Online Assignment of Heterogeneous Tasks in Crowdsourcing Markets

# Online Assignment of Heterogeneous Tasks in Crowdsourcing Markets

Sepehr Assadi, Justin Hsu, Shahin Jabbari
Department of Computer and Information Science, University of Pennsylvania
Supported in part by NSF grants CCF-1116961 and IIS-1447470.Supported in part by a grant from the Simons Foundation (#360368 to Justin Hsu) and NSF grant CNS-1065060.
###### Abstract

We investigate the problem of heterogeneous task assignment in crowdsourcing markets from the point of view of the requester, who has a collection of tasks. Workers arrive online one by one, and each declare a set of feasible tasks they can solve, and desired payment for each feasible task. The requester must decide on the fly which task (if any) to assign to the worker, while assigning workers only to feasible tasks. The goal is to maximize the number of assigned tasks with a fixed overall budget.

We provide an online algorithm for this problem and prove an upper bound on the competitive ratio of this algorithm against an arbitrary (possibly worst-case) sequence of workers who want small payments relative to the requester’s total budget. We further show an almost matching lower bound on the competitive ratio of any algorithm in this setting. Finally, we propose a different algorithm that achieves an improved competitive ratio in the random permutation model, where the order of arrival of the workers is chosen uniformly at random. Apart from these strong theoretical guarantees, we carry out experiments on simulated data which demonstrates the practical applicability of our algorithms.

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## 1 Introduction

Crowdsourcing markets have seen a remarkable rise in recent years, as more and more markets use the internet to connect people with tasks to solve, to people willing to solve tasks in exchange for payment. While these tasks were originally simple tasks that could be accomplished while sitting at a computer—say, labeling images or cleaning data—recent systems handle complex tasks in the real world. Services like TaskRabbit let users hire workers to run errands, like picking up a package; ride-sharing applications like Lyft or Uber provide drivers on-demand to transport customers; other services offer meal or grocery delivery, house cleaning, and even on-demand massages111www.zeel.com to customers.

Accordingly, an important challenge for task requesters in crowdsourcing (and more general) platforms is handling more and more heterogenous workers and tasks. A worker may not be able to solve all the tasks—say, they may only be able to translate certain languages, or they may only be able to handle time-sensitive tasks on a deadline far into the future. Among the solvable tasks, a worker may want different payment for different tasks; a short task can require a small payment, while a more complex task may warrant a higher payment.

At the same time, requesters must cope with a highly dynamic flow of workers. Since it is common for workers to work for several different platforms simultaneously, the pool of available workers is constantly changing. The requester may not have the luxury of seeing all the workers, and then selecting the right workers. Instead, requesters may need to select workers in an online fashion, where workers arrive one by one and must be hired (or not) as soon as they arrive.

We take aim at both of these challenges in crowdsourcing by considering the following task assignment problem: if workers (i) have an arbitrary set of feasible tasks, (ii) demand heterogeneous payments by bidding for feasible tasks, and (iii) arrive online, how can a requester assign workers to tasks in order to maximize the number of completed tasks given a fixed overall budget?

Singer and Mittal [8] were the first to study a simpler version of this problem. In their setting, each worker has a single bid for all the tasks; all the tasks are treated homogeneously, and workers are assumed to be able to solve all the tasks (if they receive a payment which is at least equal to their bids). We generalize their setting in two significant ways: First, we allow workers to specify only a subset of feasible tasks that they can handle. Second, we work in a setting with heterogeneous tasks: workers can have different preferences over tasks. As we will show, this heterogeneity significantly complicates the task assignment problem; algorithms for the heterogeneous case may look nothing like their counterparts in the homogeneous case.

As a warm up, we start by providing two algorithms for the offline problem – where the requester knows the sequence of workers and their bids up front: first, the optimal algorithm via min-cost flows, and second, a fixed-threshold constant factor approximation algorithm inspired by the algorithm of Singer and Mittal [8]. Then, we move to the online setting, where we draw on techniques from the online knapsack literature.

As is typical for online algorithms, we measure the performance of our algorithm via the competitive ratio, i.e., the ratio of the number of tasks assigned by the best offline algorithm to the number of tasks assigned by the online algorithm on the same problem. When the bids are bounded in , the bids of the workers are small compared to the budget of the requester, and the order is worst-case, we give an algorithm that achieves an competitive ratio when .222In particular, when , which is arguably the case in most practical applications, our algorithm achieves an optimal competitive ratio of . The central idea is to use a moving threshold, and take all workers with bid below the threshold. As the budget is depleted, the threshold steadily decreases. Our algorithm is inspired by Zhou, Chakrabarty, and Lukose [13], who give algorithms for the online knapsack problem. We also show a lower bound: for any (possibly randomized) online algorithm, there exists an input that forces a competitive ratio of at least .

