201 research outputs found
The Knapsack Subproblem of the Algorithm to Compute the Erdos-Selfridge Function
This thesis summarizes the methodology of a new algorithm to compute the Erdos-Selfridge function which uses a wheel sieve, shows that a knapsack algorithm can be used to minimize the work needed to compute these values by selecting a subset of rings for use in the wheel, and compares the results of several different knapsack algorithms in this particular scenario
Knapsack and Subset Sum with Small Items
Knapsack and Subset Sum are fundamental NP-hard problems in combinatorial optimization. Recently there has been a growing interest in understanding the best possible pseudopolynomial running times for these problems with respect to various parameters. In this paper we focus on the maximum item size s and the maximum item value v. We give algorithms that run in time O(n + sÂł) and O(n + vÂł) for the Knapsack problem, and in time Ă(n + s^{5/3}) for the Subset Sum problem. Our algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance. In these variants n denotes the (possibly much smaller) number of distinct items. Our results follow from combining and optimizing several diverse lines of research, notably proximity arguments for integer programming due to Eisenbrand and Weismantel (TALG 2019), fast structured (min,+)-convolution by Kellerer and Pferschy (J. Comb. Optim. 2004), and additive combinatorics methods originating from Galil and Margalit (SICOMP 1991)
Guaranteeing Envy-Freeness under Generalized Assignment Constraints
We study fair division of goods under the broad class of generalized
assignment constraints. In this constraint framework, the sizes and values of
the goods are agent-specific, and one needs to allocate the goods among the
agents fairly while further ensuring that each agent receives a bundle of total
size at most the corresponding budget of the agent. Since, in such a constraint
setting, it may not always be feasible to partition all the goods among the
agents, we conform -- as in recent works -- to the construct of charity to
designate the set of unassigned goods. For this allocation framework, we obtain
existential and computational guarantees for envy-free (appropriately defined)
allocation of divisible and indivisible goods, respectively, among agents with
individual, additive valuations for the goods.
We deem allocations to be fair by evaluating envy only with respect to
feasible subsets. In particular, an allocation is said to be feasibly envy-free
(FEF) iff each agent prefers its bundle over every (budget) feasible subset
within any other agent's bundle (and within the charity). The current work
establishes that, for divisible goods, FEF allocations are guaranteed to exist
and can be computed efficiently under generalized assignment constraints.
In the context of indivisible goods, FEF allocations do not necessarily
exist, and hence, we consider the fairness notion of feasible envy-freeness up
to any good (FEFx). We show that, under generalized assignment constraints, an
FEFx allocation of indivisible goods always exists. In fact, our FEFx result
resolves open problems posed in prior works. Further, for indivisible goods and
under generalized assignment constraints, we provide a pseudo-polynomial time
algorithm for computing FEFx allocations, and a fully polynomial-time
approximation scheme (FPTAS) for computing approximate FEFx allocations.Comment: 29 page
Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression
We present an exact method, based on an arc-flow formulation with side
constraints, for solving bin packing and cutting stock problems --- including
multi-constraint variants --- by simply representing all the patterns in a very
compact graph. Our method includes a graph compression algorithm that usually
reduces the size of the underlying graph substantially without weakening the
model. As opposed to our method, which provides strong models, conventional
models are usually highly symmetric and provide very weak lower bounds.
Our formulation is equivalent to Gilmore and Gomory's, thus providing a very
strong linear relaxation. However, instead of using column-generation in an
iterative process, the method constructs a graph, where paths from the source
to the target node represent every valid packing pattern.
The same method, without any problem-specific parameterization, was used to
solve a large variety of instances from several different cutting and packing
problems. In this paper, we deal with vector packing, graph coloring, bin
packing, cutting stock, cardinality constrained bin packing, cutting stock with
cutting knife limitation, cutting stock with binary patterns, bin packing with
conflicts, and cutting stock with binary patterns and forbidden pairs. We
report computational results obtained with many benchmark test data sets, all
of them showing a large advantage of this formulation with respect to the
traditional ones
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