We consider the computational complexity of a problem modeling bribery in the context of voting systems. In the scenario of SWAP BRIBERY, each voter assigns a certain price for swapping the positions of two consecutive candidates in his preference ranking. The question is whether it is possible, without exceeding a given budget, to bribe the voters in a way that the preferred candidate wins in the election.We initiate a parameterized and multivariate complexity analysis of SWAP BRIBERY, focusing on the case of k-approval. We investigate how different cost functions affect the computational complexity of the problem. We identify a special case of k-approval for which the problem can be solved in polynomial time, whereas we prove NP-hardness for a slightly more general scenario. We obtain fixed-parameter tractability as well as W[1]-hardness results for certain natural parameters.
In practical applications, algorithms for the classic version of the hospitals residents problem (the many-one version of the stable marriage problem) may have to be extended to accommodate the needs of couples who wish to be allocated to (geographically) compatible places. Such an extension has been in operation in the National Resident Matching Problem (NRMP) matching scheme in the United States for a number of years. In this setting, a stable matching need not exist, and it is an NP-complete problem to decide if one does. However, the only previous empirical study in this context (focused on the NRMP algorithm), together with information from NRMP, suggest that, in practice, stable matchings do exist and that an appropriate heuristic can be used to find such a matching.The study presented here was motivated by the recent decision to accommodate couples in the Scottish Foundation Allocation Scheme (SFAS), the Scottish equivalent of the NRMP. Here, the problem is a special case, since hospital preferences are derived from a "master list" of resident scores, but we show that the existence problem remains NP-complete in this case. We describe the algorithm used in SFAS and contrast it with a version of the algorithm that forms the basis of the NRMP approach. We also propose a third simpler algorithm based on satisfying blocking pairs, and an FPT algorithm when the number of couples is viewed as a parameter. We present an empirical study of the performance of a number of variants of these algorithms using a range of datasets. The results indicate that, not surprisingly, increasing the ratio of couples to single applicants typically makes it harder to find a stable matching (and, by inference, less likely that a stable matching exists). However, the likelihood of finding a stable matching is very high for realistic values of this ratio, and especially so for particular variants of the algorithms.
In the k-APEX problem the task is to find at most k vertices whose deletion makes the given graph planar. The graphs for which there exists a solution form a minor closed class of graphs, hence by the deep results of Robertson and Seymour (J. Comb. Theory, Ser. B 63(1):65-110, 1995; J. Comb. Theory, Ser. B 92(2): 2004), there is a cubic algorithm for every fixed value of k. However, the proof is extremely complicated and the constants hidden by the big-O notation are huge. Here we give a much simpler algorithm for this problem with quadratic running time, by iteratively reducing the input graph and then applying techniques for graphs of bounded treewidth.
We consider the variant of the classical Stable Marriage problem where preference lists can be incomplete and may contain ties. In such a setting, finding a stable matching of maximum size is NP-hard. We study the parameterized complexity of this problem, where the parameter can be the number of ties, the maximum or the overall length of ties. We also investigate the applicability of a local search algorithm for the problem. Finally, we examine the possibilities for giving an FPT algorithm or an FPT approximation algorithm for finding an egalitarian or a minimum regret matching.
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