Then, we consider the random permutation setting where the bids are adversarial, but the order of workers is chosen uniformly at random. Here, we measure the performance of an online algorithm by comparing the number of tasks assigned averaged over all permutations to the number of tasks assigned by the offline optimal.

## 2 Model

Let us begin by defining our modeling more formally. We will use to index tasks and to index workers throughout. We model the problem from the perspective of a requester who has a collection of tasks.

Each worker picks a subset of tasks , along with the numeric bid for each task . Workers arrive online: the sequence of the workers and their bids are initially unknown to the requester but once a worker arrives, her bids for all the tasks are revealed to the requester. We assume that each worker can be assigned to at most one task; we can relax this assumption by having multiple copies of each worker. We denote the total number of workers by . As customary in the crowdsourcing setting we assume workers are abundant, hence is large.

We model the sequence of workers (their bids and their order of arrivals) in two different ways. First, we consider the case that we have no assumptions on the sequence of workers. This worst case scenario has been referred to as the adversarial setting in the literature. Second, we consider the random permutation model in which we still make no assumptions on the bids of the workers but we assume the order of their arrival is randomly permuted before being presented to the requester. We define these two settings in more detail in the following sections.

The requester has a budget of and has to decide on the fly which task to assign to the worker who arrives; of course, a worker can only be assigned to a task he is willing to do. If task is assigned to worker , then the requester pays worker at least the price . The goal of the requester is to maximize the number of tasks he assigns to the workers while spending at most a budget of and satisfying all the constraints of the workers (on the tasks they are willing to solve).

Similar to the online algorithms literature, we compare the performance of our online algorithms to the performance of the best offline algorithm (denoted by ), which can see all the bids and the order of arrival of workers before allocating tasks to workers. As is standard, we measure the performance via the competitive ratio: the ratio of the number of tasks assigned in the offline optimal solution to the number of tasks assigned by the online algorithm (so competitive ratio is at least , and smaller competitive ratio is more desirable).

### 2.1 Related Work

Pricing and task assignments have been previously studied in the context of mechanism design for crowdsourcing markets. Singer and Mittal [8, 9] provide mechanisms for task assignment when tasks are homogeneous and the workers are arriving from a random permutation model. Our work in the random permutation section generalizes their work to heterogeneous tasks. Ho and Vaughan [5] consider task assignment when the tasks are heterogeneous. However, they assume the existence of limited task types and the focus of their work is learning qualities of the workers from stochastic observations. Singla and Krause [11] design price mechanism for crowdsourcing markets using tools from online learning. Goel, Nikzad, and Singla [4] consider the heterogeneous task assignment in the offline setting and provide near optimal approximation algorithms. However, they focus on the mechanism design aspect of the problem—how to pay workers so that they report their bids truthfully. This line of budget feasible mechanism design is inspired by the work of Singer [10] and has been followed up in Singer and Mittal [8, 9].333While we ignore the mechanism design aspect of the problem in our formulation, as we describe at the end of the Random Permutation section, all our online algorithms (either with a slight modification or as they are) satisfy truthfulness and incentive compatibility. Also these pricing mechanisms have close connections to the stochastic online adwords problem [3] and the online primal-dual literature (see Buchbinder, Jain, and Naor [1] and references within).

Our work is inspired by variants of the online knapsack problem. In the adversarial setting with homogeneous tasks, our problem is an instance of the online knapsack problem studied by Zhou, Chakrabarty, and Lukose [13] (see references within for more information). However, it is not clear how to formulate our problem as a knapsack problem when the tasks are heterogeneous.

Variants of generalized online matching and the adwords problem are also related to our problem ( see Mehta et al. [6] and Mehta [7] for an excellent survey). The adwords problem can be described as follows. There are some bidders and each bidder has a fixed budget. Queries arrive one at the time, bidders bid for the queries and the algorithm has to decide what bidder to assign to the query. If the algorithm assigns a bidder to a query, the bidder pays the amount that is equal to her bid. The goal is to maximize the revenue. While one might attempt to formalize our task assignment problem as an instance of the online adwords problem, it is not hard to see that our constraint on the total budget of the requester cannot be written as an adwords type budget constraint for the bidders.

