On the Complexity of the Minimum Independent Set Partition Problem

Abstract

We consider the Minimum Independent Set Partition Problem ($\mathsf {MISP}$) and its dual ($\mathsf {MISPDual}$). The input is a multi-set of N vectors from $\{0,1\}^n$, where $U := \{1,\ldots ,n\}$ is the index set. In $\mathsf {MISP}$, a threshold k is given and the goal is to partition U into a minimum number of subsets such that the projected vectors on each subset of indices have multiplicity at least k, where the multiplicity is the number of times a vector repeats in the (projected) multi-set. In $\mathsf {MISPDual}$, a target number $\chi$ is given instead of k, and the goal is to partition U into $\chi$ subsets to maximize k such that each projected vector appears at least k times.

The problem is inspired from applications in private voting verification. Each of the N vectors corresponds to a voter’s preference for n contests. The n contests are partitioned into $\chi$ subsets such that each voter receives a verifiable tracking number for each subset. For each subset of contests, each voter’s tracking number together with the votes for that subset is released in some public bulletin, which can be verified by each voter. The multiplicity k of the vectors’ projection onto each subset of indices ensures that the bulletin for each subset of contests satisfies the standard privacy notion of k-anonymity.

In this paper, we show strong inapproximability results for both problems. For $\mathsf {MISP}$, we show the problem is hard to approximate to within a factor of $n^{1-\epsilon }$. For $\mathsf {MISPDual}$, we show the problem is hard to approximate to within a factor of $N^{1-\epsilon }$. Here, $\epsilon$ can be any small constant. Note that factors n and N approximation are trivial for $\mathsf {MISP}$ and $\mathsf {MISPDual}$ respectively. Hence, our results imply that any polynomial-time algorithm can almost do no better than the trivial one.

This research is partially funded by a grant from Hong Kong RGC under the contract HKU719312E.