Source-linked AI summary
Rounding the Ball LP for Fair Max-Min Diversification
Julián Mestre, Lam Khai Trinh, Anthony Wirth
TL;DR
Fair Max-Min Diversification asks for a widely spread set that satisfies prescribed group quotas, while prior ball-LP rounding offered differing guarantees for approximate and exact fairness. This paper develops new ball-LP rounding schemes, including an augmenting method for exact fairness. It obtains a 4-approximation with exact fairness in time n^O(1)2^O(k), and no rounding of the ball LP can achieve a smaller exact-fairness factor.
Problem
Fair Max-Min Diversification seeks a maximally spread set subject to prescribed group quotas, and prior ball-LP rounding had different guarantees for approximate and exact fairness.
Source: §1 · p. 1Method
The paper develops two ball-LP rounding schemes, including greedy rounding and an exact-fairness augmentation adapted from Haxell's theorem.
Source: §1.2 · p. 2, §1.1 · p. 2Results
A factor 2 with (1−ε)-fairness improves on the prior factor 6, while an exactly fair factor 4 runs in n^O(1)2^O(k); no ball-LP rounding achieves a smaller exact-fairness factor.
Source: §1.2 · p. 2, §5 · p. 14Takeaways & Limitations
The exact-fairness factor 4 does not depend on the number of groups, and is tight for rounding the ball LP.
Source: §1.2 · p. 2, §5 · p. 14The exact-fairness algorithm runs in n^O(1)2^O(k), so it is polynomial only when k = O(log n); a constant-factor polynomial-time algorithm independent of m remains open.
Source: §5 · p. 14
Abstract
from arXiv · showhide
Given $n$ points in a metric space, partitioned into groups, $X_1,\dots,X_m$, and integer quotas, $k_1,\dots,k_m$, summing to $k$, the Fair Max-Min Diversification problem asks for a set of $k$ points, exactly $k_i$ from each group $X_i$, maximizing the minimum pairwise distance. Addanki et al. (ICDT 2022) described a ball LP for this problem and rounding algorithms that yield a factor 2 approximation whose fairness holds only in expectation and a factor 6 approximation with relaxed fairness guarantees, as well as an $(m+1)$-approximation with exact fairness. We introduce two new algorithms. The first method refines the rounding of Addanki et al., yielding a 2-approximate solution that is $\varepsilon$-fair with high probability, meaning that from every group $X_i$, at least $(1-\varepsilon) k_i$ points are chosen. The second method in polynomial time returns a 4-approximation to the optimal value of the exact fairness version. In time $n^{O(1)} 2^{O(k)}$, which is fixed-parameter tractable in $k$, we achieve an exactly fair 4-approximation. Our method adapts the augmenting procedure behind Haxell's theorem (Graphs Combin., 1995). This approximation factor does not depend on $m$. Moreover, we show that no rounding of the ball LP achieves a smaller factor with exact fairness.
1 Introduction
Fair Max-Min Diversification seeks a diverse sample while meeting group quotas. The paper improves ball-LP rounding guarantees for approximate and exact fairness.
1 Introduction
- The problem selects a maximally spread set while meeting prescribed quotas for each group, capturing goals such as proportional or equal representation.The introduction gives a city restaurant map as an example of balancing geographic spread with representation across cuisines.Source: §1 · p. 1
1.1 Related Work
- Addanki et al. introduced the ball LP; their rounding gave factor 2 with fairness in expectation, factor 6 with relaxed fairness, and factor m + 1 with exact fairness.Moumoulidou et al. [11] had earlier studied the fair variant and supplied approximation algorithms for it.Source: §1.1 · p. 2
1.2 Our Results
- The new greedy rounding achieves diversity factor 2 with (1−ε)-fairness, improving on Addanki et al.'s [1] factor 6 under the condition k_i ≥ 4ε^-2 ln(2m) for every group.It selects one point at a time and discards that point's ball; the resulting self-similar process supports a direct concentration bound without assuming independent selection events.Source: §1.2 · p. 2
- An iterative augmentation yields a 4-approximate exactly fair solution in time n^O(1)2^O(k), with a factor independent of the number of groups.The approach adapts Haxell's augmenting-path idea and is a strict improvement over the (m + 1)-approximation when m ≥ 4.Source: §1.2 · p. 2, §1.1 · p. 2
2 Preliminaries
The preliminaries define the quota-constrained metric-space objective and set up a threshold-based conflict graph and ball LP for rounding.
