Fair Allocation based on Diminishing Differences

Ranking alternatives is a natural way for humans to explain their preferences. It is being used in many settings, such as school choice (NY, Boston), Course allocations, and the Israeli medical lottery. In some cases (such as the latter two), several ``items'' are given to each participant. Without having any information on the underlying cardinal utilities, arguing about fairness of allocation requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where a X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation if it exists. Using simulations, we show that with high probability, a necessarily-proportional allocation does not exist but a necessarily-DD-proportional allocation exists, and moreover, that allocation is proportional according to the underlying cardinal utilities.

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Fair Allocation with Diminishing Differences

Journal of Artificial Intelligence Research ◽

10.1613/jair.1.11994 ◽

2020 ◽

Vol 67 ◽

Author(s):

Erel Segal-Halevi ◽

Avinatan Hassidim ◽

Haris Aziz

Keyword(s):

School Choice ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Conservative Extension ◽

Score Functions ◽

Full Characterization ◽

Proportional Allocation ◽

The Difference ◽

Natural Way

Ranking alternatives is a natural way for humans to explain their preferences. It is used in many settings, such as school choice, course allocations and residency matches. Without having any information on the underlying cardinal utilities, arguing about the fairness of allocations requires extending the ordinal item ranking to ordinal bundle ranking. The most commonly used such extension is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if its score is better according to all additive score functions. SD is a very conservative extension, by which few allocations are necessarily fair while many allocations are possibly fair. We propose to make a natural assumption on the underlying cardinal utilities of the players, namely that the difference between two items at the top is larger than the difference between two items at the bottom. This assumption implies a preference extension which we call diminishing differences (DD), where X is preferred over Y if its score is better according to all additive score functions satisfying the DD assumption. We give a full characterization of allocations that are necessarily-proportional or possibly-proportional according to this assumption. Based on this characterization, we present a polynomial-time algorithm for finding a necessarily-DD-proportional allocation whenever it exists. Using simulations, we compare the various fairness criteria in terms of their probability of existence, and their probability of being fair by the underlying cardinal valuations. We find that necessary-DD-proportionality fares well in both measures. We also consider envy-freeness and Pareto optimality under diminishing-differences, as well as chore allocation under the analogous condition --- increasing-differences.

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Supply Chain Batching Problem with Identical Orders and Lifespan

Mathematical Problems in Engineering ◽

10.1155/2015/716248 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9

Author(s):

Shanlin Li ◽

Maoqin Li ◽

Hong Yan

Keyword(s):

Dynamic Programming ◽

Processing Time ◽

Recursion Formula ◽

Dynamic Programming Algorithm ◽

Setup Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Programming Algorithm ◽

Short Lifespan ◽

The Difference

In the real world, there are a large number of supply chains that involve the short lifespan products. In this paper, we consider an integrated production and distribution batch scheduling problem on a single machine for the orders with a short lifespan, because it may be cheaper or faster to process and distribute orders in a batch than to process and distribute them individually. Assume that the orders have the identical processing time and come from the same location, and the batch setup time is a constant. The problem is to choose the number of batches and batch sizes to minimize the total delivery time without violating the order lifespan. We first give a backward dynamic programming algorithm, but it is not an actually polynomial-time algorithm. Then we propose a constant time partial dynamic programming algorithm by doing further research into the recursion formula in the algorithm. Further, using the difference characteristics of the optimal value function, a specific calculating formula to solve the problem with the setup time being integer times of the processing time is obtained.

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Complete Characterization of Incorrect Orthology Assignments in Best Match Graphs

Journal of Mathematical Biology ◽

10.1007/s00285-021-01564-8 ◽

2021 ◽

Vol 82 (3) ◽

Cited By ~ 2

Author(s):

David Schaller ◽

Manuela Geiß ◽

Peter F. Stadler ◽

Marc Hellmuth

Keyword(s):

Gene Transfer ◽

Horizontal Gene Transfer ◽

A Priori ◽

Gene Tree ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Complete Characterization ◽

Species Trees ◽

Genome Scale

AbstractGenome-scale orthology assignments are usually based on reciprocal best matches. In the absence of horizontal gene transfer (HGT), every pair of orthologs forms a reciprocal best match. Incorrect orthology assignments therefore are always false positives in the reciprocal best match graph. We consider duplication/loss scenarios and characterize unambiguous false-positive (u-fp) orthology assignments, that is, edges in the best match graphs (BMGs) that cannot correspond to orthologs for any gene tree that explains the BMG. Moreover, we provide a polynomial-time algorithm to identify all u-fp orthology assignments in a BMG. Simulations show that at least $$75\%$$ 75 % of all incorrect orthology assignments can be detected in this manner. All results rely only on the structure of the BMGs and not on any a priori knowledge about underlying gene or species trees.

