scholarly journals The Parameterized Complexity of Connected Fair Division

2021 ◽  
Author(s):  
Argyrios Deligkas ◽  
Eduard Eiben ◽  
Robert Ganian ◽  
Thekla Hamm ◽  
Sebastian Ordyniak
Keyword(s):  
Lower Bounds ◽  
Parameterized Complexity ◽  
Fundamental Problem ◽  
Fair Division ◽  
Division Problem ◽  
Additional Parameter ◽  
Fixed Parameter Tractability ◽  
Connected Subgraph ◽  
Fixed Parameter ◽  
New Algorithms

We study the Connected Fair Division problem (CFD), which generalizes the fundamental problem of fairly allocating resources to agents by requiring that the items allocated to each agent form a connected subgraph in a provided item graph G. We expand on previous results by providing a comprehensive complexity-theoretic understanding of CFD based on several new algorithms and lower bounds while taking into account several well-established notions of fairness: proportionality, envy-freeness, EF1 and EFX. In particular, we show that to achieve tractability, one needs to restrict both the agents and the item graph in a meaningful way. We design (XP)-algorithms for the problem parameterized by (1) clique-width of G plus the number of agents and (2) treewidth of G plus the number of agent types, along with corresponding lower bounds. Finally, we show that to achieve fixed-parameter tractability, one needs to not only use a more restrictive parameterization of G, but also include the maximum item valuation as an additional parameter.

scholarly journals Fixed-parameter tractability and lower bounds for stabbing problems

2013 ◽  
Vol 46 (7) ◽  
pp. 839-860 ◽  
Author(s):  
Panos Giannopoulos ◽  
Christian Knauer ◽  
Günter Rote ◽  
Daniel Werner
Keyword(s):  
Lower Bounds ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter

Chapter 17. Fixed-Parameter Tractability

2021 ◽  
Author(s):  
Marko Samer ◽  
Stefan Szeider
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Complexity Analysis ◽  
Fixed Parameter Tractability ◽  
Graph Representations ◽  
Fine Grained ◽  
Input Size ◽  
Backdoor Sets ◽  
Fixed Parameter ◽  
Problem Instances

Parameterized complexity is a new theoretical framework that considers, in addition to the overall input size, the effects on computational complexity of a secondary measurement, the parameter. This two-dimensional viewpoint allows a fine-grained complexity analysis that takes structural properties of problem instances into account. The central notion is “fixed-parameter tractability” which refers to solvability in polynomial time for each fixed value of the parameter such that the order of the polynomial time bound is independent of the parameter. This chapter presents main concepts and recent results on the parameterized complexity of the satisfiability problem and it outlines fundamental algorithmic ideas that arise in this context. Among the parameters considered are the size of backdoor sets with respect to various tractable base classes and the treewidth of graph representations of satisfiability instances.

scholarly journals Parameterized Algorithms for Finding a Collective Set of Items

2020 ◽  
Vol 34 (02) ◽  
pp. 1838-1845
Author(s):  
Robert Bredereck ◽  
Piotr Faliszewski ◽  
Andrzej Kaczmarczyk ◽  
Dušan Knop ◽  
Rolf Niedermeier
Keyword(s):  
Parameterized Complexity ◽  
Weighted Average ◽  
Fixed Parameter Tractability ◽  
Parameterized Algorithms ◽  
Utility Value ◽  
Average Operator ◽  
Fixed Parameter ◽  
Ordered Weighted Average ◽  
Integer Weight

We extend the work of Skowron et al. (AIJ, 2016) by considering the parameterized complexity of the following problem. We are given a set of items and a set of agents, where each agent assigns an integer utility value to each item. The goal is to find a set of k items that these agents would collectively use. For each such collective set of items, each agent provides a score that can be described using an OWA (ordered weighted average) operator and we seek a set with the highest total score. We focus on the parameterization by the number of agents and we find numerous fixed-parameter tractability results (however, we also find some W[1]-hardness results). It turns out that most of our algorithms even apply to the setting where each agent has an integer weight.

scholarly journals On the Computation of Fully Proportional Representation

2013 ◽  
Vol 47 ◽  
pp. 475-519 ◽  
Author(s):  
N. Betzler ◽  
A. Slinko ◽  
J. Uhlmann
Keyword(s):  
Polynomial Time ◽  
Parameterized Complexity ◽  
Proportional Representation ◽  
Fixed Parameter Tractability ◽  
Np Hard ◽  
Winner Determination ◽  
Negative Results ◽  
Polynomial Time Algorithms ◽  
Fixed Parameter

