scholarly journals Flipping Modules to Minimize Maximum Wire Length

VLSI Design ◽  
1995 ◽  
Vol 3 (1) ◽  
pp. 37-41
Author(s):  
Kyunrak Chong ◽  
Sartaj Sahni
Keyword(s):  
Linear Time ◽  
Np Hard ◽  
Wire Length

We show that obtaining the optimal orientations of modules to minimize the length of the longest wire is NP-hard. If each module is permitted only two possible orientations, this can be done in linear time. When all four orientations are permissible and wires are restricted to connect modules whose separation is bounded by some constant, the problem also can be solved in linear time.

scholarly journals Channel Density Minimization by Pin Permutation

VLSI Design ◽  
1994 ◽  
Vol 2 (2) ◽  
pp. 171-183
Author(s):  
Yang Cai ◽  
D. F. Wong ◽  
Jason Cong
Keyword(s):  
Optimal Algorithm ◽  
Linear Time ◽  
Layout Design ◽  
Strong Sense ◽  
Np Hard ◽  
Wire Length ◽  
Channel Density ◽  
Time Optimal ◽  
Two Sides ◽  
Intergrated Circuits

We present in this paper a linear time optimal algorithm for minimizing the density of a channel (with exits) by permuting the terminals on the two sides of the channel. This compares favorably with the previously known near-optimal algorithm presented in [6] that runs in superlinear time. Our algorithm has important applications in hierarchical layout design of intergrated circuits. We also show that the problem of minimizing wire length by permuting terminals is NP-hard in the strong sense.

scholarly journals Oriented Coloring on Recursively Defined Digraphs

Algorithms ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 87 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Directed Graphs ◽  
Graph Isomorphism ◽  
Oriented Graph ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Np Hard ◽  
Graph Isomorphism Problem ◽  
Acyclic Digraph

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

scholarly journals On star family packing of graphs

2021 ◽  
Author(s):  
Mengya Li ◽  
Wensong Lin
Keyword(s):  
Positive Integer ◽  
Approximation Algorithm ◽  
Linear Time ◽  
Complete Bipartite Graph ◽  
Packing Problem ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Np Hard ◽  
Input Graph ◽  
Positive Integers

Let $\mathcal{H}$ be a family of graphs. An $\mathcal{H}$-packing of a graph $G$ is a set $\{G_1,G_2,\dots,G_k\}$ of disjoint subgraphs of $G$ such that each $G_j$ is isomorphic to some element of $\mathcal{H}$. An $\mathcal{H}$-packing of a graph $G$ that covers the maximum number of vertices of $G$ is called a maximum $\mathcal{H}$-packing of $G$. The $\mathcal{H}$-packing problem seeks to find a maximum $\mathcal{H}$-packing of a graph. Let $i$ be a positive integer. An $i$-star is a complete bipartite graph $K_{1,i}$. This paper investigates the $\mathcal{H}$-packing problem with $\mathcal{H}$ being a family of stars. For an arbitrary family $\mathcal{S}$ of stars, we design a linear-time algorithm for the $\mathcal{S}$-packing problem in trees. Let $t$ be a positive integer. An $\mathcal{H}$-packing is called a $t^+$-star packing if $\mathcal{H}$ consists of all $i$-stars with $i\ge t$. We show that the $t^+$-star packing problem for $t\ge 2$ is NP-hard in bipartite graphs. As a consequence, the $2^+$-star packing problem is NP-hard even in bipartite graphs with maximum degree at most $4$. Let $T$ and $t$ be two positive integers with $T>t$. An $\mathcal{H}$-packing is called a $T\setminus t$-star packing if $\mathcal{H}=\{K_{1,1},K_{1,2},\dots,K_{1,T}\}\setminus \{K_{1,t}\}$. For $t\ge 2$, we present a $\frac{t}{t+1}$-approximation algorithm for the $T\setminus t$-star packing problem that runs in $\mathcal{O}(mn^{1/2})$ time, where $n$ is the number of vertices and $m$ the number of edges of the input graph. We also design a $\frac{1}{2}$-approximation algorithm for the $2^+$-star packing problem that runs in $\mathcal{O}(m)$ time, where $m$ is the number of edges of the input graph. As a consequence, every connected graph with at least $3$ vertices has a $2^+$-star packing that covers at least half of its vertices.

