
Fair and Efficient Allocations of Chores under Bivalued Preferences
We study the problem of fair and efficient allocation of a set of indivi...
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Fair and Efficient Allocations under Subadditive Valuations
We study the problem of allocating a set of indivisible goods among agen...
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EFX Exists for Three Agents
We study the problem of allocating a set of indivisible items among agen...
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PROPm Allocations of Indivisible Goods to Multiple Agents
We study the classic problem of fairly allocating a set of indivisible g...
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Fair Allocation of Conflicting Items
We study fair allocation of indivisible items, where the items are furni...
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Fair Ride Allocation on a Line
With the advent of the ridesharing platform, the carpooling has become ...
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FalseNameProof Facility Location on Discrete Structures
We consider the problem of locating a single facility on a vertex in a g...
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Improving EFX Guarantees through Rainbow Cycle Number
We study the problem of fairly allocating a set of indivisible goods among n agents with additive valuations. Envyfreeness up to any good (EFX) is arguably the most compelling fairness notion in this context. However, the existence of EFX allocations has not been settled and is one of the most important problems in fair division. Towards resolving this problem, many impressive results show the existence of its relaxations, e.g., the existence of 0.618EFX allocations, and the existence of EFX at most n1 unallocated goods. The latter result was recently improved for three agents, in which the two unallocated goods are allocated through an involved procedure. Reducing the number of unallocated goods for arbitrary number of agents is a systematic way to settle the big question. In this paper, we develop a new approach, and show that for every ε∈ (0,1/2], there always exists a (1ε)EFX allocation with sublinear number of unallocated goods and high Nash welfare. For this, we reduce the EFX problem to a novel problem in extremal graph theory. We introduce the notion of rainbow cycle number R(·). For all d ∈ℕ, R(d) is the largest k such that there exists a kpartite digraph G =(∪_i ∈ [k] V_i, E), in which 1) each part has at most d vertices, i.e.,  V_i ≤ d for all i ∈ [k], 2) for any two parts V_i and V_j, each vertex in V_i has an incoming edge from some vertex in V_j and viceversa, and 3) there exists no cycle in G that contains at most one vertex from each part. We show that any upper bound on R(d) directly translates to a sublinear bound on the number of unallocated goods. We establish a polynomial upper bound on R(d), yielding our main result. Furthermore, our approach is constructive, which also gives a polynomialtime algorithm for finding such an allocation.
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