332 research outputs found
Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT
In directed graphs, we investigate the problems of finding: 1) a minimum
feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2)
a feedback vertex set inducing an acyclic graph (also called the Vertex
2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a
minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS).
We show that these problems are strongly related to (variants of) Monotone
3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in
positive form. As a consequence, we deduce several NP-completeness results on
restricted versions of these problems. In particular, we define the 2-Choice
version of an optimization problem to be its restriction where the optimum
value is known to be either D or D+1 for some integer D, and the problem is
reduced to decide which of D or D+1 is the optimum value. We show that the
2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones
Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of
Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth
assignment minimize the number of variables set to true.
Finally, we propose two classes of directed graphs for which Acyclic FVS is
polynomially solvable, namely flow reducible graphs (for which MFVS is already
known to be polynomially solvable) and C1P-digraphs (defined by an adjacency
matrix with the Consecutive Ones Property)
Unravelling Expressive Delegations: Complexity and Normative Analysis
We consider binary group decision-making under a rich model of liquid
democracy recently proposed by Colley, Grandi, and Novaro (2022): agents submit
ranked delegation options, where each option may be a function of multiple
agents' votes; e.g., "I vote yes if a majority of my friends vote yes." Such
ballots are unravelled into a profile of direct votes by selecting one entry
from each ballot so as not to introduce cyclic dependencies. We study
delegation via monotonic Boolean functions, and two unravelling procedures:
MinSum, which minimises the sum of the ranks of the chosen entries, and its
egalitarian counterpart, MinMax. We provide complete computational dichotomies:
MinSum is hard to compute (and approximate) as soon as any non-trivial
functions are permitted, and polynomial otherwise; for MinMax the easiness
results extend to arbitrary-arity logical ORs and ANDs taken in isolation, but
not beyond. For the classic model of delegating to individual agents, we give
asymptotically near-tight algorithms for carrying out the two procedures and
efficient algorithms for finding optimal unravellings with the highest vote
count for a given alternative. These algorithms inspire novel tie-breaking
rules for the setup of voting to change a status quo. We then introduce a new
axiom, which can be viewed as a variant of the participation axiom, and use
algorithmic techniques developed earlier in the paper to show that it is
satisfied by MinSum and a lexicographic refinement of MinMax (but not MinMax
itself).Comment: To appear in AAAI'2
The Parameterized Complexity of Degree Constrained Editing Problems
This thesis examines degree constrained editing problems within the framework of parameterized complexity. A degree constrained editing problem takes as input a graph and a set of constraints and asks whether the graph can be altered in at most k editing steps such that the degrees of the remaining vertices are within the given constraints. Parameterized complexity gives a framework for examining
problems that are traditionally considered intractable and developing efficient exact algorithms for them, or showing that it is unlikely that they have such algorithms, by introducing an additional component to the input, the parameter, which gives additional information about the structure of the problem. If the problem has an algorithm that is exponential in the parameter, but polynomial, with constant degree, in the size of the input, then it is considered to be fixed-parameter tractable.
Parameterized complexity also provides an intractability framework for identifying problems that are likely to not have such an algorithm.
Degree constrained editing problems provide natural parameterizations in terms of the total cost k of vertex deletions, edge deletions and edge additions allowed, and
the upper bound r on the degree of the vertices remaining after editing. We define a class of degree constrained editing problems, WDCE, which generalises several well know problems, such as Degree r Deletion, Cubic Subgraph, r-Regular Subgraph, f-Factor and General Factor. We show that in general if both k and r are part of the parameter, problems in the WDCE class are fixed-parameter tractable, and if parameterized by k or r alone, the problems are intractable in a parameterized sense.
We further show cases of WDCE that have polynomial time kernelizations, and in particular when all the degree constraints are a single number and the editing
operations include vertex deletion and edge deletion we show that there is a kernel with at most O(kr(k + r)) vertices. If we allow vertex deletion and edge addition,
we show that despite remaining fixed-parameter tractable when parameterized by k and r together, the problems are unlikely to have polynomial sized kernelizations, or
polynomial time kernelizations of a certain form, under certain complexity theoretic assumptions.
We also examine a more general case where given an input graph the question is whether with at most k deletions the graph can be made r-degenerate. We show that in this case the problems are intractable, even when r is a constant
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