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The Fisher market is one of the most fundamental models for resource allocation. However, Fisher markets are less amenable for resource allocation settings when agents have additional linear constraints beyond the budget constraints of buyers and the capacity constraints of goods. Thus, in this work, we introduce a modified Fisher market, where agents may have additional linear constraints, and study the properties of the resulting equilibria. To set equilibrium prices, we introduce a budget-adjusted social optimization problem (BA-SOP), whose optimal dual variables correspond to the equilibrium prices. Since solving BA-SOP can be computationally intensive and requires centralized knowledge of all agents' utilities, we propose a new class of distributed algorithms based on the Alternating Direction Method of Multipliers (ADMM) to compute equilibrium prices. Our ADMM approach has strong convergence guarantees and provides a general-purpose method for computing market equilibria for Fisher markets with homogeneous linear constraints and classical Fisher markets. [ABSTRACT FROM AUTHOR]

A 2- (v , k , λ) block design (P , B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G , +) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G , the embedding of P in G is also such that, conversely, any zero-sum k -subset of P is a block in B , then (P , B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2- (16 , 4 , 2) design (F 4 × F 4 , B 2) , which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG (4 , 2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem. [ABSTRACT FROM AUTHOR]

Let G be the additive group of a finite field. J. Li and D. Wan determined the exact number of solutions of the subset sum problem over G , by giving an explicit formula for the number of subsets of G of prescribed size whose elements sum up to a given element of G. They also determined a closed-form expression for the case where the subsets are required to contain only nonzero elements. In this paper we give an alternative proof of the two formulas. Our argument is purely combinatorial, as in the original proof by Li and Wan, but follows a different and somehow more "natural" approach. We also indicate some new connections with coding theory and combinatorial designs. [ABSTRACT FROM AUTHOR]