper bounded by $(1-\alpha _{N})/(1-\alpha _{N}^{N})$, where $\alpha _{N}\in [0,1)$ is a given constant.

Robustness and Approximation for the Linear Contract Design

In the edge-cloud environment, offloading technique decides the task to be executed either at the cloud or at the edge. Offloading can improve the quality of service and the efficiency of the system. In most previous works on the offloading problem, the communication costs between tasks on both cloud side or the edge side are often ignored. We consider a general offloading model where the communication costs between any two tasks is non-zero and asymmetric. Moreover, due to the resource limitation on the edge side, we assume that the number of tasks executed on the edge side is bounded by a fixed constant k. This generalized offloading problem is NP-hard in minimizing the total cost with cardinality constraint. Based on semidefinite program, we give an approximation algorithm with the performance guarantee of $2/\pi $.

Approximation Algorithm for the Offloading Problem in Edge Computing

Fair division captures many real-world scenarios and plays an important role in many research fields including computer science, economy, operations research, etc. For the problem of indivisible goods allocation, it is well-known that an allocation satisfying the maximum Nash Social Welfare (Max-NSW) is envy-free up to one good (EF1), which is an important fairness concept. However, EF1 is often considered to be somewhat weak since one may remove the most valuable good. In this paper, we investigate the relationship between the Max-NSW allocation and a stronger fairness concept, envy-free up to any good (EFX), which means the envyness disappears after the removal of the least valuable good. We focus on the budget-feasible variant in which each agent has a budget to cover the total cost of goods she gets. We show that a Max-NSW allocation guarantees $$\frac{1}{4}$$ -EFX when agents’ value are in the Binary $$\{0,1\}$$ . Moreover, we provide an algorithm to find a budget-feasible EFX allocation for the Binary variant. 

Guichen Gao

Papers

Online data caching in edge computing

Robustness and Approximation for the Linear Contract Design

Approximation Algorithm for the Offloading Problem in Edge Computing

Principal–agent problem under the linear contract

EFX Under Budget Constraint