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Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽

10.1007/s00453-020-00794-4 ◽

2021 ◽

Author(s):

Britta Dorn ◽

Ronald de Haan ◽

Ildikó Schlotter

Keyword(s):

Control Problem ◽

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Upper Bounds ◽

Fair Allocation ◽

Indivisible Goods ◽

Lower And Upper Bounds ◽

Proportional Allocation ◽

Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

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Liveness in broadcast networks

Computing ◽

10.1007/s00607-021-00986-y ◽

2021 ◽

Author(s):

Peter Chini ◽

Roland Meyer ◽

Prakash Saivasan

Keyword(s):

Model Checking ◽

Polynomial Time ◽

Message Passing ◽

Linear Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Final State ◽

Broadcast Network ◽

Broadcast Networks

AbstractWe study liveness and model checking problems for broadcast networks, a system model of identical clients communicating via message passing. The first problem that we consider is Liveness Verification. It asks whether there is a computation such that one clients visits a final state infinitely often. The complexity of the problem has been open. It was shown to be $$\texttt {P}$$ P -hard but in $$\texttt {EXPSPACE}$$ EXPSPACE . We close the gap by a polynomial-time algorithm. The latter relies on a characterization of live computations in terms of paths in a suitable graph, combined with a fixed-point iteration to efficiently check the existence of such paths. The second problem is Fair Liveness Verification. It asks for a computation where all participating clients visit a final state infinitely often. We adjust the algorithm to also solve fair liveness in polynomial time. Both problems can be instrumented to answer model checking questions for broadcast networks against linear time temporal logic specifications. The first problem in this context is Fair Model Checking. It demands that for all computations of a broadcast network, all participating clients satisfy the specification. We solve the problem via the Vardi–Wolper construction and a reduction to Liveness Verification. The second problem is Sparse Model Checking. It asks whether each computation has a participating client that satisfies the specification. We reduce the problem to Fair Liveness Verification.

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On the complexity of the balanced vertex ordering problem

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.383 ◽

2007 ◽

Vol Vol. 9 no. 1 (Graph and Algorithms) ◽

Author(s):

Jan Kára ◽

Jan Kratochvil ◽

David R. Wood

Keyword(s):

Polynomial Time ◽

Maximum Degree ◽

Graph Drawing ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Np Hard ◽

Vertex Ordering ◽

The Difference ◽

Balanced Ordering ◽

Applied Math

Graphs and Algorithms International audience We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.

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Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Algorithms ◽

10.3390/a14040110 ◽

2021 ◽

Vol 14 (4) ◽

pp. 110

Author(s):

David Schaller ◽

Manuela Geiß ◽

Marc Hellmuth ◽

Peter F. Stadler

Keyword(s):

Phylogenetic Tree ◽

Polynomial Time ◽

Binary Tree ◽

A Priori ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Minimum Cardinality ◽

Mathematical Phylogenetics ◽

Special Case

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

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Efficient Dynamic Flow Algorithms for Evacuation Planning Problems with Partial Lane Reversal

Mathematics ◽

10.3390/math7100993 ◽

2019 ◽

Vol 7 (10) ◽

pp. 993 ◽

Cited By ~ 6

Author(s):

Urmila Pyakurel ◽

Hari Nandan Nath ◽

Stephan Dempe ◽

Tanka Nath Dhamala

Keyword(s):

Road Network ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Evacuation Planning ◽

Dynamic Flow ◽

The Road ◽

Flow Problems ◽

Dynamic Flows ◽

The Difference ◽

Planning Problems

Contraflow technique has gained a considerable focus in evacuation planning research over the past several years. In this work, we design efficient algorithms to solve the maximum, lex-maximum, earliest arrival, and quickest dynamic flow problems having constant attributes and their generalizations with partial contraflow reconfiguration in the context of evacuation planning. The partial static contraflow problems, that are foundations to the dynamic flows, are also studied. Moreover, the contraflow model with inflow-dependent transit time on arcs is introduced. A strongly polynomial time algorithm to compute approximate solution of the quickest partial contraflow problem on two terminal networks is presented, which is substantiated by numerical computations considering Kathmandu road network as an evacuation network. Our results show that the quickest time to evacuate a flow of value 100,000 units is reduced by more than 42% using the partial contraflow technique, and the difference is more with the increase in the flow value. Moreover, the technique keeps the record of the portions of the road network not used by the evacuees.

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What's the Matter with Tie-Breaking? Improving Efficiency in School Choice

The American Economic Review ◽

10.1257/aer.98.3.669 ◽

2008 ◽

Vol 98 (3) ◽

pp. 669-689 ◽

Cited By ~ 160

Author(s):

Aytek Erdil ◽

Haluk Ergin

Keyword(s):

United States ◽

School Choice ◽

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

The United States ◽

Stable Mechanism ◽

Deferred Acceptance Algorithm ◽

Deferred Acceptance ◽

Stable Improvement

In several school choice districts in the United States, the student proposing deferred acceptance algorithm is applied after indifferences in priority orders are broken in some exogenous way. Although such a tie-breaking procedure preserves stability, it adversely affects the welfare of the students since it introduces artificial stability constraints. Our main finding is a polynomial-time algorithm for the computation of a student-optimal stable matching when priorities are weak. The idea behind our construction relies on a new notion which we call a stable improvement cycle. We also investigate the strategic properties of the student-optimal stable mechanism. (JEL C78, D82, I21)

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Uniquely monopolar-partitionable block graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2073 ◽

2014 ◽

Vol Vol. 16 no. 2 (PRIMA 2013) ◽

Author(s):

Xuegang Chen ◽

Jing Huang

Keyword(s):

Polynomial Time Algorithm ◽

Time Algorithm ◽

Special Issue ◽

Block Graphs ◽

International Audience ◽

Split Graphs ◽

Np Complete ◽

Vertex Partitions ◽

General Graphs

Special issue PRIMA 2013 International audience As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

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