An Approximation Algorithm for Monotone Submodular Cost Allocation

In this paper, we consider the minimum submodular cost allocation (MSCA) problem. The input of MSCA is $k$ non-negative submodular functions $f_1,f_2,\ldots,f_k$ on the ground set $N$ given by evaluation oracles, and the goal is to partition $N$ into $k$ (possibly empty) sets $S_1,S_2,\ldots,S_k$ so that $\sum_{i=1}^k f_i(S_i)$ is minimized. In this paper, we focus on the case when $f_1,f_2,\ldots,f_k$ are monotone, which coincides with the submodular facility location problem considered by Svitkina and Tardos. We show that the integrality gap of a natural LP-relaxation for MSCA with monotone submodular functions is at most $k/2$, yielding a $k/2$-approximation algorithm. We also prove a nearly matching lower bound: the integrality gap is at least $k/2-ε$ for any constant $ε>0$ when $k$ is fixed.

💡 Research Summary

The paper studies the Minimum Submodular Cost Allocation (MSCA) problem, where one is given a ground set N and k non‑negative submodular functions f₁,…,f_k (accessible via evaluation oracles) and must partition N into k (possibly empty) subsets S₁,…,S_k so as to minimize the total cost Σ_i f_i(S_i). When all functions are monotone (i.e., f_i(S) ≤ f_i(T) for S⊆T), the problem coincides with the submodular facility location problem introduced by Svitkina and Tardos; the authors refer to this special case as Minimum Monotone Submodular Cost Allocation (MMSCA).

Previous work showed that for general submodular functions the LP relaxation has an unbounded integrality gap (unless P=NP), and for monotone functions the best known upper bound on the integrality gap of the natural LP (LP_mono) was min{k, 1 + ln n}, yielding a corresponding approximation ratio. Moreover, when k is part of the input, a reduction from set cover implies a (1 − o(1))·ln n hardness, matching the 1 + ln n algorithm. However, for small or fixed k the exact gap remained open.

The authors present two complementary results that essentially close this gap.

Upper bound (Theorem 1.1).
They prove that the integrality gap of LP_mono is at most k/2 for any k ≥ 2. The proof is constructive and leads directly to a k/2‑approximation algorithm (Theorem 1.2). The main steps are:

1. Optimal LP solution and chain structure.
   Using the ellipsoid method, they obtain an optimal fractional solution (y₁*,…,y_k*) where, after uncrossing, the support of each y_i* forms a chain C₁ⁱ⊃C₂ⁱ⊃…⊃C_{d_i}ⁱ.

2. Integer scaling.
   They define a common denominator M (a multiple of k(k‑1) and all fractional values) and set z_i = M·y_i*. This yields integer multiplicities for each set in the chain.

3. Replication of sets.
   For each i they create M copies U₁ⁱ,…,U_Mⁱ, where the first z_i(C₁ⁱ) copies equal C₁ⁱ, the next z_i(C₂ⁱ) copies equal C₂ⁱ, and so on. Lemma 2.1 guarantees the multiplicities.

4. Selection of a small fraction.
   They consider only the top 2/k fraction of the indices j (i.e., the largest sets) for each i. Because M = k(k‑1)·m, the total number of selected indices is 2·m·(k‑1).

5. Covering tuples.
   They construct 2·m·(k‑1) ordered k‑tuples (a₁,…,a_k) with distinct indices such that the union of the corresponding U_{a_j}ⁱ over i equals N. Lemma 2.4 shows that if such tuples exist, the integrality gap is at most k/2.

6. Explicit construction.
   Using modular arithmetic they define the tuples explicitly, proving they satisfy the required distinctness and covering properties.

With these tuples, they pick the one minimizing Σ_i f_i(U_{a_j}ⁱ) and then extract a genuine partition S₁,…,S_k ⊆ U_{a_j}ⁱ. Monotonicity ensures Σ_i f_i(S_i) ≤ Σ_i f_i(U_{a_j}ⁱ), and the analysis yields Σ_i f_i(S_i) ≤ (k/2)·OPT_LP, establishing the gap bound.

Lower bound (Theorem 1.3).
When log n ≥ k, they construct a family of monotone submodular functions for which the LP gap is at least k/2 − ε for any constant ε > 0 (provided k is fixed). The construction uses a collection of “large” sets of size Θ(log n) and defines each f_i to be the cardinality of the intersection with a designated family, scaled appropriately. The fractional LP can spread mass thinly across many small sets, achieving a low objective, while any integral solution must pick a relatively large set for each i, incurring a cost that is roughly a factor k/2 higher. The analysis shows the gap approaches k/2 as n grows, matching the upper bound up to an arbitrarily small additive ε.

Implications and connections.

• For k = 2, LP_mono is the dual of the unweighted polymatroid intersection problem, which is known to be totally dual integral; thus the k/2 bound (i.e., 1) is tight and recovers the classic result.
• The result improves upon the previous min{k, 1 + ln n} bound, especially when k is much smaller than ln n, delivering a substantially better approximation ratio.
• The techniques blend linear programming, uncrossing arguments, integer scaling, and combinatorial constructions based on modular arithmetic, illustrating a novel way to control the structure of fractional solutions.

Overall, the paper settles the integrality gap of the natural LP for monotone submodular cost allocation when k is small, providing both a tight theoretical bound and a practical k/2‑approximation algorithm that outperforms earlier logarithmic‑factor methods. This advances our understanding of submodular optimization under partition constraints and opens the door to further refinements for related problems such as monotone submodular multiway partition and submodular facility location.