23 research outputs found
Adapting Stable Matchings to Forced and Forbidden Pairs
We introduce the problem of adapting a stable matching to forced and
forbidden pairs. Specifically, given a stable matching , a set of
forced pairs, and a set of forbidden pairs, we want to find a stable
matching that includes all pairs from , no pair from , and that is as
close as possible to . We study this problem in four classical stable
matching settings: Stable Roommates (with Ties) and Stable Marriage (with
Ties). As our main contribution, we develop an algorithmic technique to
"propagate" changes through a stable matching. This technique is at the core of
our polynomial-time algorithm for adapting Stable Roommates matchings to forced
pairs. In contrast to this, we show that the same problem for forbidden pairs
is NP-hard. However, our propagation technique allows for a fixed-parameter
tractable algorithm with respect to the number of forbidden pairs when both
forced and forbidden pairs are present. Moreover, we establish strong
intractability results when preferences contain ties
Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems
When computing stable matchings, it is usually assumed that the preferences
of the agents in the matching market are fixed. However, in many realistic
scenarios, preferences change over time. Consequently, an initially stable
matching may become unstable. Then, a natural goal is to find a matching which
is stable with respect to the modified preferences and as close as possible to
the initial one. For Stable Marriage/Roommates, this problem was formally
defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI
'20]. As they showed that Incremental Stable Roommates and Incremental Stable
Marriage with Ties are NP-hard, we focus on the parameterized complexity of
these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We
show that Incremental Stable Roommates is W[1]-hard parameterized by the number
of changes in the preferences, yet admits an intricate XP-algorithm, and we
show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by
the number of ties. Furthermore, we analyze the influence of the degree of
"similarity" between the agents' preference lists, identifying several
polynomial-time solvable and fixed-parameter tractable cases, but also proving
that Incremental Stable Roommates and Incremental Stable Marriage with Ties
parameterized by the number of different preference lists are W[1]-hard.Comment: Accepted to MFCS'2
Fully Polynomial-Time Algorithms Parameterized by Vertex Integrity Using Fast Matrix Multiplication
Parameterized Complexity of Stable Roommates with Ties and Incomplete Lists Through the Lens of Graph Parameters
We continue and extend previous work on the parameterized complexity analysis of the NP-hard Stable Roommates with Ties and Incomplete Lists problem, thereby strengthening earlier results both on the side of parameterized hardness as well as on the side of fixed-parameter tractability. Other than for its famous sister problem Stable Marriage which focuses on a bipartite scenario, Stable Roommates with Incomplete Lists allows for arbitrary acceptability graphs whose edges specify the possible matchings of each two agents (agents are represented by graph vertices). Herein, incomplete lists and ties reflect the fact that in realistic application scenarios the agents cannot bring all other agents into a linear order. Among our main contributions is to show that it is W[1]-hard to compute a maximum-cardinality stable matching for acceptability graphs of bounded treedepth, bounded tree-cut width, and bounded feedback vertex number (these are each time the respective parameters). However, if we "only" ask for perfect stable matchings or the mere existence of a stable matching, then we obtain fixed-parameter tractability with respect to tree-cut width but not with respect to treedepth. On the positive side, we also provide fixed-parameter tractability results for the parameter feedback edge set number
No Polynomial Kernels for Knapsack
This paper focuses on kernelization algorithms for the fundamental Knapsack
problem. A kernelization algorithm (or kernel) is a polynomial-time reduction
from a problem onto itself, where the output size is bounded by a function of
some problem-specific parameter. Such algorithms provide a theoretical model
for data reduction and preprocessing and are central in the area of
parameterized complexity. In this way, a kernel for Knapsack for some parameter
reduces any instance of Knapsack to an equivalent instance of size at most
in polynomial time, for some computable function . When
then we call such a reduction a polynomial kernel.
Our study focuses on two natural parameters for Knapsack: The number of
different item weights , and the number of different item profits
. Our main technical contribution is a proof showing that Knapsack does
not admit a polynomial kernel for any of these two parameters under standard
complexity-theoretic assumptions. Our proof discovers an elaborate application
of the standard kernelization lower bound framework, and develops along the way
novel ideas that should be useful for other problems as well. We complement our
lower bounds by showing the Knapsack admits a polynomial kernel for the
combined parameter