Binary MMS-Feasible Valuations

• The paper shows that binary additive valuations yield exact WMMS allocations through a polynomial-time algorithm, and binary submodular (matroid rank) valuations achieve exact MMS and APS equality.
• Binary XOS valuations, by contrast, lose exact feasibility and offer only a worst-case 1/n-WMMS guarantee due to inherent structural limitations in binary marginals.
• Algorithmic, strategic, and communication analyses emphasize that structural differences in valuation classes critically impact fair allocation outcomes under both symmetric and asymmetric settings.

Searching arXiv for the specified paper and closely related work on binary valuations, MMS/WMMS, APS, and matroid-rank settings. arXiv search query: (Chen et al., 14 Jan 2026) Binary XOS WMMS asymmetric agents arXiv search query: (Kulkarni et al., 2023) APS MMS binary marginals XOS matroid rank arXiv search query: (Suksompong et al., 2022) weighted Nash welfare binary valuations Binary-valued MMS-feasible valuations form a research umbrella for valuation classes over indivisible goods in which maximin-share-type guarantees become exact, approximately exact, or provably unattainable under binary structural restrictions. In the recent literature, “binary” has two distinct formal meanings: additive singleton values in $\{0,1\}$, and more general binary marginals $v(S\cup\{g\})-v(S)\in\{0,1\}$. That distinction is decisive. Binary additive and binary submodular valuations exhibit strong exact-feasibility phenomena, while binary XOS valuations already lose exact WMMS/MMS feasibility and admit only worst-case approximation guarantees (Chen et al., 14 Jan 2026, Kulkarni et al., 2023).

1. Formal scope and basic notions

The canonical setting is fair allocation of indivisible goods among $n$ agents. For symmetric entitlements, agent $i$’s maximin share is

$$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$

For asymmetric entitlements $b_i$, the weighted maximin share is

$$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$

When $b_1=\cdots=b_n=\frac1n$, WMMS reduces exactly to MMS. The weighted formulation is therefore the natural extension of MMS to asymmetric agents (Chen et al., 14 Jan 2026).

The phrase “binary-valued” is not uniform across papers. In additive models, binary means $v_i(g)\in\{0,1\}$ for each good $g$, so $v(S\cup\{g\})-v(S)\in\{0,1\}$0 is the number of liked goods in $v(S\cup\{g\})-v(S)\in\{0,1\}$1. In XOS and submodular work, binary instead refers to binary marginals: $v(S\cup\{g\})-v(S)\in\{0,1\}$2 This does not mean $v(S\cup\{g\})-v(S)\in\{0,1\}$3 for every bundle. For XOS valuations, the representation is

$v(S\cup\{g\})-v(S)\in\{0,1\}$4

where each $v(S\cup\{g\})-v(S)\in\{0,1\}$5 is additive. Binary XOS is thus strictly broader than binary additive, and binary submodular valuations coincide with matroid rank functions (Chen et al., 14 Jan 2026, Kulkarni et al., 2023).

A second benchmark that repeatedly interacts with MMS is Any Price Share (APS), defined through a price simplex and known to dominate MMS for monotone valuations: $v(S\cup\{g\})-v(S)\in\{0,1\}$6 The extent to which APS and MMS coincide, or diverge, is one of the main structural separators among binary classes (Kulkarni et al., 2023).

2. Exact MMS-feasible binary classes

The cleanest exact-feasibility result is for binary additive valuations. In the asymmetric setting, if agent $v(S\cup\{g\})-v(S)\in\{0,1\}$7 has a demand set

$v(S\cup\{g\})-v(S)\in\{0,1\}$8

then exact WMMS allocations always exist and can be found in polynomial time. The same result immediately yields exact MMS in the symmetric special case. The allocation algorithm uses a balancing rule based on normalized load $v(S\cup\{g\})-v(S)\in\{0,1\}$9, and the stated runtime is $n$0 (Chen et al., 14 Jan 2026).

A second exact class is binary submodular valuations, equivalently matroid rank functions. In that class, APS and MMS collapse exactly: $n$1 Consequently, exact APS allocations exist and can be computed efficiently; since exact MMS allocations are already known for matroid rank valuations, the same allocation is also exact APS. The literature summarized in this line of work further states that such exact allocations can be computed while maximizing social welfare (Kulkarni et al., 2023).