When comparing an online algorithm with an offline algorithm, we first need to precisely specify the inputs on which we make the comparison. Let us first consider the worst case for the requester: adversarial inputs. In this setting, there is a fixed input and order, and we compare an online algorithm in this single input to an offline algorithm on the same input via the competitive ratio. We are interested in bounding this ratio in the worst case, i.e., the max over all inputs.

Of course, if we really do not make any assumption on the input, we cannot hope to compete with an offline algorithm as the competitive ratio may be arbitrarily high (see, for instance,  Zhou, Chakrabarty, and Lukose [13, Section 1.2]).

Consequently, we restrict the adversary’s power by making one main assumption on the relationship between worker bids and the budget: the ratio of the largest bid to the smallest bid should be small compared to the budget. More precisely, we can scale worker bids so that , and we write . We will frequently consider to be small—this is appropriate for the crowdsourcing problems we have in mind, where (i) the scale of the budget is much larger than the scale of payments to workers and (ii) the tasks are not extremely difficult, so no worker charges an exorbitantly high price. We refer to this assumption as large market assumption because when the bids are small compared to the budget, the requester can hire a large number of workers before exhausting his budget. This is in line with the common assumption in crowdsourcing that is large.

Before diving into the technical details, let us consider which range of parameters and competitive ratios is interesting. Since the bids are restricted in , achieving a competitive ratio of is trivial. So, we will be mainly interested in the following question: Can we design an algorithm that has a competitive ratio much smaller than for any sequence of workers?

The rest of this section is organized as follows. We first investigate the offline problem, describing how the problem can be solved optimally. Then we propose a simpler algorithm for the offline setting; the performance is a constant factor off from optimal, but the simpler algorithm will be useful in Section 4, when we will work in the random permutation setting. We then move to the online setting, giving an algorithm with competitive ratio of for any sequence of workers. Finally, we give a lower bound showing that any algorithm has competitive ratio of at least in the worst case.

### 3.1 The Offline Problem

Let us start by investigating the offline problem where the sequence of workers and their bids are known up front. We first show how the offline optimal assignment can be computed. Then we propose a simpler algorithm that approximates the offline optimal assignment by a factor of 4. We show later in the paper that how this second algorithm, while suboptimal, can be converted to an online algorithm.

The offline problem is a well-known problem in the crowdsourcing literature [10] and can be solved by a reduction to the min-cost flow problem defined as follows. Consider a graph with costs and capacities on the edges and two nodes marked as source and target. Given this graph, for a demand value of flow, the goal of the min-cost flow problem is to route this amount of flow from the source to the target while minimizing the total flow cost over the edges, where the flow cost over an edge is equal to the amount of the flow passing the edge multiply by the cost of the edge.

We now briefly describe an algorithm for solving the offline problem optimally which uses min-cost flow algorithms as a subroutine. Note that this algorithm is generally known in the literature and we provide it here for the sake of completeness. In our problem, we want to maximize the number of assigned tasks given a fixed budget. In order to do so, we first construct the following instance of the min-cost flow problem. We start with a bipartite graph with workers on one side and tasks on the other. We then draw an edge from a worker to a task if the worker is willing to solve that task and let the cost of this edge to be equal to the bid of the worker for the task. Additionally, we attach a source node pointing to all the workers and connect all the tasks to an added target node. Finally, we set the capacity of all edges to be .

First notice that any feasible flow in this graph corresponds to an assignment of tasks and workers. Moreover, for any integer , the minimum cost of a flow with value corresponds to the minimum budget required for assigning worker-task pairs.444Since the amount of flow routed from the source to the target and the capacity of the edges are integral, the min-cost flow problem has an integral solution. Consequently, we can search over all possible , solve the min-cost flow problem on the described graph and demand flow of , and return the maximum value of where the minimum cost flow is at most the available budget.

While the above approach achieves the optimal solution, we will see in the random permutation section that using a fixed threshold to decide which workers to hire can be very useful. Hence, we next provide a simpler algorithm with this feature; while this simpler algorithm does not guarantee an optimal solution anymore, we show in Theorem 1 that its solution is within a factor of at most from .

The algorithm, which we refer to as Offline Approximation Algorithm () (see Algorithm 2) is mainly using the subroutine Fixed Threshold Policy (). In this subroutine, given a threshold value , the algorithm goes over workers one by one and assigns a task to an unassigned worker if the bid of the worker for that task is not bigger than . In case there are more than one unassigned task that the worker bids or less for, the algorithm break ties arbitrarily. The subroutine is described in Algorithm 1.555While we use as an offline algorithm in this section, the sequential nature of the algorithm allows to be used when workers arrive online (one by one). We exploit this feature of in Section 4.

then searches over all possible values of to find the proper threshold. Although the search space is continuous, it is sufficient for the algorithm to restrict its search to the bids of the workers. Hence, the running time of is polynomial in and . We show that the number of assignments of the is least a quarter of the for any sequence of workers.

