Binary XOS Valuations

• Binary XOS valuations are set functions defined as the maximum over binary additive clauses, ensuring 0-1 marginal increments and complement-free preferences.
• They enable efficient approximations for Nash social welfare and fairness metrics, underpinning robust mechanism design in allocation problems.
• Key properties like non-wasteful extractability and gradual truthfulness facilitate practical, algorithmic approaches despite potential exponential clause complexity.

A binary XOS valuation is a set function $v:2^M \rightarrow \mathbb{Z}_+$ on a ground set $M$ of $m$ goods, defined by expressing $v$ as the pointwise maximum over (possibly exponentially many) additive clauses with binary item weights. Binary XOS valuations constitute a broad, complement-free class that generalizes both binary additive and matroid-rank valuations and play a central role in fair division, mechanism design, and complexity analyses of allocation problems. They exhibit strong combinatorial structure—bounded, integral marginals and subadditivity—but notably need not be submodular. This entry surveys definitions, structural properties, placement within valuation hierarchies, mechanism design implications, and algorithmic aspects.

1. Formal Definitions and Representation

A set function $v:2^M\to\mathbb{Z}_+$ is a binary XOS valuation if it admits any of the following (equivalent) representations:

• Clause form: There exists a family $\mathcal{F}\subseteq 2^M$ such that for every $S\subseteq M$,

$v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$

Each $F\in\mathcal{F}$ is called an additive “clause,” with value given by simple cardinality on intersection with $S$.

• Additive maximum form: There is a (possibly exponential-sized) collection of additive functions $M$0 with weights $M$1 so that for all $M$2,

$M$3

• Binary marginals property: For every $M$4 and $M$5,

$M$6

• Extractability property: For every $M$7, there exists $M$8 with $M$9.

Each of these explicitly appears and is shown equivalent in (Barman et al., 2021) and (Barman et al., 2021). These canonical representations are critical for structural and algorithmic arguments.

2. Placement in Valuation Hierarchies

Binary XOS valuations strictly generalize several familiar classes:

Class	Definition	Containment in Binary XOS
Additive	$m$0, $m$1	Special case (one clause per $m$2)
Matroid-rank	Submodular, $m$3–$m$4 marginals; $m$5 = matroid rank of $m$6	Special case ($m$7 = independent sets)
General XOS	$m$8, $m$9	Binary XOS imposes $v$0
Binary subadditive	$v$1: $v$2 and subadditive	Binary XOS is strictly contained

Matroid-rank (i.e., binary submodular) valuations are precisely those binary XOS valuations that are submodular; their hereditary structure implies powerful combinatorial properties not shared by all binary XOS functions (Barman et al., 2021, Kulkarni et al., 2023).

3. Structural Properties

Key structural results include:

• Nonnegativity and monotonicity: Immediate from the form as a max of nonnegative, monotonic clauses.
• Subadditivity: Holds since XOS (fractionally subadditive) implies $v$3.
• Binary marginals: The value jump on adding any good is $v$4 or $v$5.
• Non-wasteful extractability (doubling): For any $v$6, one can find $v$7 with $v$8; new goods can always be split off efficiently, a property heavily exploited algorithmically (Barman et al., 2021).
• Closure under restriction: Restricting $v$9 to any $v:2^M\to\mathbb{Z}_+$0 (intersecting every clause $v:2^M\to\mathbb{Z}_+$1 with $v:2^M\to\mathbb{Z}_+$2) yields a new binary XOS function.
• Not closed under union: Arbitrary unions of binary XOS functions may lose clause structure.
• Potential exponential clause size: An arbitrary binary XOS function may need exponentially many clauses in $v:2^M\to\mathbb{Z}_+$3 for explicit representation, in contrast to the polynomial representability of matroid-rank functions via independence oracles.

4. Distinction from Matroid-Rank Functions

All matroid-rank valuations (binary, submodular with $v:2^M\to\mathbb{Z}_+$4–$v:2^M\to\mathbb{Z}_+$5 marginals) are binary XOS, but the converse fails: binary XOS functions need not be submodular.

Counterexample: On $v:2^M\to\mathbb{Z}_+$6, $v:2^M\to\mathbb{Z}_+$7 with clauses $v:2^M\to\mathbb{Z}_+$8 yields $v:2^M\to\mathbb{Z}_+$9, $\mathcal{F}\subseteq 2^M$0, violating submodularity at $\mathcal{F}\subseteq 2^M$1 (Barman et al., 2021, Kulkarni et al., 2023).

Matroid-rank functions possess additional hereditary clause structure, enabling strong results such as equality of AnyPrice Share (APS) and Maximin Share (MMS) (Kulkarni et al., 2023). This equality collapses generically in the larger binary XOS class.

