8 research outputs found
Sprague-Grundy values and complexity for LCTR
Given a Young diagram on boxes as a non-increasing sequence of integers,
we consider the impartial combinatorial game LCTR in which moves consist of
removing either the left column or top row of boxes. We show that for both
normal and mis\`ere play, the optimal strategy can consist mostly of mirroring
the opponent's moves. This allows for computing the Sprague-Grundy value of the
given game in time units, where time unit allows for reading an
integer, or performing a basic arithmetic operation. This improves on the
previous bound of time units, due to by Ili\'c (2019), which can be
obtained by an improvement of the Sprague-Grundy recursion.Comment: 24 pages, 7 figures, 1 tabl
Minimizing Maximum Dissatisfaction in the Allocation of Indivisible Items under a Common Preference Graph
We consider the task of allocating indivisible items to agents, when the
agents' preferences over the items are identical. The preferences are captured
by means of a directed acyclic graph, with vertices representing items and an
edge , meaning that each of the agents prefers item over item .
The dissatisfaction of an agent is measured by the number of items that the
agent does not receive and for which it also does not receive any more
preferred item. The aim is to allocate the items to the agents in a fair way,
i.e., to minimize the maximum dissatisfaction among the agents. We study the
status of computational complexity of that problem and establish the following
dichotomy: the problem is NP-hard for the case of at least three agents, even
on fairly restricted graphs, but polynomially solvable for two agents. We also
provide several polynomial-time results with respect to different underlying
graph structures, such as graphs of width at most two and tree-like structures
such as stars and matchings. These findings are complemented with fixed
parameter tractability results related to path modules and independent set
modules. Techniques employed in the paper include bottleneck assignment
problem, greedy algorithm, dynamic programming, maximum network flow, and
integer linear programming.Comment: 26 pages, 2 figure
Strong cliques in diamond-free graphs
A strong clique in a graph is a clique intersecting all inclusion-maximal
stable sets. Strong cliques play an important role in the study of perfect
graphs. We study strong cliques in the class of diamond-free graphs, from both
structural and algorithmic points of view. We show that the following five
NP-hard or co-NP-hard problems remain intractable when restricted to the class
of diamond-free graphs: Is a given clique strong? Does the graph have a strong
clique? Is every vertex contained in a strong clique? Given a partition of the
vertex set into cliques, is every clique in the partition strong? Can the
vertex set be partitioned into strong cliques? On the positive side, we show
that the following two problems whose computational complexity is open in
general can be solved in linear time in the class of diamond-free graphs: Is
every maximal clique strong? Is every edge contained in a strong clique? These
results are derived from a characterization of diamond-free graphs in which
every maximal clique is strong, which also implies an improved Erd\H{o}s-Hajnal
property for such graphs.Comment: An extended abstract of this work was accepted at WG 202
Sprague-Grundy function of symmetric hypergraphs
We consider a generalization of the classical game of called hypergraph
. Given a hypergraph \cH on the ground set of
piles of stones, two players alternate in choosing a hyperedge H \in \cH and
strictly decreasing all piles . The player who makes the last move is
the winner. Recently it was shown that for many classes of hypergraphs the
Sprague-Grundy function of the corresponding game is given by the formula
introduced originally by Jenkyns and Mayberry (1980). In this paper we
characterize symmetric hypergraphs for which the Sprague-Grundy function is
described by the same formula