###### Theorem 1.

Let and denote the number of tasks assigned by the and the offline optimal algorithm for a sequence of workers , respectively. Then for any , .

Before proving Theorem 1, we state the following useful lemma.

###### Lemma 1 (Singer and Mittal [9, Lemma 3.1]).

Let be a sorted sequence of positive numbers in an increasing order such that . Then .

The proof of Lemma 1 is the result of the following two simple observations: (i) the sum of the second half of the numbers is at most and (ii) each of the numbers in the second half is at least as big as the median of the sequence.

###### Proof of Theorem 1.

Consider all the (worker, task) pairs in the offline optimal where a pair simply denotes that the task is assigned to the worker. Sort these pairs in an increasing order of the bids. Let denote the median of the bids. By Lemma 1, we can assign half of these (worker, task) pairs, pay all the workers price (clearly an upper bound on the bid of every considered worker) and be sure not to exceed the budget. However, there are two problems: (i) we do not know which (worker, task) pairs are in the optimal solution and hence, (ii) we do not know the value .

To deal with problem (i), suppose we somehow knew . We can then assign a worker to any task where the worker has a bid of at most and continue until (a) we exhaust the budget or (b) there are no more workers or tasks left.

In case (a), the algorithm made at least assignments and by Lemma 1, we know this number is at least . For case (b), consider the bipartite graph between the tasks and the workers described earlier for the min-cost flow problem construction and remove all the edges with cost bigger than . Since is the median of the bids in the optimal solution, we know that in this graph, there exists a matching of size at least between the workers and the tasks. On the other hand, since we know terminated in this case because there are no tasks or workers left, we know that has arrived at a maximal matching. Finally, since the size of any maximal matching is at least half of the size of the maximum matching, we know the number of assignments of is at least a half of the number of assignments of . So if we knew .

To deal with problem (ii) (not knowing ), we run the for all the values that can take and return the maximum number of assignments as our solution.666Note that this might result in using a threshold which is different than but since picks a that maximizes the number of assignments we know the number of assignments made by using is at least as large as the number of assignments made by using .

#### A brief detour: the homogeneous case.

As we discussed in the introduction, we make a strong distinction between the setting with heterogeneous tasks and the setting with homogeneous tasks. To highlight this difference, we show that in the homogeneous case, the following simple algorithm computes the offline optimal: sort all the workers by their bids and (if possible) assigns a task to the sorted workers until the budget is exhausted or there are no more tasks or workers left.

It is easy to verify that this greedy construction indeed computes the best offline assignments when the workers are homogeneous. However, this method will not result in the optimal offline assignment when the tasks are heterogeneous. For example consider the following toy problem with two workers and two tasks where each workers is willing to do both tasks. Let the bids of worker and to be and for the two tasks, respectively. With a budget of , the optimal offline assignment is to assign task to worker and task to worker . However, the homogeneous greedy algorithm will assign task to worker and will not have enough budget left to assign a task to worker .

### 3.2 The Online Problem

Let’s now move to the online setting, where the workers arrive online and our algorithm must decide which (if any) task to assign to a worker before seeing the remaining workers. We propose an online algorithm and prove an upper bound on the competitive ratio of our algorithm which holds against any sequence of workers. Our upper bound crucially depends on the large market assumption where we assume that the bids of the workers are small compared to the budget. So throughout this section, we assume the bids are bounded in where for some small value of .

Our Online Heterogeneous Algorithm () is inspired by an algorithm from the online knapsack literature proposed by Zhou, Chakrabarty, and Lukose [13], with an analysis modified for our setting. The idea is to use a potential function based on the fraction of budget spent so far as an input. The function acts as a price threshold: the algorithm assigns a task to a worker only if the bid of the worker for any of the remaining unassigned tasks is below the value of the potential function—intuitively, as the budget shrinks, the algorithm becomes pickier about which workers to hire. Once a worker is selected, the algorithm greedily assigns a task arbitrarily from the remaining set of tasks. See Algorithm 3 for a pseudo-code.777In Algorithm 3, is the base of natural logarithm.

###### Theorem 2.

Let and denote the number of tasks assigned by the and the offline optimal algorithm for a sequence of workers , respectively. Then for any with bids in such that ,

###### Proof.

Fix a sequence of workers . Let be the set of (worker, task) pairs assigned by the where and index workers and tasks, respectively. Also let be the offline optimal. We want to bound .

