On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

Mak‐Hau, Vicky

doi:10.1007/s10878-015-9932-4

Cited by 42 publications

(22 citation statements)

References 31 publications

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“…The EE-MTZ model [Mak-Hau 2015], another compact formulation, uses the variables and constraints of the extended edge formulation to model cycles and a variant of the Miller-Tucker-Zemlin model for the traveling salesman problem to model chains. The same paper introduces the exponentially-sized SPLIT-MTZ model, which adds redundant constraints to the edge formulation in order to tighten the LP relaxation.…”

Section: Prior Researchmentioning

confidence: 99%

“…The pricing problem is to bring at least one new positive price cycle into the model, or prove that none exist. Multiple methods exist for solving the pricing problem in kidney exchange [Abraham et al 2007;Glorie et al 2014;Mak-Hau 2015;Plaut et al 2016a]; in our experimental section, we use the cycle pricer of Glorie et al [2014] with the bugfix of Plaut et al [2016a].…”

Section: Practical Implementation Of the Picef Modelmentioning

confidence: 99%

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Position-Indexed Formulations for Kidney Exchange

Dickerson

Manlove

Plaut

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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A kidney exchange is an organized barter market where patients in need of a kidney swap willing but incompatible donors. Determining an optimal set of exchanges is theoretically and empirically hard. Traditionally, exchanges took place in cycles, with each participating patient-donor pair both giving and receiving a kidney. The recent introduction of chains, where a donor without a paired patient triggers a sequence of donations without requiring a kidney in return, increased the efficacy of fielded kidney exchanges-while also dramatically raising the empirical computational hardness of clearing the market in practice. While chains can be quite long, unbounded-length chains are not desirable: planned donations can fail before transplant for a variety of reasons, and the failure of a single donation causes the rest of that chain to fail, so parallel shorter chains are better in practice.In this paper, we address the tractable clearing of kidney exchanges with short cycles and chains that are long but bounded. This corresponds to the practice at most modern fielded kidney exchanges. We introduce three new integer programming formulations, two of which are compact. Furthermore, one of these models has a linear programming relaxation that is exactly as tight as the previous tightest formulation (which was not compact) for instances in which each donor has a paired patient. On real data from the UNOS nationwide exchange in the United States and the NLDKSS nationwide exchange in the United Kingdom, as well as on generated realistic large-scale data, we show that our new models are competitive with all existing solvers-in many cases outperforming all other solvers by orders of magnitude.Finally, we note that our position-indexed chain-edge formulation can be modified in a straightforward way to take post-match edge failure into account, under the restriction that edges have equal probabilities of failure. Post-match edge failure is a primary source of inefficiency in presently-fielded kidney exchanges. We show how to implement such failure-aware matching in our model, and also extend the state-of-the-art general branch-and-price-based non-compact formulation for the failure-aware problem to run its pricing problem in polynomial time.

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Section: Prior Researchmentioning

confidence: 99%

Section: Practical Implementation Of the Picef Modelmentioning

confidence: 99%

Position-Indexed Formulations for Kidney Exchange

Dickerson

Manlove

Plaut

et al. 2016

Proceedings of the 2016 ACM Conference on Economics and Computation

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“…Limitations on the maximum length of cycles may be weakened in the future. It is reported that a 9-cycle exchange was performed successfully in 2015 [20]. In this section, we discuss the KEP with L ≥ 3 parameterized by the size of kidney transplants.…”

Section: A Randomized Algorithm Based On Algebraic Techniquementioning

confidence: 99%

“…Mak-Hau [19] studied the edge formulations for the KEP with and without chains and presented some polyhedral results. For a detailed review of various IP approaches to solving the KEP, see a recent survey of Mak-Hau [20].…”

Section: Introductionmentioning

confidence: 99%

Randomized Parameterized Algorithms for the Kidney Exchange Problem

Lin

Wang

et al. 2019

Algorithms

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In order to increase the potential kidney transplants between patients and their incompatible donors, kidney exchange programs have been created in many countries. In the programs, designing algorithms for the kidney exchange problem plays a critical role. The graph theory model of the kidney exchange problem is to find a maximum weight packing of vertex-disjoint cycles and chains for a given weighted digraph. In general, the length of cycles is not more than a given constant L (typically 2 ≤ L ≤ 5), and the objective function corresponds to maximizing the number of possible kidney transplants. In this paper, we study the parameterized complexity and randomized algorithms for the kidney exchange problem without chains from theory. We construct two different parameterized models of the kidney exchange problem for two cases L = 3 and L ≥ 3 , and propose two randomized parameterized algorithms based on the random partitioning technique and the randomized algebraic technique, respectively.

