142 research outputs found
Degreewidth: a New Parameter for Solving Problems on Tournaments
In the paper, we define a new parameter for tournaments called degreewidth
which can be seen as a measure of how far is the tournament from being acyclic.
The degreewidth of a tournament denoted by is the minimum value
for which we can find an ordering of the
vertices of such that every vertex is incident to at most backward arcs
(\textit{i.e.} an arc such that ). Thus, a tournament is
acyclic if and only if its degreewidth is zero.
Additionally, the class of sparse tournaments defined by Bessy et al. [ESA
2017] is exactly the class of tournaments with degreewidth one.
We first study computational complexity of finding degreewidth. Namely, we
show it is NP-hard and complement this result with a -approximation
algorithm. We also provide a cubic algorithm to decide if a tournament is
sparse.
Finally, we study classical graph problems \textsc{Dominating Set} and
\textsc{Feedback Vertex Set} parameterized by degreewidth. We show the former
is fixed parameter tractable whereas the latter is NP-hard on sparse
tournaments. Additionally, we study \textsc{Feedback Arc Set} on sparse
tournaments
On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width
The relational width of a finite structure, if bounded, is always (1,1) or
(2,3). In this paper we study the relational width of first-order expansions of
finitely bounded homogeneous binary cores where binary cores are structures
with equality and some anti-reflexive binary relations such that for any two
different elements a, b in the domain there is exactly one binary relation R
with (a, b) in R.
Our main result is that first-order expansions of liberal finitely bounded
homogeneous binary cores with bounded strict width have relational width (2,
MaxBound) where MaxBound is the size of the largest forbidden substructure, but
is not less than 3, and liberal stands for structures that do not forbid
certain finite structures of small size. This result is built on a new approach
and concerns a broad class of structures including reducts of homogeneous
digraphs for which the CSP complexity classification has not yet been obtained.Comment: A long version of an extended abstract that appeared in LICS 202
The "No Justice in the Universe" phenomenon: why honesty of effort may not be rewarded in tournaments
In 2000 Allen Schwenk, using a well-known mathematical model of matchplay
tournaments in which the probability of one player beating another in a single
match is fixed for each pair of players, showed that the classical
single-elimination, seeded format can be "unfair" in the sense that situations
can arise where an indisputibly better (and thus higher seeded) player may have
a smaller probability of winning the tournament than a worse one. This in turn
implies that, if the players are able to influence their seeding in some
preliminary competition, situations can arise where it is in a player's
interest to behave "dishonestly", by deliberately trying to lose a match. This
motivated us to ask whether it is possible for a tournament to be both honest,
meaning that it is impossible for a situation to arise where a rational player
throws a match, and "symmetric" - meaning basically that the rules treat
everyone the same - yet unfair, in the sense that an objectively better player
has a smaller probability of winning than a worse one. After rigorously
defining our terms, our main result is that such tournaments exist and we
construct explicit examples for any number n >= 3 of players. For n=3, we show
(Theorem 3.6) that the collection of win-probability vectors for such
tournaments form a 5-vertex convex polygon in R^3, minus some boundary points.
We conjecture a similar result for any n >= 4 and prove some partial results
towards it.Comment: 26 pages, 2 figure
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