These two domains already show that binaryity can be highly favorable, but only under substantial structural restrictions. Binary additivity yields exact weighted and unweighted maximin guarantees; binary submodularity yields exact MMS and exact APS, but the positive theorem is fundamentally tied to matroid rank structure rather than to binary marginals alone (Chen et al., 14 Jan 2026, Kulkarni et al., 2023).

Valuation class	Exact MMS/WMMS status	Representative statement
Binary additive	Exact WMMS and exact MMS	Polynomial-time exact WMMS allocation (Chen et al., 14 Jan 2026)
Binary submodular / matroid rank	Exact MMS; $n$2	Efficient exact APS/MMS allocation (Kulkarni et al., 2023)
Binary XOS	Not exact in worst case	Only $n$3-WMMS guaranteed (Chen et al., 14 Jan 2026)

3. Binary XOS and the breakdown of exact feasibility

The sharpest negative result concerns binary XOS valuations. For general XOS valuations, there always exists a $n$4-WMMS allocation: $n$5 The paper then proves that this factor is tight even when all valuations have binary marginals: for every $n$6, there exists an instance of $n$7 agents with binary XOS valuations where no allocation is better than $n$8-WMMS (Chen et al., 14 Jan 2026).

The lower-bound instance uses $n$9 goods and asymmetric entitlements

$i$0

Agents $i$1 value a bundle at $i$2 iff it contains at least one of $i$3; agent $i$4 either values the whole block $i$5 additively or values each remaining good $i$6 as a separate singleton clause of value $i$7. In this construction,

$i$8

yet any allocation that gives positive value to the first $i$9 agents forces

$$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$0

hence

$$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$1

The theorem is formally a weighted result, but it establishes the central separation: binary XOS does not exhibit the exact-feasibility behavior enjoyed by binary additive valuations (Chen et al., 14 Jan 2026).

The same line of work gives an APS-side approximation result for binary XOS in the asymmetric setting: a polynomial-time $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$2-APS allocation, which matches the known upper bound. In the symmetric case, APS dominates MMS, so this implies a polynomial-time $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$3-MMS allocation. This is an approximation theorem, not an exact-feasibility theorem (Chen et al., 14 Jan 2026).

4. APS, MMS, and class separations under binary structure

APS is especially informative because it dominates MMS for every monotone valuation but behaves very differently across binary classes. For matroid rank valuations, the strongest possible relation holds: $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$4 The proof uses capping at the APS value, matroid exchange, and union-matroid rank conditions to derive $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$5 disjoint bases, which then realize an MMS partition of value equal to APS. This makes binary submodularity a particularly robust exact share-based domain (Kulkarni et al., 2023).

Binary XOS breaks this equivalence. In that class,

$$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$6

and the upper bound is almost tight: there exists an instance with three agents and six goods, all with identical binary XOS valuations, for which

$$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$7

The construction uses two additive binary clauses, one supported on $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$8 and one on $$MMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j).$$9, so that APS can fractionally support many value-$b_i$0 bundles while MMS is bottlenecked by integral partitioning. From this bound, a known $b_i$1-MMS algorithm yields a polynomial-time $b_i$2-APS allocation, while the lower-bound instance implies that no better than $b_i$3-APS allocation may exist even with identical valuations (Kulkarni et al., 2023).

The asymmetric setting introduces a further separation. APS guarantees do not translate into WMMS guarantees when entitlements are unequal: there are additive instances where, for some agent,

$b_i$4

for any $b_i$5. This shows that APS and WMMS can diverge dramatically outside the symmetric case, and explains why binary XOS requires a separate weighted analysis rather than a transfer from APS bounds (Chen et al., 14 Jan 2026).

5. Algorithmic, strategic, and communication perspectives

Existence, implementability, and strategic realizability separate sharply in binary domains. For XOS WMMS, the $b_i$6-guarantee is given constructively, but the construction assumes access to WMMS values, WMMS partitions, and additive clauses maximizing selected bundles. The result is therefore best read as a constructive existence theorem rather than a full polynomial-time algorithm under standard oracle access. By contrast, the binary additive case comes with a genuine polynomial-time exact algorithm (Chen et al., 14 Jan 2026).

Mechanism design introduces another boundary. Under matroid-rank valuations, exact MMS allocations, and even MMS plus Pareto-efficient allocations, exist by prior work cited in the literature, but there is no mechanism that is simultaneously truthful, index-oblivious, Pareto efficient, and maximin fair. The impossibility persists even if Pareto efficiency is weakened to local efficiency. The same paper proves a characterization of truthful non-wasteful mechanisms—truthfulness iff gradualness—and notes that the proof extends from matroid-rank valuations to binary XOS functions. The overall message is that existential MMS-feasibility does not imply truthful exact implementation (Barman et al., 2021).

