1,256 research outputs found
The Computational Complexity of the Game of Set and its Theoretical Applications
The game of SET is a popular card game in which the objective is to form Sets
using cards from a special deck. In this paper we study single- and multi-round
variations of this game from the computational complexity point of view and
establish interesting connections with other classical computational problems.
Specifically, we first show that a natural generalization of the problem of
finding a single Set, parameterized by the size of the sought Set is W-hard;
our reduction applies also to a natural parameterization of Perfect
Multi-Dimensional Matching, a result which may be of independent interest.
Second, we observe that a version of the game where one seeks to find the
largest possible number of disjoint Sets from a given set of cards is a special
case of 3-Set Packing; we establish that this restriction remains NP-complete.
Similarly, the version where one seeks to find the smallest number of disjoint
Sets that overlap all possible Sets is shown to be NP-complete, through a close
connection to the Independent Edge Dominating Set problem. Finally, we study a
2-player version of the game, for which we show a close connection to Arc
Kayles, as well as fixed-parameter tractability when parameterized by the
number of rounds played
On the complexity of the multiple stack TSP, kSTSP
The multiple Stack Travelling Salesman Problem, STSP, deals with the collect
and the deliverance of n commodities in two distinct cities. The two cities are
represented by means of two edge-valued graphs (G1,d2) and (G2,d2). During the
pick-up tour, the commodities are stored into a container whose rows are
subject to LIFO constraints. As a generalisation of standard TSP, the problem
obviously is NP-hard; nevertheless, one could wonder about what combinatorial
structure of STSP does the most impact its complexity: the arrangement of the
commodities into the container, or the tours themselves? The answer is not
clear. First, given a pair (T1,T2) of pick-up and delivery tours, it is
polynomial to decide whether these tours are or not compatible. Second, for a
given arrangement of the commodities into the k rows of the container, the
optimum pick-up and delivery tours w.r.t. this arrangement can be computed
within a time that is polynomial in n, but exponential in k. Finally, we
provide instances on which a tour that is optimum for one of three distances
d1, d2 or d1+d2 lead to solutions of STSP that are arbitrarily far to the
optimum STSP
A Memetic Algorithm for the Multidimensional Assignment Problem
The Multidimensional Assignment Problem (MAP or s-AP in the case of s
dimensions) is an extension of the well-known assignment problem. The most
studied case of MAP is 3-AP, though the problems with larger values of s have
also a number of applications. In this paper we propose a memetic algorithm for
MAP that is a combination of a genetic algorithm with a local search procedure.
The main contribution of the paper is an idea of dynamically adjusted
generation size, that yields an outstanding flexibility of the algorithm to
perform well for both small and large fixed running times. The results of
computational experiments for several instance families show that the proposed
algorithm produces solutions of very high quality in a reasonable time and
outperforms the state-of-the art 3-AP memetic algorithm.Comment: 14 page
Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints
For a given graph with positive integral cost and delay on edges,
distinct vertices and , cost bound and delay bound , the bi-constraint path (BCP) problem is to compute disjoint
-paths subject to and . This problem is known NP-hard, even when
\cite{garey1979computers}. This paper first gives a simple approximation
algorithm with factor-, i.e. the algorithm computes a solution with
delay and cost bounded by and respectively. Later, a novel improved
approximation algorithm with ratio
is developed by constructing
interesting auxiliary graphs and employing the cycle cancellation method. As a
consequence, we can obtain a factor- approximation algorithm by
setting and a factor- algorithm by
setting . Besides, by setting , an
approximation algorithm with ratio , i.e. an algorithm with
only a single factor ratio on cost, can be immediately obtained. To
the best of our knowledge, this is the first non-trivial approximation
algorithm for the BCP problem that strictly obeys the delay constraint.Comment: 12 page
The complexity of graph contractions
For a fixed pattern graph H, let H-CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. We continue a line of research that was started in 1987 by Brouwer & Veldman, and we determine the computational complexity of H-CONTRACTIBILITY for certain classes of pattern graphs. In particular, we pin-point the complexity for all graphs H with five vertices.
Interestingly, in all cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP-complete, the pattern graph H does not have a dominating vertex
Embedding Vertices at Points: Few Bends Suffice for Planar Graphs
The existing literature gives efficient algorithms for mapping trees or less restrictively outerplanar graphs on a given set of points in a plane, so that the edges are drawn planar and as straight lines. We relax the latter requirement and allow very few bends on each edge while considering general plane graphs. Our results show two algorithms for mapping four-connected plane graphs with at most one bend per edge and for mapping general plane graphs with at most two bends per edge. Furthermore we give a point set, where for arbitrary plane graphs it is NP-complete to decide whether there is an mapping such that each edge has at most one bend
Fast algorithms for min independent dominating set
We first devise a branching algorithm that computes a minimum independent
dominating set on any graph with running time O*(2^0.424n) and polynomial
space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A
branch-and-reduce algorithm for finding a minimum independent dominating set in
graphs, Proc. WG'06). We then show that, for every r>3, it is possible to
compute an r-((r-1)/r)log_2(r)-approximate solution for min independent
dominating set within time O*(2^(nlog_2(r)/r))
On Approximating Restricted Cycle Covers
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L. The weight of a cycle cover of an edge-weighted
graph is the sum of the weights of its edges.
We come close to settling the complexity and approximability of computing
L-cycle covers. On the one hand, we show that for almost all L, computing
L-cycle covers of maximum weight in directed and undirected graphs is APX-hard
and NP-hard. Most of our hardness results hold even if the edge weights are
restricted to zero and one.
On the other hand, we show that the problem of computing L-cycle covers of
maximum weight can be approximated within a factor of 2 for undirected graphs
and within a factor of 8/3 in the case of directed graphs. This holds for
arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change
Phase transitions for the cavity approach to the clique problem on random graphs
We give a rigorous proof of two phase transitions for a disordered system
designed to find large cliques inside Erdos random graphs. Such a system is
associated with a conservative probabilistic cellular automaton inspired by the
cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure
The zero exemplar distance problem
Given two genomes with duplicate genes, \textsc{Zero Exemplar Distance} is
the problem of deciding whether the two genomes can be reduced to the same
genome without duplicate genes by deleting all but one copy of each gene in
each genome. Blin, Fertin, Sikora, and Vialette recently proved that
\textsc{Zero Exemplar Distance} for monochromosomal genomes is NP-hard even if
each gene appears at most two times in each genome, thereby settling an
important open question on genome rearrangement in the exemplar model. In this
paper, we give a very simple alternative proof of this result. We also study
the problem \textsc{Zero Exemplar Distance} for multichromosomal genomes
without gene order, and prove the analogous result that it is also NP-hard even
if each gene appears at most two times in each genome. For the positive
direction, we show that both variants of \textsc{Zero Exemplar Distance} admit
polynomial-time algorithms if each gene appears exactly once in one genome and
at least once in the other genome. In addition, we present a polynomial-time
algorithm for the related problem \textsc{Exemplar Longest Common Subsequence}
in the special case that each mandatory symbol appears exactly once in one
input sequence and at least once in the other input sequence. This answers an
open question of Bonizzoni et al. We also show that \textsc{Zero Exemplar
Distance} for multichromosomal genomes without gene order is fixed-parameter
tractable if the parameter is the maximum number of chromosomes in each genome.Comment: Strengthened and reorganize
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