Let and denote the fraction of the budget used by the when worker arrives and upon termination, respectively. We will analyze the (worker, task) pairs in three stages. First, consider the common (worker, task) pairs which we denote by . Let denote the total bid of such workers. Since each worker in the common part who is assigned to task is picked by the , it must be that . Therefore,

 |S∩S∗|≥∑(i,j)∈S∩S∗bijϕ(xi). (1)

Second, consider . This can happen either because (i) the worker is assigned to a task different than by , or (ii) the worker is not assigned to any task by . We know the number of pairs that satisfy the condition (i) is at most because all such workers are assigned to a task by both and the offline optimal.

For the pairs that satisfy condition (ii), either (a) or (b) . It is easy to see that in case (a), assigned task to some other worker who arrived before . Again we know this can happen at most times. For case (b), since is non-increasing then for all such pairs . Since the offline optimal spent a budget of for hiring workers in , then it has a budget of at most to hire workers in case (b). Since all the pairs in case (b) have , the offline optimal can hire at most workers. Adding up cases (i) and (ii), we can bound

 |S∗∖S|≤2|S|+(B−W)ϕ(X). (2)

Finally, consider . Since for all such pairs, we know

 |S∖S∗|≥∑(i,j)∈S∖S∗bijϕ(xi). (3)

Putting all the pieces together, Equations 3, 2 and 1 yield

 OPT(σ)−2⋅ALGOHA(σ)ALGOHA(σ)=|S∩S∗|+|S∗∖S|−2|S||S∩S∗|+|S∖S∗|≤∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∗∖S|−2|S|∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗|. (4)

To see why the inequality holds, first note that we know by Equation 1. Now if then we are done. Otherwise,

 |S∗| ≥3|S|⟹|S∩S∗|+|S∗∖S|≥|S∖S∗|+|S∩S∗|+2|S|⟹|S∗∖S|−2|S|≥|S∖S∗|.

Hence, the inequality holds because if and then .

We bound the right hand side of Equation 4 as follows.

 ∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∗∖S|−2|S|∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| ≤∑(i,j)∈S∩S∗bij/ϕ(xi)+2|S|+(B−W)/ϕ(X)−2|S|∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| =∑(i,j)∈S∩S∗bij/ϕ(xi)+(B−W)/ϕ(X)∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| ≤∑(i,j)∈S∩S∗bij/ϕ(X)+(B−W)/ϕ(X)∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| =W/ϕ(X)+(B−W)/ϕ(X)∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| =B/ϕ(X)∑(i,j)∈S∩S∗bij/ϕ(xi)+|S∖S∗| ≤B/ϕ(X)∑(i,j)∈S∩S∗bij/ϕ(xi)+∑(i,j)∈S∖S∗bij/ϕ(xi) =B/ϕ(X)∑(i,j)∈Sbij/ϕ(xi) =1ϕ(X)∑(i,j)∈SΔ(xi)/ϕ(xi),

where . The first inequality is due to Equation 2, the second is due to monotonicity of and the third inequality is due to Equation 3. Since , assigns task to worker and increases the fraction of the budget consumed by , so .

We now estimate the sum with an integral. If ,

 ∑(i,j)∈SΔ(xi)1ϕ(xi)≥∫X−ϵ01ϕ(xi)dx.

Letting , we have . Similar to [13], we bound the integral as follows.

 ∫X−ϵ01ϕ(xi)dx =∫c01Rdx+∫X−ϵc1ϕ(xi)dx =cR+1Re⋅11+ln(R)((Re)X−ϵ−(Re)c) =(cR−1Re⋅11+ln(R)(Re)c)+1Re⋅1ln(R)+1(Re)X−ϵ =1Re⋅11+ln(R)(Re)X−ϵ =1ϕ(X)⋅(Re)−ϵ1+ln(R).

It is easy to show by algebraic manipulation that the first term in the 3rd line is equal to . Therefore,

 OPT(σ)−2⋅ALGOHA(σ)ALGOHA(σ)≤1ϕ(X)∑(i,j)∈SΔ(xi)/ϕ(xi)≤(Re)ϵ(ln(R)+1).

Thus, , as desired. ∎

### 3.3 A Lower Bound on the Competitive Ratio

To wrap up this section, we show that if the large market assumption is the only assumption on the sequence of workers, then no algorithm (even randomized) can achieve a constant competitive ratio. We prove the result for the special case that all the tasks are homogeneous.