Communication complexity reveals a different sense in which binary additive valuations are special. For equally entitled agents with binary additive valuations, the expected average number of bits transmitted per agent for randomized protocols computing an MMS allocation is

$b_i$7

with total randomized communication

$b_i$8

The lower bound holds even when all agents have the same binary valuation function. A nearby class already breaks this compressibility: for 2-valued additive valuations, the paper gives a lower bound of

$b_i$9

on the expected average number of bits transmitted per agent (Feige, 2024).

Weighted Nash welfare adds useful indirect structure in the binary additive setting, but not direct MMS theorems. The lexicographically refined MWNW rule studied for binary additive valuations is resource-monotone, population-monotone, group-strategyproof, and computable in polynomial time, with runtime

$$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$0

However, that work does not define MMS or prove any MMS guarantee; its relevance to binary-valued MMS-feasibility is therefore structural rather than theorem-level (Suksompong et al., 2022).

6. Online and mixed-domain extensions

Online models expose how fragile binary MMS-feasibility becomes under irrevocability. In the model with sequentially arriving agents and additive valuations, no online algorithm can ensure any non-trivial MMS approximation without suitable information about future valuation functions, even with only two agents and binary valuations. Under the $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$1-type model, adversarial arrivals admit a $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$2-MMS algorithm but also a lower bound ruling out any $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$3-MMS-competitive algorithm, even for binary valuations. Under stochastic arrivals, the picture improves substantially: an asymptotic, arbitrarily close-to-$$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$4-MMS competitive guarantee is achievable under the paper’s stated distributional assumptions (Kulkarni et al., 3 Mar 2025).

A different online model studies sequentially arriving items allocated to offline agents. There the exact-feasibility frontier is class-dependent. For goods, additive binary valuations admit deterministic online allocations that are exact MMS, EF1, and max-USW; submodular binary valuations admit $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$5-MMS, $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$6-EF1, and $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$7-max-USW, and no deterministic non-wasteful algorithm can do better than $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$8-MMS. For chores, additive binary costs admit exact MMS, EF1, and min-USC, whereas supermodular binary costs admit no deterministic complete algorithm achieving $$WMMS_i^N(M)=\max_{S\in \Pi_n}\min_{j\in[n]} v_i(S_j)\frac{b_i}{b_j}.$$9-MMS for any $b_1=\cdots=b_n=\frac1n$0. The same paper also identifies mixed binary-plus-bivalued classes where exact MMS remains possible, and two-agent bivalued settings where only approximation guarantees survive (Wang et al., 30 May 2025).

Mixed divisible/indivisible environments do not provide direct MMS theorems, but they show that binary structure alone does not trivialize fair allocation once continuous and discrete resources are combined. Under binary additive valuations for mixed goods, the feasible utility region is generally neither convex nor $b_1=\cdots=b_n=\frac1n$1-convex; fair-allocation optimization is NP-hard even when all indivisible goods are identical; and a polynomial-time algorithm is obtained only when all divisible goods are identical. The main structural contribution is a proximity theorem stating that if $b_1=\cdots=b_n=\frac1n$2 is a $b_1=\cdots=b_n=\frac1n$3-fair integral utility vector and $b_1=\cdots=b_n=\frac1n$4 is a fair relaxed one, then

$b_1=\cdots=b_n=\frac1n$5

This suggests that binary mixed domains retain strong additive structure, but not the universal exact-feasibility behavior seen in pure binary additive allocation (Kawase et al., 2023).

Taken together, these results yield a precise landscape. Binary additive valuations are an exact MMS- and WMMS-feasible class, with efficient algorithms and unusually low communication. Binary submodular valuations remain exact in the symmetric offline sense through the identity $b_1=\cdots=b_n=\frac1n$6. Binary XOS valuations do not preserve this behavior: in weighted settings the best worst-case guarantee is exactly $b_1=\cdots=b_n=\frac1n$7, and even APS separates from MMS by an almost factor-$b_1=\cdots=b_n=\frac1n$8 gap. Online and mixed settings then show that binary structure is powerful but not sufficient by itself; feasibility depends critically on additivity, matroid structure, temporal model, and whether fairness is weighted, strategic, or communication-constrained (Chen et al., 14 Jan 2026, Kulkarni et al., 2023).