8 research outputs found
High-Multiplicity Fair Allocation Using Parametric Integer Linear Programming
Using insights from parametric integer linear programming, we significantly
improve on our previous work [Proc. ACM EC 2019] on high-multiplicity fair
allocation. Therein, answering an open question from previous work, we proved
that the problem of finding envy-free Pareto-efficient allocations of
indivisible items is fixed-parameter tractable with respect to the combined
parameter "number of agents" plus "number of item types." Our central
improvement, compared to this result, is to break the condition that the
corresponding utility and multiplicity values have to be encoded in unary
required there. Concretely, we show that, while preserving fixed-parameter
tractability, these values can be encoded in binary, thus greatly expanding the
range of feasible values.Comment: 15 pages; Published in the Proceedings of ECAI-202
Allocation in Practice
How do we allocate scarcere sources? How do we fairly allocate costs? These
are two pressing challenges facing society today. I discuss two recent projects
at NICTA concerning resource and cost allocation. In the first, we have been
working with FoodBank Local, a social startup working in collaboration with
food bank charities around the world to optimise the logistics of collecting
and distributing donated food. Before we can distribute this food, we must
decide how to allocate it to different charities and food kitchens. This gives
rise to a fair division problem with several new dimensions, rarely considered
in the literature. In the second, we have been looking at cost allocation
within the distribution network of a large multinational company. This also has
several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on
Artificial Intelligence (KI 2014), Springer LNC
Online Fair Division: A Survey
We survey a burgeoning and promising new research area that considers the
online nature of many practical fair division problems. We identify wide
variety of such online fair division problems, as well as discuss new
mechanisms and normative properties that apply to this online setting. The
online nature of such fair division problems provides both opportunities and
challenges such as the possibility to develop new online mechanisms as well as
the difficulty of dealing with an uncertain future.Comment: Accepted by the 34th AAAI Conference on Artificial Intelligence (AAAI
2020
Multi-agent Online Scheduling: MMS Allocations for Indivisible Items
We consider the problem of fairly allocating a sequence of indivisible items
that arrive online in an arbitrary order to a group of n agents with additive
normalized valuation functions. We consider both the allocation of goods and
chores and propose algorithms for approximating maximin share (MMS)
allocations. When agents have identical valuation functions the problem
coincides with the semi-online machine covering problem (when items are goods)
and load balancing problem (when items are chores), for both of which optimal
competitive ratios have been achieved. In this paper, we consider the case when
agents have general additive valuation functions. For the allocation of goods,
we show that no competitive algorithm exists even when there are only three
agents and propose an optimal 0.5-competitive algorithm for the case of two
agents. For the allocation of chores, we propose a (2-1/n)-competitive
algorithm for n>=3 agents and a square root of 2 (approximately
1.414)-competitive algorithm for two agents. Additionally, we show that no
algorithm can do better than 15/11 (approximately 1.364)-competitive for two
agents.Comment: 29 pages, 1 figure (to appear in ICML 2023
Elements of dynamic and 2-SAT programming: paths, trees, and cuts
This thesis presents faster (in terms of worst-case running times) exact algorithms for special cases of graph problems through dynamic programming and 2-SAT programming. Dynamic programming describes the procedure of breaking down a problem recursively into overlapping subproblems, that is, subproblems with common subsubproblems. Given optimal solutions to these subproblems, the dynamic program then combines them into an optimal solution for the original problem. 2-SAT programming refers to the procedure of reducing a problem to a set of 2-SAT formulas, that is, boolean formulas in conjunctive normal form in which each clause contains at most two literals. Computing whether such a formula is satisfiable (and computing a satisfying truth assignment, if one exists) takes linear time in the formula length. Hence, when satisfying truth assignments to some 2-SAT formulas correspond to a solution of the original problem and all formulas can be computed efficiently, that is, in polynomial time in the input size of the original problem, then the original problem can be solved in polynomial time. We next describe our main results. Diameter asks for the maximal distance between any two vertices in a given undirected graph. It is arguably among the most fundamental graph parameters. We provide both positive and negative parameterized results for distance-from-triviality-type parameters and parameter combinations that were observed to be small in real-world applications. In Length-Bounded Cut, we search for a bounded-size set of edges that intersects all paths between two given vertices of at most some given length. We confirm a conjecture from the literature by providing a polynomial-time algorithm for proper interval graphs which is based on dynamic programming. k-Disjoint Shortest Paths is the problem of finding (vertex-)disjoint paths between given vertex terminals such that each of these paths is a shortest path between the respective terminals. Its complexity for constant k > 2 has been an open problem for over 20 years. Using dynamic programming, we show that k-Disjoint Shortest Paths can be solved in polynomial time for each constant k. The problem Tree Containment asks whether a phylogenetic tree T is contained in a phylogenetic network N. A phylogenetic network (or tree) is a leaf-labeled single-source directed acyclic graph (or tree) in which each vertex has in-degree at most one or out-degree at most one. The problem stems from computational biology in the context of the tree of life (the history of speciation). We introduce a particular variant that resembles certain types of uncertainty in the input. We show that if each leaf label occurs at most twice in a phylogenetic tree N, then the problem can be solved in polynomial time and if labels can occur up to three times, then the problem becomes NP-hard. Lastly, Reachable Object is the problem of deciding whether there is a sequence of rational trades of objects among agents such that a given agent can obtain a certain object. A rational trade is a swap of objects between two agents where both agents profit from the swap, that is, they receive objects they prefer over the objects they trade away. This problem can be seen as a natural generalization of the well-known and well-studied Housing Market problem where the agents are arranged in a graph and only neighboring agents can trade objects. We prove a dichotomy result that states that the problem is polynomial-time solvable if each agent prefers at most two objects over its initially held object and it is NP-hard if each agent prefers at most three objects over its initially held object. We also provide a polynomial-time 2-SAT program for the case where the graph of agents is a cycle