Logarithmic Approximations for Fair k-Set Selection

TL;DR

This paper introduces the fair k-set selection FKSS problem, showing it can be modeled as selecting $k$ right-side vertices in a bipartite graph to minimize the maximum weighted neighbor sum on the left. It establishes NP-hardness for $\Delta=3$, polynomial solvability for $\Delta=2$ and laminar set families, and provides LP-based rounding algorithms that achieve $O\big(\frac{\log n}{\log \log n}\big)$-approximation on general graphs and $O(\log\Delta)$-approximation on graphs with maximum degree $\Delta$, with matching integrality gaps up to constants. The methods include independent and dependent rounding, and a Lovász Local Lemma based approach for degree-bounded graphs, extending to the weighted setting with preserved guarantees. The results illuminate the trade-offs between graph structure and approximation quality and open directions for tighter bounds and matroid-based extensions with potential practical impact in fair resource allocation and representative subset selection.

Abstract

We study the fair k-set selection problem where we aim to select $k$ sets from a given set system such that the (weighted) occurrence times that each element appears in these $k$ selected sets are balanced, i.e., the maximum (weighted) occurrence times are minimized. By observing that a set system can be formulated into a bipartite graph $G:=(L\cup R, E)$, our problem is equivalent to selecting $k$ vertices from $R$ such that the maximum total weight of selected neighbors of vertices in $L$ is minimized. The problem arises in a wide range of applications in various fields, such as machine learning, artificial intelligence, and operations research. We first prove that the problem is NP-hard even if the maximum degree $Δ$ of the input bipartite graph is $3$, and the problem is in P when $Δ=2$. We then show that the problem is also in P when the input set system forms a laminar family. Based on intuitive linear programming, we show that a dependent rounding algorithm achieves $O(\frac{\log n}{\log \log n})$-approximation on general bipartite graphs, and an independent rounding algorithm achieves $O(\logΔ)$-approximation on bipartite graphs with a maximum degree $Δ$. We demonstrate that our analysis is almost tight by providing a hard instance for this linear programming. Finally, we extend all our algorithms to the weighted case and prove that all approximations are preserved.

Figures (4)

• Figure 1: An example of a laminar family and its corresponding tree. The original laminar family is shown on the left and marked as black lines. We add some dummy sets (whole ground element sets and all singleton sets), and they are marked as red lines. The corresponding tree is shown on the right, where black lines and sets are from the original instance, and red lines and sets are constructed dummy sets.
• Figure 2: The integrality gap instance to \ref{['Feas-LP']}. The figure shows an example with the demand $k=2$. Hence, there are $4$ vertices in $R$ and $\binom{4}{2}=6$ vertices in $L$; see subfigure (i). The subfigure (ii) shows an optimal integral solution. By our construction, any vertex set of $R$ with cardinality $2$ is an optimal integral solution. In the example, we pick the first and third vertex of $R$, denoted by $S$. The solid lines of subfigure (ii) show the vertex-induced graph $G[S]$. The disagreement of all vertices in $L$ is marked inside of each vertex. The subfigure (iii) shows an optimal fractional solution in which the demand $k$ is equally distributed to each vertex in $R$. Thus, the disagreement of all vertices in $L$ is equal to each other, which is equal to $1$.
• Figure 3: An example for the bad events (\ref{['def:badevents']}) and their dependency graph (\ref{['def:dependency-graph']}). The remaining graph $G'$ has four vertices in $L'$ and four vertices in $R'$. Assuming that the maximum degree $\Delta$ of the original graph is $2$. The bad events are shown in the subfigure (i). For each vertex $v$ in $R'$, there is a random variable $X_v\in\set{0,1}$ indicating whether $v$ is added to the solution. All bad events in $\mathcal{A}\cup\mathcal{B}$ are defined over $\mathcal{X}:=\set{X_v}_{v\in R}$. For each vertex $u\in L'$, we have one bad event $A_u$ in $\mathcal{A}$, and $A_u$ is determined by a set $\mathcal{A}_u$ of random variables in $\mathcal{X}$. For example, bad event $A_1$ is determined by $\set{X_1,X_2}$ which are neighbors of vertex $1$. We group all vertices in $R'$ into several subgroups, each of which consists of at most $\Delta$ vertices in $R'$. Then, we shall first define a bad event for each subgroup whose size is exactly $\Delta$. For the last subgroup, whose size may be smaller than $\Delta$, we might have a bad event depending on some condition. The dependency graph of $\mathcal{A}\cup\mathcal{B}$ is shown in the subfigure (ii). The dependency graph describes the relationship between all bad events. For example, the occurrence of bad event $B_1$ depends on the occurrence of bad events $A_1,A_2,A_3$.
• Figure 4: Illustration for \ref{['alg:degree=2']} and the proof of \ref{['lem:degree=2:decision']}. The black vertices are vertices in $L$, and all other vertices are in $R$. \ref{['alg:degree=2']} first assigns an index to each vertex from $R$ according to some fix direction. Depending on the parity of $\lvert R\rvert$ and whether the connected component is a cycle, \ref{['alg:degree=2']} shall pick the different number of vertices from $R$ to satisfy the disagreement requirement. The shadow vertices are the vertices picked by \ref{['alg:degree=2']}.

Theorems & Definitions (80)

• Theorem 1
• proof : Proof of \ref{['thm:degree=3:hardness']}
• Theorem 2
• Definition 1: Red-Blue
• Lemma 1: DBLP:journals/corr/abs-2308-09501
• proof : Proof of \ref{['thm:weight:degree=2']}
• Theorem 3
• Lemma 2
• proof
• proof : Proof of \ref{['thm:laminar']}