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“…In this paper, we will present arc-based integer programming models of the Cardinality Constrained Multicycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP) and study their polyhedral structures, in the hope of finding strong constraints to be applied in a branch-and-bound framework for finding exact solutions in computationally efficient ways. Preliminary results of the exponential-size arc based formulations for the CCMcP and the CCCCP are presented in a companion paper Mak-Hau (2015a).…”

Section: Introductionmentioning

confidence: 99%

Polyhedral results for the Cardinality Constrained Multi-cycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP)

Mak‐Hau

2015

Weber, T., McPhee, M.J. And Anderssen, R.S. (Eds) MODSIM2015, 21st International Congress on Modelling and Simulation

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In recent years, there has been studies on the cardinality constrained multi-cycle problems on directed graphs, some of which considered chains co-existing on the same digraph whilst others did not. These studies were inspired by the optimal matching of kidneys known as the Kidney Exchange Problem (KEP). In a KEP, a vertex on the digraph represents a donor-patient pair who are related, though the kidney of the donor is incompatible to the patient. When there are multiple such incompatible pairs in the kidney exchange pool, the kidney of the donor of one incompatible pair may in fact be compatible to the patient of another incompatible pair. If Donor A's kidney is suitable for Patient B, and vice versa, then there will be arcs in both directions between Vertex A to Vertex B. Such exchanges form a 2-cycle. There may also be cycles involving 3 or more vertices. As all exchanges in a kidney exchange cycle must take place simultaneously, (otherwise a donor can drop out from the program once his/her partner has received a kidney from another donor), due to logistic and human resource reasons, only a limited number of kidney exchanges can occur simultaneously, hence the cardinality of these cycles are constrained. In recent years, kidney exchange programs around the world have altruistic donors in the pool. A sequence of exchanges that starts from an altruistic donor forms a chain instead of a cycle. We therefore have two underlying combinatorial optimization problems: Cardinality Constrained Multi-cycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP). The objective of the KEP is either to maximize the number of kidney matches, or to maximize a certain weighted function of kidney matches. In a CCMcP, a vertex can be in at most one cycle whereas in a CCCCP, a vertex can be part of (but in no more than) a cycle or a chain. The cardinality of the cycles are constrained in all studies. The cardinality of the chains, however, are considered unconstrained in some studies, constrained but larger than that of cycles, or the same as that of cycles in others. Although the CCMcP has some similarities to the ATSP-and VRPfamily of problems, there is a major difference: strong subtour elimination constraints are mostly invalid for the CCMcP, as we do allow smaller subtours as long as they do not exceed the size limit. The CCCCP has its distinctive feature that allows chains as well as cycles on the same directed graph. Hence, both the CCMcP and the CCCCP are interesting and challenging combinatorial optimization problems in their own rights. Most existing studies focused on solution methodologies, and as far as we aware, there is no polyhedral studies so far. In this paper, we will study the polyhedral structure of the natural arc-based integer programming models of the CCMcP and the CCCCP, both containing exponentially many constraints. We do so to pave the way for studying strong valid cuts we have found that can be applied in a Lagrangean relaxation-based branch-and-bound framework where at each n...

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On the kidney exchange problem: cardinality constrained cycle and chain problems on directed graphs: a survey of integer programming approaches

Cited by 42 publications

References 31 publications

Position-Indexed Formulations for Kidney Exchange

Position-Indexed Formulations for Kidney Exchange

Randomized Parameterized Algorithms for the Kidney Exchange Problem

Polyhedral results for the Cardinality Constrained Multi-cycle Problem (CCMcP) and the Cardinality Constrained Cycles and Chains Problem (CCCCP)

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