5. Mechanism Design and Characterization of Truthfulness

For allocation of indivisible goods among agents with binary XOS valuations (without money), a full characterization of truthful (strategy-proof) mechanisms is available (Barman et al., 2021):

• Gradualness definition: A non-wasteful deterministic mechanism $\mathcal{F}\subseteq 2^M$2 is gradual if, for every agent $\mathcal{F}\subseteq 2^M$3, profile $\mathcal{F}\subseteq 2^M$4, and assigned bundle $\mathcal{F}\subseteq 2^M$5:
  • (C₁) For any good $\mathcal{F}\subseteq 2^M$6, removing $\mathcal{F}\subseteq 2^M$7 from $\mathcal{F}\subseteq 2^M$8 reduces bundle size by at most $\mathcal{F}\subseteq 2^M$9.
  • (C₂) For any $S\subseteq M$0, restricting $S\subseteq M$1 to $S\subseteq M$2 retains the same bundle size.
• Theorem: $S\subseteq M$3 is truthful if and only if it is gradual.

The proof employs binary marginals to tie value to bundle cardinality, enabling a chain of local operations (removals/restrictions) to bound possible improvements from misreporting. This result extends all matroid-rank–based fair division mechanisms to the entire binary XOS class, provided one can access the requisite (possibly exponential) clause structure.

From a complexity perspective, checking strategy-proofness for arbitrary allocation rules remains coNP-hard even for simpler classes, but this characterization offers a sufficient (syntactic) test for binary XOS if the mechanism is explicit.

6. Algorithmic and Complexity Aspects

• Nash Social Welfare (NSW): For $S\subseteq M$4 agents with binary XOS valuations, a polynomial-time constant-factor (specifically, $S\subseteq M$5) approximation algorithm for maximizing NSW is available in the value-oracle model (Barman et al., 2021). The allocation simultaneously provides $S\subseteq M$6-approximation to total welfare and groupwise maximin share (GMMS), using only non-wasteful bundles and “doubling” to find value-extracting subsets.
• APX-hardness: Exact NSW maximization is APX-hard for binary XOS, even with identical valuations, and stronger (polynomial-factor) approximations are impossible unless P=NP.
• Fairness metrics:
  • APS vs. MMS: In the matroid-rank (i.e., binary submodular) case, APS = MMS and both can be maximized efficiently (Kulkarni et al., 2023). For general binary XOS, $S\subseteq M$7, and there exist instances achieving APS arbitrarily close to $S\subseteq M$8 (Kulkarni et al., 2023).
  • APS approximation: No approximation better than $S\subseteq M$9 is possible, and a $v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$0-APS allocation can be computed in polynomial time using a reduction to hereditary set systems (Kulkarni et al., 2023).
  • AnyPrice Share (APS) under asymmetric entitlements: For asymmetric agents, a $v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$1-APS guarantee is best-possible and can be obtained in polynomial time, closing a previous gap for general XOS (Chen et al., 14 Jan 2026).
  • Weighted maximin share: A $v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$2-WMMS allocation always exists for binary XOS; this is tight, in contrast to the binary additive case where exact WMMS allocation is possible in polynomial time (Chen et al., 14 Jan 2026).
• Representation and query access: Efficient algorithms often rely on value-oracle access due to potentially exponential clauses; explicit representations are infeasible in general (Barman et al., 2021). The “extractability” property enables subroutine primitives such as extracting non-wasteful bundles of prescribed size in $v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$3 time given a value oracle (Barman et al., 2021).

Task	Binary XOS Complexity	Matroid-rank/ Additive
NSW approximation (constant factor)	Poly-time (Barman et al., 2021)	Poly-time, optimal possible
Exact NSW	APX-hard	Poly-time (matroid-rank only)
APS approx (symmetric/asymmetric)	$v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$4 / $v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$5 poly-time	$v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$6 (exact), poly-time
WMMS guarantee	$v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$7, tight	$v(S) = \max_{F\in\mathcal{F}}\,|F\cap S|$8, tight
Explicit representation	Exponential (in general)	Poly-time/compact

7. Implications and Open Directions

Binary XOS valuations provide a robust, combinatorially tractable framework for complement-free preferences that is strictly more expressive than matroid-rank but remains structurally amenable to constant-factor approximations for Nash welfare and various fairness criteria. The binary marginal property enables strong algorithmic primitives—non-wasteful bundle extraction, doubling, gradualness—that collapse in more general XOS or subadditive settings.

Nevertheless, significant gaps remain between algorithmic upper bounds and existential lower bounds for some metrics (such as APS approximability), and computationally efficient, simple representations remain elusive outside the matroid-rank fragment. Truthfulness characterization via gradualness extends powerful mechanism design paradigms but is only practical for explicit or oracle-represented mechanisms.

Recent work demonstrates that the binary-marginal restriction sharply delineates the frontier of tractability and fairness guarantees in fair allocation—stringent enough to admit strong guarantees, but weak enough to capture essential combinatorial hardness phenomena (Barman et al., 2021, Barman et al., 2021, Kulkarni et al., 2023, Chen et al., 14 Jan 2026).