2 Preliminaries
- An instance has a finite metric space partitioned into groups with quotas, and seeks a quota-respecting set of maximum diversity.The total quota is k = Σ_i k_i, with k ≥ 2.Source: §2 · p. 3
2.1 The conflict graph and the ball LP
- At threshold γ, the conflict graph joins pairs closer than γ; a set is independent exactly when its diversity is at least γ.The open ball B(p, r) contains points at distance less than r from p.Source: §2.1 · p. 3
- The ball LP combines fairness and ball constraints; its feasible thresholds can be searched in polynomial time over O(n^2) candidate values.Every γ ≤ ℓ* is feasible, and the largest feasible threshold is itself feasible.Source: §2.1 · p. 3
3 2-approximation with relaxed fairness
Greedy rounding of the ball LP gives a diversity factor of 2 with ε-fairness under a quota-size condition, and its fairness guarantee can be boosted by rerunning until the output is ε-fair. The section develops a self-similar rounding process and concentration analysis to establish this guarantee.
3 2-approximation with relaxed fairness
- Diversity factor 2 holds always, while the output is ε-fair with probability at least 1/2 when every quota is at least 4ε^-2 ln(2m), improving Addanki et al.’s factor 6 guarantee.Repeated runs until an ε-fair output is obtained can boost the success probability arbitrarily close to 1.Source: §3 · p. 4
- No polynomial-time algorithm achieves a diversity factor below 2 while selecting at least βk_i points per group for any β > 0, unless P = NP.Thus the factor-2 guarantee cannot be improved even by relaxing fairness further.Source: §3 · p. 4
3.1 Greedy rounding
- Each greedy-rounding choice removes its entire closed neighborhood, leaving the same process on the remaining set; unlike Addanki et al.’s rule, discarded points do not discard others.The process returns an independent set, and the section gives a coupling under which its output contains Addanki et al.’s output.Source: §3.1 · pp. 4–5
3.2 The concentration inequality
- Because inclusion events are dependent, the analysis bounds an exponential moment by induction on the remaining set, then applies Markov’s inequality to obtain a tail bound.The resulting bounds are expressed in terms of x(A), rather than E|S ∩ A|.Source: §3.2 · p. 5
3.3 Proof of Theorem 3
- Capping groups that exceed their quotas preserves diversity and the lower fairness bounds, but this capping argument relies on groups being disjoint.For overlapping groups, one deletion can reduce two group counts at once.Source: §3.3 · p. 8
4 4-approximately diverse with exact fairness
The section gives an exactly fair factor-4 rounding of the ball LP, with a polynomial-time value estimate and an nO(1)2O(k)-time solution. Its iterative augmentation adapts Haxell’s procedure, and a tight instance shows that no exact-fair ball-LP rounding can improve the factor.
4 4-approximately diverse with exact fairness
- A polynomial-time 4-estimation algorithm and an nO(1)2O(k)-time algorithm return, respectively, a value within factor 4 and an exactly fair set of diversity at least ℓ*/4.The set is obtained by rounding at threshold γ/4.Source: §4 · p. 8
4.1 Tight instance
- An instance with n=12, m=4, and quotas ki=1 has optimum diversity 1 while the ball LP is feasible at γ=4.Thus no exact-fair ball-LP rounding can improve the factor 4.Source: §4.1 · p. 8, Figure 1 · p. 9
4.2 Iterative augmentation
- The augmentation builds an independent set one point at a time, rearranging earlier choices when a candidate cannot be added directly.Unlike the combinatorial argument behind Haxell’s theorem, its progress argument uses the ball-LP fractional solution and metric geometry.Source: §4.2 · pp. 9, 11
4.3 Correctness
- Each move either adds a point and ends the phase or preserves a good state, ensuring the partial solution continues to respect group quotas.The proof establishes that moves are available and preserve good states.Source: §4.2 · p. 10, §4.3 · pp. 11–12
4.4 Complexity
- At most 2k moves occur per phase, and at most k phases are needed, yielding nO(1)2O(k) total time.The move bound follows from strictly decreasing state signatures.Source: §4.4 · p. 14, §4.5 · p. 14
4.5 Proof of Lemma 7
- The procedure ends with an independent set containing exactly ki points from every group Xi, establishing the rounding guarantee.Each phase adds one point, and the total number of phases is at most k.Source: §4 · p. 8, §4.5 · p. 14
5 Concluding remarks
The concluding remarks contrast the limits of approximate and exact fairness for ball-LP rounding. They also leave open whether a stronger LP can achieve factor 2 with exact fairness or a constant factor in polynomial time.
5 Concluding remarks
- Whether exact fairness admits a constant-factor approximation independent of m in polynomial time remains open.The best known algorithm cited here has factor m+1.Source: §5 · p. 14
- No exact-fair rounding of the ball LP can achieve a factor below 4, while a stronger LP might still attain factor 2.Whether the stronger-LP guarantee is possible remains open.Source: §5 · p. 14