Biaoshuai Tao

Panfeng Liu, Biaoshuai Tao

This paper investigates the influence maximization problem under the Independent Cascade(IC) and Linear Threshold (LT) models. While this problem is known to be APX-hard on general graphs, we explore its computational limits by focusing on Directed Acyclic Graphs (DAGs) and more restricted tree structures. Our primary result demonstrates that influence maximization remains APX-hard on DAGs under the LT model, suggesting that the absence of cycles is insufficient to achieve a polynomial-time approximation scheme (PTAS). In contrast, we show that the problem becomes tractable when the topology is further restricted to out-arborescences and in-arborescences. Specifically, for out-arborescences, we show that the IC model and the LT model are equivalent, and we develop exact polynomial-time algorithms based on dynamic programming that leverage the unique path properties of these structures. For in-arborescences, it is known that the problem is polynomial-time solvable under the LT model, and it is NP-hard under the IC model. We complement these results by presenting a fully polynomial-time approximation scheme (FPTAS) for the IC model.

Yizhou Ai, Biaoshuai Tao

We study the approval-based multiwinner election problem where a set of $n$ voters cast approval-based ballots to a set of $m$ candidates, and we are to select a winner committee consisting of $k$ candidates. We consider two axioms: strong justified representation (SJR) and average justified representation (AJR). A winner committee satisfies SJR if the satisfaction for each voter in every $\ell$-cohesive group is at least $\ell$. AJR is a weaker axiom that requires the average satisfaction for each $\ell$-cohesive group to be at least $\ell$. It is well known that a winner committee satisfying AJR may not exist (and neither does SJR). In this paper, we study the computational complexity of the following decision problem: given an approval-based multiwinner election instance, decide if there exists a winner committee satisfying SJR/AJR. We prove that this problem is $Θ_2^p$-complete for SJR, and $Σ_2^p$-complete for AJR. Our results indicate that the decision problem with SJR is more amenable to SAT-based implementations, whereas the decision problem with AJR is substantially harder. As byproducts, we derive some results that are interesting in their own right. Firstly, we show that adding one more adaptive query to an NP oracle on top of polynomially many non-adaptive NP queries does not add more computational power, and the resulting complexity class is still $Θ_2^p$. Secondly, we construct a set system that can be useful in other applications, especially when doing reductions from typical satisfiability problems such as 3SAT.

Yuqing Kong, Chris Peikert, Grant Schoenebeck et al.

We consider a setting where a verifier with limited computation power delegates a resource intensive computation task---which requires a $T\times S$ computation tableau---to two provers where the provers are rational in that each prover maximizes their own payoff---taking into account losses incurred by the cost of computation. We design a mechanism called the Minimal Refereed Mechanism (MRM) such that if the verifier has $O(\log S + \log T)$ time and $O(\log S + \log T)$ space computation power, then both provers will provide a honest result without the verifier putting any effort to verify the results. The amount of computation required for the provers (and thus the cost) is a multiplicative $\log S$-factor more than the computation itself, making this schema efficient especially for low-space computations.