The proof is very similar to the lower bound for the competitive ratio in a variant of the online knapsack problem studied by Zhou, Chakrabarty, and Lukose [13]. In this variant, items arrive one by one online and we have a knapsack with some known capacity. Each item has a weight and a utility parameter and it is assumed that the utility to weight ratio for all the items are within a bounded range. Furthermore, it is assumed that the weight of each item is small compared to the capacity of the knapsack (similar to our large market assumption). The algorithm have to decide whether to pick an item or not upon arrival and the goal is to maximize the utility of the picked items while satisfying the knapsack capacity constraint. Since our setting when the tasks are homogeneous is a special case of the online knapsack variant, the lower bound for that problem might not necessarily provide us with a lower bound. However, we show that with a bit of care, we can achieve the same lower bound using a similar construction.

The main idea of the lower bound is to construct hard sequences of worker arrivals. A hard sequence can be described as follows. The sequence starts with workers with maximum bid, , and then the following workers progressively have smaller and smaller bids compared to the preceding workers. Then at some random point until the end of the sequence only workers with bid appear in the sequence. Intuitively, these sequences are hard because no algorithm can foresee whether the bids will decrease (so it should wait for cheaper workers) or increase (so it should spend its budget on the current workers).

\thmt@toks\thmt@toks

For any (possibly randomized) online algorithm, there exists a set of sequences of worker arrivals satisfying the large market assumption (all bids in and ) such that the competitive ratio of the algorithm on the sequence is at least .

###### Theorem 3.

Since the proof is modification of a proof by Zhou, Chakrabarty, and Lukose [13], we defer the details to Appendix A.

Finally, it is worth mentioning that if we consider the limit where worker’s bids are very small compared to the budget (), Theorem 2 shows that Algorithm 3 has an optimal competitive ratio approaching , the best possible as proven by Section 3.3.

## 4 Random Permutation Setting

Now that we have considered the worst-case scenario of inputs, let us consider more “well-distributed” inputs. We consider the random permutation model [3]. In the random permutation model there is no assumption about the bids of the workers (they can still be chosen by an adversary) but we measure the competitive ratio a bit differently: we take all possible permutations of these workers, and take the average competitive ratio over all permutations. Intuitively, this assumption makes the task assignment easier because the premium workers get distributed evenly in the sequence.

As pointed out by Devanur and Hayes [3], the random permutation model can be considered as drawing bids from an unknown distribution without replacement. Hence, the random permutation model is very similar to the model that assumes the bids of the workers are drawn from an unknown distribution.

Throughout this section, we assume that the number of available workers, , is large and known to us. Also, we restrict our attention to inputs where the offline optimal algorithm assigns at least a constant fraction of workers, i.e., for all sequences .

For the case that the workers are homogeneous, Singer and Mittal [9] provide an algorithm with competitive ratio of . In this section, we extend their result to the case the tasks are heterogeneous. We also improve on their competitive ratio, though the setting is a bit different: as they were concerned with mechanism design properties, it was important for Singer and Mittal [9] to carefully tune their payments to (i) incentivize workers to bid honestly and (ii) compensate all workers, even workers at the very beginning. Our algorithm only satisfies the first property. We discuss more about this at the end of this section.

If we knew the sequence of the workers and their bids, we could run to compute a threshold price and number of assignments where we know is at least a quarter of the optimal number of assignments by Theorem 1. However, since we are in the random permutation model, we can estimate this threshold with high probability by observing a subset of workers. This idea is summarized as the Random Permutation Algorithm () in Algorithm 4.

The algorithm observes the first half of workers, assign no tasks but computes a threshold by running with a half of the budget on the sequence of workers on the first half. Given an input parameter , the algorithm then uses the threshold and runs on the second half of the workers with the remaining half of the budget.888While we can use the whole budget on the second half of the workers (as we did in our experiments) because the algorithm assigns no tasks to the first half of the workers, this will only decrease the competitive ratio by a factor of half. While using the threshold instead of might decrease the performance of our algorithm, we show that is higher than with high probability and use this observation to make the analysis of the competitive ratio of easier.

###### Theorem 4.

Let , and suppose the number of workers is at least

 n=Ω(1αlog(1δ)),

and for every input. For any sequence of workers , let and denote the number of tasks assigned by the and the offline optimal algorithm for a sequence of workers , respectively. Then,

 OPT(σ)≤8(1+α)2/(1−α)⋅ALGRPA(σ),

with probability at least for all .

###### Proof.

If we knew the sequence of the workers and their bids, we could run to compute a price and number of assignments . We know by Theorem 1. Let us refer to these workers hired by as good workers.

