88,725 research outputs found
A solvable class of quadratic 0–1 programming
AbstractWe show that the minimum of the pseudo-Boolean quadratic function ƒ(x) = xTQx + cTx can be found in linear time when the graph defined by Q is transformable into a combinatorial circuit of AND, OR, NAND, NOR or NOT logic gates. A novel modeling technique is used to transform the graph defined by Q into a logic circuit. A consistent labeling of the signals in the logic circuit from the set {0, 1} corresponds to the global minimum of ƒ and the labeling is determined through logic simulation of the circuit. Our approach establishes a direct and constructive relationship between pseudo-Boolean functions and logic circuits.In the restricted case when all the elements of Q are nonpositive, the minimum of ƒ can be obtained in polynomial time [15]. We show that the problem of finding the minimum of ƒ, even in the special case when all the elements of Q are positive, is NP-complete
Searching for Realizations of Finite Metric Spaces in Tight Spans
An important problem that commonly arises in areas such as internet
traffic-flow analysis, phylogenetics and electrical circuit design, is to find
a representation of any given metric on a finite set by an edge-weighted
graph, such that the total edge length of the graph is minimum over all such
graphs. Such a graph is called an optimal realization and finding such
realizations is known to be NP-hard. Recently Varone presented a heuristic
greedy algorithm for computing optimal realizations. Here we present an
alternative heuristic that exploits the relationship between realizations of
the metric and its so-called tight span . The tight span is a
canonical polytopal complex that can be associated to , and our approach
explores parts of for realizations in a way that is similar to the
classical simplex algorithm. We also provide computational results illustrating
the performance of our approach for different types of metrics, including
-distances and two-decomposable metrics for which it is provably possible
to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure
Kirchhoff's Circuit Law Applications to Graph Simplification in Search Problems
This paper proposes a new analysis of graph using the concept of electric
potential, and also proposes a graph simplification method based on this
analysis. Suppose that each node in the weighted-graph has its respective
potential value. Furthermore, suppose that the start and terminal nodes in
graphs have maximum and zero potentials, respectively. When we let the level of
each node be defined as the minimum number of edges/hops from the start node to
the node, the proper potential of each level can be estimated based on
geometric proportionality relationship. Based on the estimated potential for
each level, we can re-design the graph for path-finding problems to be the
electrical circuits, thus Kirchhoff's Circuit Law can be directed applicable
for simplifying the graph for path-finding problems
Dual-Eulerian Graphs with Applications to VLSI Design
A Dual-Eulerian graph is a plane multigraph G that contains an edge list which is simultaneously an Euler tour in G and an Euler tour in the dual of G. Dual-Eulerian tours play an important role in optimizing CMOS layouts of Boolean functions. When circuits are represented by undirected multigraphs the layout area of the circuit can be optimized through finding the minimum number of disjoint dual trails that cover the graph. This paper presents an implementation of a polynomial time algorithm for determining whether or not a plane multigraph is Dual-Eulerian and for finding the Dual-Eulerian trail if it exists
External-Memory Graph Algorithms
We present a collection of new techniques for designing and analyzing efficient external-memory algorithms for graph problems and illustrate how these techniques can be applied to a wide variety of specific problems. Our results include:
Proximate-neighboring. We present a simple
method for deriving external-memory lower bounds
via reductions from a problem we call the “proximate neighbors” problem. We use this technique to derive non-trivial lower bounds for such problems as list ranking, expression tree evaluation, and connected components. PRAM simulation. We give methods for efficiently
simulating PRAM computations in external memory, even for some cases in which the PRAM algorithm is not work-optimal. We apply this to derive a number of optimal (and simple) external-memory graph algorithms.
Time-forward processing. We present a general
technique for evaluating circuits (or “circuit-like”
computations) in external memory. We also usethis in a deterministic list ranking algorithm.
Deterministic 3-coloring of a cycle. We give
several optimal methods for 3-coloring a cycle,
which can be used as a subroutine for finding large
independent sets for list ranking. Our ideas go
beyond a straightforward PRAM simulation, and
may be of independent interest.
External depth-first search. We discuss a method
for performing depth first search and solving related
problems efficiently in external memory. Our
technique can be used in conjunction with ideas
due to Ullman and Yannakakis in order to solve
graph problems involving closed semi-ring computations even when their assumption that vertices fit in main memory does not hold.
Our techniques apply to a number of problems, including list ranking, which we discuss in detail, finding Euler tours, expression-tree evaluation, centroid decomposition of a tree, least-common ancestors, minimum spanning tree verification, connected and biconnected components, minimum spanning forest, ear decomposition, topological sorting, reachability, graph drawing, and visibility representation
A taxonomy of problems with fast parallel algorithms
The class NC consists of problems solvable very fast (in time polynomial in log n) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC1-reducibility is introduced and used throughout (problem R is NC1-reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S). Problems complete with respect to this reducibility are given for many of the subclasses of NC. A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC2 (solvable by uniform Boolean circuits of depth O(log2 n) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York)
Pendekatan Matriks Ketetanggaan Berbobot untuk Solusi Minimum Spanning Tree (MST)
Minimum Spanning Tree (MST) or often called Minimum Weighting Spanning Tree (MWST) is a path or edge search algorithm that connects all vertices in a connected graph and does not form a circuit with a minimum weight. The classic algorithm used to solve MST problems is the Prim’s and Kruskal’s algorithms. The problem of minimum spanning tree is a problem related to optimization in finding the minimum weighted edge that can connect all vertices). MST is widely used in computer science such as to determine access points, build networks and many more. The purpose of this reserarch is to find new solutions that can provide alternative MST solutions besides classical algorithms such as Prim and Kruskal. This experimental research uses a weighted adjacency matrix approach, with weight taken from the minimum side in each pair of matrix to produce a minimum spanning tree. This new solution can be an alternative in solving the minimum spanning tree problems
DNA origami and the complexity of Eulerian circuits with turning costs
Building a structure using self-assembly of DNA molecules by origami folding
requires finding a route for the scaffolding strand through the desired
structure. When the target structure is a 1-complex (or the geometric
realization of a graph), an optimal route corresponds to an Eulerian circuit
through the graph with minimum turning cost. By showing that it leads to a
solution to the 3-SAT problem, we prove that the general problem of finding an
optimal route for a scaffolding strand for such structures is NP-hard. We then
show that the problem may readily be transformed into a Traveling Salesman
Problem (TSP), so that machinery that has been developed for the TSP may be
applied to find optimal routes for the scaffolding strand in a DNA origami
self-assembly process. We give results for a few special cases, showing for
example that the problem remains intractable for graphs with maximum degree 8,
but is polynomial time for 4-regular plane graphs if the circuit is restricted
to following faces. We conclude with some implications of these results for
related problems, such as biomolecular computing and mill routing problems
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