scholarly journals Converging from branching to linear metrics on Markov chains

2017 ◽  
Vol 29 (1) ◽  
pp. 3-37 ◽  
Author(s):  
GIORGIO BACCI ◽  
GIOVANNI BACCI ◽  
KIM G. LARSEN ◽  
RADU MARDARE
Keyword(s):  
Model Checking ◽  
Markov Chains ◽  
Temporal Logic ◽  
Polynomial Time ◽  
Linear Time ◽  
Linear Temporal Logic ◽  
Np Hard ◽  
Branching Time ◽  
Threshold Problem ◽  
Ltl Model Checking

We study two well-known linear-time metrics on Markov chains (MCs), namely, the strong and strutter trace distances. Our interest in these metrics is motivated by their relation to the probabilistic linear temporal logic (LTL)-model checking problem: we prove that they correspond to the maximal differences in the probability of satisfying the same LTL and LTL−X(LTL without next operator) formulas, respectively.The threshold problem for these distances (whether their value exceeds a given threshold) is NP-hard and not known to be decidable. Nevertheless, we provide an approximation schema where each lower and upper approximant is computable in polynomial time in the size of the MC.The upper approximants are bisimilarity-like pseudometrics (hence, branching-time distances) that converge point-wise to the linear-time metrics. This convergence is interesting in itself, because it reveals a non-trivial relation between branching and linear-time metric-based semantics that does not hold in equivalence-based semantics.

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Np Hard ◽  
Sensor Coverage ◽  
Entire Plane ◽  
Line Segments ◽  
Minimum Number

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

scholarly journals On the Maximal Shortest Paths Cover Number

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1592
Author(s):  
Iztok Peterin ◽  
Gabriel Semanišin
Keyword(s):  
Shortest Path ◽  
Linear Time ◽  
Shortest Paths ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Minimum Cardinality ◽  
Graph Parameters

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S⊆V(G) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ(G). We show that it is NP-hard to determine ξ(G). We establish a connection between ξ(G) and several other graph parameters. We present a linear time algorithm that computes exact value for ξ(T) of a tree T.

scholarly journals Latest Algorithms on Particular Graph Classes

2020 ◽  
pp. 21-35
Author(s):  
Phan Thuan DO ◽  
Ba Thai PHAM ◽  
Viet Cuong THAN
Keyword(s):  
Optimization Problems ◽  
Linear Time ◽  
Maximum Clique ◽  
Independent Set ◽  
Maximum Independent Set ◽  
Np Hard ◽  
Induced Matching ◽  
Graph Classes ◽  
Maximum Induced Matching ◽  
Circular Arc Graphs

Many optimization problems such as Maximum Independent Set, Maximum Clique, Minimum Clique Cover and Maximum Induced Matching are NP-hard on general graphs. However, they could be solved in polynomial time when restricted to some particular graph classes such as comparability and co-comparability graph classes. In this paper, we summarize the latest algorithms solving some classical NP-hard problems on some graph classes over the years. Moreover, we apply the -redundant technique to obtain linear time O(j j) algorithms which find a Maximum Induced Matching on interval and circular-arc graphs. Inspired of these results, we have proposed some competitive programming problems for some programming contests in Vietnam in recent years.

scholarly journals Probabilistic Deduction with Conditional Constraints over Basic Events

1999 ◽  
Vol 10 ◽  
pp. 199-241 ◽  
Author(s):  
T. Lukasiewicz
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Linear Programs ◽  
Np Hard ◽  
Global Approach ◽  
Local Approach ◽  
Nonlinear Programs ◽  
Equivalent Linear ◽  
Special Case

We study the problem of probabilistic deduction with conditional constraints over basic events. We show that globally complete probabilistic deduction with conditional constraints over basic events is NP-hard. We then concentrate on the special case of probabilistic deduction in conditional constraint trees. We elaborate very efficient techniques for globally complete probabilistic deduction. In detail, for conditional constraint trees with point probabilities, we present a local approach to globally complete probabilistic deduction, which runs in linear time in the size of the conditional constraint trees. For conditional constraint trees with interval probabilities, we show that globally complete probabilistic deduction can be done in a global approach by solving nonlinear programs. We show how these nonlinear programs can be transformed into equivalent linear programs, which are solvable in polynomial time in the size of the conditional constraint trees.

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

1993 ◽  
Vol 03 (01) ◽  
pp. 85-105 ◽  
Author(s):  
SVANTE CARLSSON ◽  
BENGT J. NILSSON ◽  
SIMEON NTAFOS
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Art Gallery ◽  
Art Galleries ◽  
Art Gallery Problem ◽  
Natural Connection ◽  
Route Problem ◽  
Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

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