We first claim that if we use a price of instead of , the number of assignments will decrease by a factor of . This is because we can still assign a task to workers who assigned a task to with a threshold of , but now we also have access to workers with bids in which may cause us to exhaust the budget faster.

Second, we can estimate from the first half of the workers. Call this estimate . We claim that at least of the good workers are in the second half of the sequence with high probability, and

 p≤(1+α)^p≤(1+α)p/(1−α). (5)

This together with the first claim and the 4-approximation of will give us the competitive ratio claimed in the statement of the theorem.

To complete the proof, we first show that there are enough good workers in the second half of the sequence. Since we are in the random permutation model, then with high probability, we know the number of good workers in the first half of the workers is in . Chvatal [2] show that this probability is bounded by

 1−2Q∑i=Q(1+α)/2(iQ/2)(n/2−in−Q)(nn/2) ={1−O(e−αQ2/n) if Q≥n2,1−O(e−α(n−Q)2Q/n2) if Q

To prove that Equation 5 holds, we consider two cases.

1. The number of good workers in the first half is . In this case because we can obviously use the budget to assign tasks to all the good workers. And if there is leftover budget, we might be able to assign tasks to workers with bid greater than . So,

 ^pQ2(1−α) ≤B2=pQ2⟹^p≤1(1−α)p,

since by the last line in Algorithm 2. Combined with , the condition in Equation 5 holds in this case.

2. The number of good workers in the first half is .

In this case . Suppose by contradiction that . This means the workers hired by the with bids of at most is at least . We know because all good workers have bids of most . So,

 B2^p=pQ2^p ≥Q2⟹p≥^p,

 ^pQ2(1+α) ≥B2=pQ2⟹^p≥11+αp.

Combined with , the condition in Equation 5 also holds in this case, concluding the proof.

#### A Note On Incentive Compatibility.

A problem closely related to ours is to design incentive compatible mechanisms in a setting where the workers arrive online but their bids are private and unknown to the requester [9]. While our focus in this paper was mainly on designing competitive online algorithms, we point out that both of our online algorithms, (Algorithm 3) and (Algorithm 4), can lead to incentive compatible mechanisms in a straightforward way. We briefly describe this relation here and omit the formal proofs.

For any worker , let denote the largest price that satisfies the potential upon the arrival of worker . Suppose in , instead of checking the bids of each worker against the potential function , the requester simply offers to worker (without knowing the worker’s true bids). One can simply check that by allocating this budget to the -th worker instead of her bid (which is never bigger than by definition) in , the proof of Theorem 2 still holds and consequently the modified still achieves the promised competitive ration. However, in this variant of , the dominant strategy for a worker is to reveal her true bid since her utility (i.e, the payment she receives) is independent of her true bid ( if and otherwise). Similarly, we can argue this property for the (with the modification that we pay the workers equal to the fixed threshold price not their true bid) also. Although we are dealing with heterogeneous tasks, the analysis of incentive compatibility is similar to Singer and Mittal [9] which only consider homogenous tasks.

## 5 Experiments

In this section we describe some experiments to examine the performance of (Algorithm 3) and (Algorithm 4) on synthetic data.999We slightly modified the in our experiments by using all the budget (instead of half) on the second half of the workers. Note that this modification can only improve the performance in our implementation compared to our theoretical guarantee for Algorithm 4.

We conduct two sets of experiments. The first set focuses on the performance under homogeneous and adversarially chosen bids (similar to the hard distribution in Section 3.3). In the second set of experiments, we use a sequence of heterogeneous workers and bids produced from a family of distributions meant to model input in practical settings.

The input in this setting is created similar to the hard distribution in Section 3.3 with groups of workers that are willing to solve any of the tasks with the same bid, but the bids may vary across different groups of workers. Since these instances are hard inputs from the adversarial setting (as shown in Section 3.3), they are a good benchmark for comparing the algorithms in adversarial setting.

Formally, the input is created as follows. We vary in the range from to (on all powers of two) and set . The bids of the workers are , where in each experiment, is chosen uniformly at random from ; for each group with bid , there will be workers. We always pad the input with a set of workers with bid to ensure that the total number of workers in each instance is the same (i.e., we set . We set the number of tasks to be equal to the number of workers (so a worker will never run out of tasks to solve no matter what the previous workers have done). Finally, for each value of , we create the input times and compute the average competitive ratio of . Note that since in this distribution, there is essentially no suitable worker in the second half, performs very poorly and hence we do not include it in this part of the experiment. We then ran the same experiment (for both and ) on the same input but this time we permute the workers randomly; Figure 1 shows the result for this experiment.

