78,170 research outputs found
Complexity of Chess Domination Problems
We study different domination problems of attacking and non-attacking rooks
and queens on polyominoes and polycubes of all dimensions. Our main result
proves that maximal domination is NP-complete for non-attacking queens and for
non-attacking rooks on polycubes of dimension three and higher. We also analyse
these problems for polyominoes and convex polyominoes, conjecture the
complexity classes and provide a computer tool for investigation. We have also
computed new values for classical queen domination problems on chessboards
(square polyominoes). For our computations, we have translated the problem into
an integer linear programming instance. Finally, using this computational
implementation and the game engine Godot, we have developed a video game of
minimal domination of queens and rooks on randomly generated polyominoes.Comment: 19 pages, 20 figures, 4 tables. Theorem 1 now for d>2, added results
on approximation, fixed typos, reorganised some proof
A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits
We give a nontrivial algorithm for the satisfiability problem for cn-wire
threshold circuits of depth two which is better than exhaustive search by a
factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first
nontrivial satisfiability algorithm for cn-wire threshold circuits of depth
two. The independently interesting problem of the feasibility of sparse 0-1
integer linear programs is a special case. To our knowledge, our algorithm is
the first to achieve constant savings even for the special case of Integer
Linear Programming. The key idea is to reduce the satisfiability problem to the
Vector Domination Problem, the problem of checking whether there are two
vectors in a given collection of vectors such that one dominates the other
component-wise.
We also provide a satisfiability algorithm with constant savings for depth
two circuits with symmetric gates where the total weighted fan-in is at most
cn.
One of our motivations is proving strong lower bounds for TC^0 circuits,
exploiting the connection (established by Williams) between satisfiability
algorithms and lower bounds. Our second motivation is to explore the connection
between the expressive power of the circuits and the complexity of the
corresponding circuit satisfiability problem
The tractability frontier of well-designed SPARQL queries
We study the complexity of query evaluation of SPARQL queries. We focus on
the fundamental fragment of well-designed SPARQL restricted to the AND,
OPTIONAL and UNION operators. Our main result is a structural characterisation
of the classes of well-designed queries that can be evaluated in polynomial
time. In particular, we introduce a new notion of width called domination
width, which relies on the well-known notion of treewidth. We show that, under
some complexity theoretic assumptions, the classes of well-designed queries
that can be evaluated in polynomial time are precisely those of bounded
domination width
Maker-Breaker total domination game
Maker-Breaker total domination game in graphs is introduced as a natural
counterpart to the Maker-Breaker domination game recently studied by Duch\^ene,
Gledel, Parreau, and Renault. Both games are instances of the combinatorial
Maker-Breaker games. The Maker-Breaker total domination game is played on a
graph by two players who alternately take turns choosing vertices of .
The first player, Dominator, selects a vertex in order to totally dominate
while the other player, Staller, forbids a vertex to Dominator in order to
prevent him to reach his goal.
It is shown that there are infinitely many connected cubic graphs in which
Staller wins and that no minimum degree condition is sufficient to guarantee
that Dominator wins when Staller starts the game. An amalgamation lemma is
established and used to determine the outcome of the game played on grids.
Cacti are also classified with respect to the outcome of the game. A connection
between the game and hypergraphs is established. It is proved that the game is
PSPACE-complete on split and bipartite graphs. Several problems and questions
are also posed.Comment: 21 pages, 5 figure
The Complexity of Iterated Strategy Elimination
We consider the computational complexity of the question whether a certain
strategy can be removed from a game by means of iterated elimination of
dominated strategies. In particular, we study the influence of different
definitions of domination and of the number of different payoff values. In
addition, the consequence of restriction to constant-sum games is shown
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
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