We investigate two systems of fully proportional representation suggested by Chamberlin Courant and Monroe. Both systems assign a representative to each voter so that the "sum of misrepresentations" is minimized. The winner determination problem for both systems is known to be NP-hard, hence this work aims at investigating whether there are variants of the proposed rules and/or specific electorates for which these problems can be solved efficiently. As a variation of these rules, instead of minimizing the sum of misrepresentations, we considered minimizing the maximal misrepresentation introducing effectively two new rules. In the general case these "minimax" versions of classical rules appeared to be still NP-hard. We investigated the parameterized complexity of winner determination of the two classical and two new rules with respect to several parameters. Here we have a mixture of positive and negative results: e.g., we proved fixed-parameter tractability for the parameter the number of candidates but fixed-parameter intractability for the number of winners. For single-peaked electorates our results are overwhelmingly positive: we provide polynomial-time algorithms for most of the considered problems. The only rule that remains NP-hard for single-peaked electorates is the classical Monroe rule.

scholarly journals Semantic Width and the Fixed-Parameter Tractability of Constraint Satisfaction Problems

2020 ◽  
Author(s):  
Hubie Chen ◽  
Georg Gottlob ◽  
Matthias Lanzinger ◽  
Reinhard Pichler
Keyword(s):  
Constraint Satisfaction ◽  
Fundamental Problem ◽  
Constraint Satisfaction Problems ◽  
Fixed Parameter Tractability ◽  
Scheduling Problems ◽  
Fixed Parameter Tractable ◽  
Database Theory ◽  
Formal Framework ◽  
Fixed Parameter

Constraint satisfaction problems (CSPs) are an important formal framework for the uniform treatment of various prominent AI tasks, e.g., coloring or scheduling problems. Solving CSPs is, in general, known to be NP-complete and fixed-parameter intractable when parameterized by their constraint scopes. We give a characterization of those classes of CSPs for which the problem becomes fixed-parameter tractable. Our characterization significantly increases the utility of the CSP framework by making it possible to decide the fixed-parameter tractability of problems via their CSP formulations. We further extend our characterization to the evaluation of unions of conjunctive queries, a fundamental problem in databases. Furthermore, we provide some new insight on the frontier of PTIME solvability of CSPs. In particular, we observe that bounded fractional hypertree width is more general than bounded hypertree width only for classes that exhibit a certain type of exponential growth. The presented work resolves a long-standing open problem and yields powerful new tools for complexity research in AI and database theory.

scholarly journals Maximin Share Allocations on Cycles

2020 ◽  
Vol 69 ◽  
pp. 613-655
Author(s):  
Miroslaw Truszczynski ◽  
Zbigniew Lonc
Keyword(s):  
Resource Allocation ◽  
Social Choice ◽  
Fundamental Problem ◽  
Fair Division ◽  
Connected Subgraph ◽  
Indivisible Goods ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Multi Agent ◽  
Share Value

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

scholarly journals Solving Integer Quadratic Programming via Explicit and Structural Restrictions

2019 ◽  
Vol 33 ◽  
pp. 1477-1484
Author(s):  
Eduard Eiben ◽  
Robert Ganian ◽  
Dusan Knop ◽  
Sebastian Ordyniak
Keyword(s):  
Quadratic Programming ◽  
Lower Bounds ◽  
Parameterized Complexity ◽  
Integer Quadratic Programming ◽  
New Algorithms ◽  
Variable Interactions

We study the parameterized complexity of Integer Quadratic Programming under two kinds of restrictions: explicit restrictions on the domain or coefficients, and structural restrictions on variable interactions. We argue that both kinds of restrictions are necessary to achieve tractability for Integer Quadratic Programming, and obtain four new algorithms for the problem that are tuned to possible explicit restrictions of instances that we may wish to solve. The presented algorithms are exact, deterministic, and complemented by appropriate lower bounds.

scholarly journals Maximin Share Allocations on Cycles

2018 ◽  
Author(s):  
Zbigniew Lonc ◽  
Miroslaw Truszczynski
Keyword(s):  
Social Choice ◽  
Polynomial Time ◽  
Complete Graph ◽  
Original Problem ◽  
Fundamental Problem ◽  
Fair Division ◽  
Connected Subgraph ◽  
Indivisible Goods ◽  
Share Value

The problem of fair division of indivisible goods is a fundamental problem of social choice. Recently, the problem was extended to the setting when goods form a graph and the goal is to allocate goods to agents so that each agent's bundle forms a connected subgraph. Researchers proved that, unlike in the original problem (which corresponds to the case of the complete graph in the extended setting), in the case of the goods-graph being a tree, allocations offering each agent a bundle of or exceeding her maximin share value always exist. Moreover, they can be found in polynomial time. We consider here the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide results on allocations guaranteeing each agent a certain portion of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

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