10,447 research outputs found
Packing odd -joins with at most two terminals
Take a graph , an edge subset , and a set of
terminals where is even. The triple is
called a signed graft. A -join is odd if it contains an odd number of edges
from . Let be the maximum number of edge-disjoint odd -joins.
A signature is a set of the form where and is even. Let be the minimum cardinality a -cut
or a signature can achieve. Then and we say that
packs if equality holds here.
We prove that packs if the signed graft is Eulerian and it
excludes two special non-packing minors. Our result confirms the Cycling
Conjecture for the class of clutters of odd -joins with at most two
terminals. Corollaries of this result include, the characterizations of weakly
and evenly bipartite graphs, packing two-commodity paths, packing -joins
with at most four terminals, and a new result on covering edges with cuts.Comment: extended abstract appeared in IPCO 2014 (under the different title
"the cycling property for the clutter of odd st-walks"
On shortest -joins and packing -cuts
We give a class of graphs with the property that for each even set T of nodes in G the minimum length of a T-join is equal to the maximum number of pairwise edge disjoint T-cuts. Our class contains the bipartite and the series-parallel graphs for which this property was derived earlier by Seymour
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
Clean clutters and dyadic fractional packings
A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three and no dyadic fractional packing of value three. Examples of clean clutters include ideal clutters, binary clutters, and clutters without an intersecting minor. Our proof is constructive and leads naturally to an albeit exponential algorithm. We improve the running time to quasi-polynomial in the rank of the input, and to polynomial in the binary cas
Single Commodity Flow Algorithms for Lifts of Graphic and Cographic Matroids
Consider a binary matroid M given by its matrix representation. We show that if M is a lift of a graphic or a cographic matroid, then in polynomial time we can either solve the single commodity flow problem for M or find an obstruction for which the Max-Flow Min-Cut relation does not hold. The key tool is an algorithmic version of Lehman's Theorem for the set covering polyhedron
Visual Inspection Algorithms for Printed Circuit Board Patterns A SURVEY
The importance of the inspection process has been magnified by the requirements of the modern manufacturing environment. In electronics mass-production manufacturing facilities, an attempt is often made to achieve 100 % quality assurance of all parts, subassemblies, and finished goods. A variety of approaches for automated visual inspection of printed circuits have been reported over the last two decades. In this survey, algorithms and techniques for the automated inspection of printed circuit boards are examined. A classification tree for these algorithms is presented and the algorithms are grouped according to this classification. This survey concentrates mainly on image analysis and fault detection strategies, these also include the state-of-the-art techniques. Finally, limitations of current inspection systems are summarized
- …