775 research outputs found
Computability of simple games: A complete investigation of the sixty-four possibilities
Classify simple games into sixteen "types" in terms of the four conventional
axioms: monotonicity, properness, strongness, and nonweakness. Further classify
them into sixty-four classes in terms of finiteness (existence of a finite
carrier) and algorithmic computability. For each such class, we either show
that it is empty or give an example of a game belonging to it. We observe that
if a type contains an infinite game, then it contains both computable ones and
noncomputable ones. This strongly suggests that computability is logically, as
well as conceptually, unrelated to the conventional axioms.Comment: 25 page
Computability of simple games: A characterization and application to the core
It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable simple games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted
Interval Query Problem on Cube-Free Median Graphs
In this paper, we introduce the \emph{interval query problem} on cube-free
median graphs. Let be a cube-free median graph and be a
commutative semigroup. For each vertex in , we are given an element
in . For each query, we are given two vertices in
and asked to calculate the sum of over all vertices belonging to a
shortest path. This is a common generalization of range query problems on
trees and grids. In this paper, we provide an algorithm to answer each interval
query in time. The required data structure is constructed in
time and space. To obtain our algorithm, we
introduce a new technique, named the \emph{stairs decomposition}, to decompose
an interval of cube-free median graphs into simpler substructures.Comment: ISAAC'21, 21 page
Lipschitz Continuous Algorithms for Graph Problems
It has been widely observed in the machine learning community that a small
perturbation to the input can cause a large change in the prediction of a
trained model, and such phenomena have been intensively studied in the machine
learning community under the name of adversarial attacks. Because graph
algorithms also are widely used for decision making and knowledge discovery, it
is important to design graph algorithms that are robust against adversarial
attacks. In this study, we consider the Lipschitz continuity of algorithms as a
robustness measure and initiate a systematic study of the Lipschitz continuity
of algorithms for (weighted) graph problems.
Depending on how we embed the output solution to a metric space, we can think
of several Lipschitzness notions. We mainly consider the one that is invariant
under scaling of weights, and we provide Lipschitz continuous algorithms and
lower bounds for the minimum spanning tree problem, the shortest path problem,
and the maximum weight matching problem. In particular, our shortest path
algorithm is obtained by first designing an algorithm for unweighted graphs
that are robust against edge contractions and then applying it to the
unweighted graph constructed from the original weighted graph.
Then, we consider another Lipschitzness notion induced by a natural mapping
that maps the output solution to its characteristic vector. It turns out that
no Lipschitz continuous algorithm exists for this Lipschitz notion, and we
instead design algorithms with bounded pointwise Lipschitz constants for the
minimum spanning tree problem and the maximum weight bipartite matching
problem. Our algorithm for the latter problem is based on an LP relaxation with
entropy regularization
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