The experiment over the truly adversarial input shows that outperforms the theoretical guarantee of , ( here) and grows roughly like , suggesting that the true performance of may be closer to our theoretical lower bound from Section 3.3.

The experiment over the randomly permuted input shows that is obtaining a constant competitive ratio even though the input does not satisfy all the properties required by Theorem 4 (e.g., assumption). Interestingly, even performs very well over the input (only slightly worse than ), and also much better than the theoretical guarantee proved in Theorem 2. Finally note that although the sequence of workers were adversarially chosen to begin with, a random permutation of this sequence makes the problem significantly easier; to the point that even which is designed to perform against worst case sequences will also achieve a constant competitive ratio.

#### Uniform Heterogeneous Bids.

We consider a similar distribution (with minor modification of parameters) as the one introduced by Goel, Nikzad, and Singla [4]. They use this setting to analyze the performance of their proposed algorithms for heterogeneous task assignment in the offline case. These inputs are aimed to capture more realistic scenarios compared to our experiments in the previous part.

The distribution is as follows. We vary in range from to and set . The workers are heterogeneous and are only willing to solve a subset of all available tasks. In particular, we create workers and tasks and create a random graph with probability of edge formation equal to between any of the (worker, task) pairs. Additionally, over each realized edge, we choose the bid of the worker for the given task uniformly at random from (independently of the bids chosen over the other edges for this worker). For each choice of , we repeat the experiment times and report the average competitive ratio of the algorithms over the runs. Figure 2 illustrates the results of this experiment.

As seen in Figure 2 both algorithms perform very well over these inputs, obtaining a competitive ratio that is nearly independent of . Interestingly, performs distinctly better than its theoretical guarantee, even outperforming . We suspect that this is because is more adaptive than , since it uses a varying price threshold while sticks to a fixed price (after discarding half the input). Hence exhausts the budget sooner than necessary, while additionally losing the contribution of half of the workers.

#### Acknowledgement

We would like to thank Steven Zhiwei Wu for helpful discussion in the earlier stages of this work. We are also grateful to the anonymous reviewers for their insightful comments.

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## Appendix A Proof of Lower Bound on Competitive Ratio

###### Proof.

We modify the proof of Theorem 2.2 by Zhou, Chakrabarty, and Lukose [13] to fit our problem formulation.

We use Yao’s minimax principle; by constructing a distribution over a set of instances and showing that no deterministic algorithm can achieve a expected competitive ratio which is better than on these instances, Yao’s principle implies that no randomized algorithm can beat this competitive ratio in the worst-case [12].

To construct the distribution, fix , let be the smallest integer such that , and define instances indexed by to as follows. contains identical workers all with bids equal to . For all , is followed by workers all with bids equal to . Since these instances have different length we pad all the instances with enough workers with bid so that all the instances have the same length.

We specify a distribution by values where denotes the probability of occurrence of instance . Let

 p0=p1=… =pk−1:=η(k+1)η+1 and pk:=1+η(k+1)η+1.

On this distribution of inputs, any deterministic algorithm is fully specified by the fraction of budget spent on hiring workers with bid ; call each fraction . Since the optimal assignment for instance is to only hire workers with bids , the inverse of the expected competitive ratio can be bounded as follows.

 k∑u=0pu∑uv=0fvB/(R(1−η)v)B/(R(1−η)u) =k∑u=0puu∑v=0fv(1−η)u−v=k∑v=0fvk∑u=vpu(1−η)u−v, (6)

where the last statement is by expanding the sums and reordering the terms.

The second sum in the RHS of Equation 6 is bounded by

 k∑u=vpu(1−η)u−v =2η(1−η)k−v+(1−η)(k+1)η+1≤2η+(1−η)(k+1)η+1=1+η(k+1)η+1, (7)

where the first equality is derived exactly similar to Zhou, Chakrabarty, and Lukose [13]. Replacing Equation 7 into the RHS of Equation 6,

 k∑v=0fvk∑u=vpu(1−η)u−v ≤1+η(k+1)η+1k∑v=0fv≤1+η(k+1)η+1

since by definition .

Now by definition of , we know , which implies . So,

 k∑v=0 fvk∑u=vpu(1−η)u−v≤1+η(k+1)η+1≤1+ηηln(R)/ln(1/(1−η))+1=O(1ln(R))

as , because . Since we bound the inverse of the competitive ratio in the above analysis, then the competitive ratio is at least . ∎

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