Truthful mechanism design via correlated tree rounding

Abstract

A powerful algorithmic technique for truthful mechanism design is the maximal-in-distributional-range (MIDR) paradigm. Unfortunately, many such algorithms use heavy algorithmic machinery, e.g., the ellipsoid method and (approximate) solution of convex programs. In this paper, we present a correlated rounding technique for designing mechanisms that are truthful in expectation. It is elementary and can be implemented quickly. The main property we rely on is that the domain offers fractional optimum solutions with a tree structure. In auctions based on the generalized assignment problem, each bidder has a publicly known knapsack constraint that captures the subsets of items that are of value to him. He has a private valuation for each item and strives to maximize the value of assigned items minus payment. For this domain we design a truthful 2-approximate MIDR mechanism for social welfare maximization. It avoids using the ellipsoid method or convex programming. In contrast to some previous work, our mechanism achieves exact truthfulness. In restricted-related scheduling with selfish machines, each job comes with a public weight, and it must be assigned to a machine from a public job-specific subset. Each machine has a private speed and strives to maximize payments minus workload of jobs assigned to it. Here we design a mechanism for makespan minimization. This is a single-parameter domain, but the approximation status of the optimization problem is similar to unrelated machine scheduling: The best known algorithm obtains a (non-truthful) 2-approximation for unrelated machines, and there is 1.5-hardness. Our mechanism matches this bound with a truthful 2-approximation.

Appendix: Optimal fractional solutions with pseudo-tree structure

For GAP it is necessary to deal with a possible cycle in the assignment graph. There exist instances of the problem for which the unique fractional optimal solution represents a cycle. To illustrate this, we use the following example:

$$\begin{aligned} \begin{array}{llll} &{}\max &{}\,\, (x_{11}+x_{21}+1.4x_{22}+x_{32}+x_{33}+x_{43}+100x_{44}+100x_{14})\\ &{}\text {s.t. } &{} \displaystyle x_{11}+x_{14} \le 1 \\ &{}&{} \displaystyle x_{21}+x_{22} \le 1 \\ &{}&{} \displaystyle x_{32}+x_{33} \le 1 \\ &{}&{} \displaystyle x_{43}+2x_{44} \le 1.25 \\ &{}&{} \displaystyle \sum _{i \in \{1,2,3,4\}} x_{ij} \le 1 \quad \quad \quad \forall j \in \{1,2,3,4\}\\ &{} &{}\quad x_{ij} \ge 0 \quad \quad \quad \forall i, j \in \{1,2,3,4\}. \end{array} \end{aligned}$$

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The assignment graph together with the optimal solution is depicted in Fig. 3. In this instance of GAP, every optimal solution will satisfy $\sum _{i\in \{1,2,3,4\}} x_{ij}= 1$ for all j: We only leave part of an item unassigned if another item can then be given away for a higher profit. The weights $b_{ij}$ of items differ by at most a factor of 2, while only for item $j_2$ there is a difference in the profit for the bidders interested in the item. The possible gain by selling $j_2$ it to bidder 2 instead of 3 is at most half the size of the reassigned fraction. Therefore, when leaving some $\varepsilon$-fraction of any item unassigned, we lose at least $\varepsilon \cdot \min \{r_{ij} \mid i,j\in \{1,2,3,4\}\}=\varepsilon$, while we gain at most $2\varepsilon (r_{22}-r_{32}) = 0.8\varepsilon$. The unique optimal solution results from assigning as much weight as possible along the cycle in counter-clockwise direction while keeping the constraint every item is fully assigned. It is given by $x^*_{11} = x^*_{21} = x^*_{22} = x^*_{32} = x^*_{33} = x^*_{43} = 1/2$, $x^*_{44} = 1/4$, and $x^*_{14} = 3/4$.

Fig. 3

In this example, the assignment graph corresponding to the unique optimal solution is a simple cycle