scholarly journals Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs

Algorithmica ◽  
2022 ◽  
Author(s):  
Boris Klemz ◽  
Günter Rote
Keyword(s):  
Bipartite Graph ◽  
Linear Time ◽  
Bipartite Graphs ◽  
Compact Representation ◽  
Maximum Cardinality ◽  
Induced Matching ◽  
Running Time ◽  
Induced Matchings ◽  
A Chain ◽  
Convex Bipartite Graphs

AbstractA bipartite graph $$G=(U,V,E)$$ G = ( U , V , E ) is convex if the vertices in V can be linearly ordered such that for each vertex $$u\in U$$ u ∈ U , the neighbors of u are consecutive in the ordering of V. An induced matchingH of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and mweighted edges, an induced matching of maximum total weight can be computed in $$O(n+m)$$ O ( n + m ) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex $$u\in U$$ u ∈ U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) time. In convex bipartite graphs, maximum-cardinality induced matchings are dual to minimum chain covers. A chain cover is a covering of the edge set by chain subgraphs, that is, subgraphs that do not contain induced matchings of more than one edge. Given a compact representation, we compute a representation of a minimum chain cover in O(n) time. If no compact representation is given, the cover can be computed in $$O(n+m)$$ O ( n + m ) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of $$O(n^2)$$ O ( n 2 ) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. 10.1016/j.tcs.2007.04.006). The weighted case has not been considered before.

scholarly journals On the induced matching problem in hamiltonian bipartite graphs

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yinglei Song
Keyword(s):  
Bipartite Graph ◽  
Parameterized Complexity ◽  
Bipartite Graphs ◽  
Matching Problem ◽  
Induced Matching ◽  
Subexponential Time ◽  
Maximum Induced Matching

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

scholarly journals The Power of Linear-Time Data Reduction for Maximum Matching

Algorithmica ◽  
2020 ◽  
Vol 82 (12) ◽  
pp. 3521-3565
Author(s):  
George B. Mertzios ◽  
André Nichterlein ◽  
Rolf Niedermeier
Keyword(s):  
Systematic Study ◽  
Data Reduction ◽  
Linear Time ◽  
Maximum Matching ◽  
Maximum Cardinality ◽  
Solution Strategy ◽  
Time Data ◽  
Undirected Graphs ◽  
Running Time ◽  
General Graphs

Abstract Finding maximum-cardinality matchings in undirected graphs is arguably one of the most central graph primitives. For m-edge and n-vertex graphs, it is well-known to be solvable in $$O(m\sqrt{n})$$ O ( m n )  time; however, for several applications this running time is still too slow. We investigate how linear-time (and almost linear-time) data reduction (used as preprocessing) can alleviate the situation. More specifically, we focus on linear-time kernelization. We start a deeper and systematic study both for general graphs and for bipartite graphs. Our data reduction algorithms easily comply (in form of preprocessing) with every solution strategy (exact, approximate, heuristic), thus making them attractive in various settings.

scholarly journals The induced matching and chain subgraph cover problems for convex bipartite graphs

2007 ◽  
Vol 381 (1-3) ◽  
pp. 260-265 ◽  
Author(s):  
Andreas Brandstädt ◽  
Elaine M. Eschen ◽  
R. Sritharan
Keyword(s):  
Bipartite Graphs ◽  
Induced Matching ◽  
Convex Bipartite Graphs

scholarly journals Complexity issues of perfect secure domination in graphs

2021 ◽  
Vol 55 ◽  
pp. 11
Author(s):  
P. Chakradhar ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Time Algorithm ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Minimum Cardinality ◽  
Split Graphs ◽  
Secure Domination ◽  
Convex Bipartite Graphs

Let G = (V, E) be a simple, undirected and connected graph. A dominating set S is called a secure dominating set if for each u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E and (S \{v}) ∪{u} is a dominating set of G. If further the vertex v ∈ S is unique, then S is called a perfect secure dominating set (PSDS). The perfect secure domination number γps(G) is the minimum cardinality of a perfect secure dominating set of G. Given a graph G and a positive integer k, the perfect secure domination (PSDOM) problem is to check whether G has a PSDS of size at most k. In this paper, we prove that PSDOM problem is NP-complete for split graphs, star convex bipartite graphs, comb convex bipartite graphs, planar graphs and dually chordal graphs. We propose a linear time algorithm to solve the PSDOM problem in caterpillar trees and also show that this problem is linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. Finally, we show that the domination and perfect secure domination problems are not equivalent in computational complexity aspects.

ON LOCALLY-BALANCED 2-PARTITIONS OF BIPARTITE GRAPHS

2020 ◽  
Vol 54 (3 (253)) ◽  
pp. 137-145
Author(s):  
Aram H. Gharibyan ◽  
Petros A. Petrosyan
Keyword(s):  
Bipartite Graph ◽  
Open Neighborhood ◽  
Bipartite Graphs ◽  
The Other ◽  
Open Neighbourhood ◽  
Np Complete ◽  
Convex Bipartite Graphs

A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow \{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with an open neighborhood}, if for every $v\in V(G)$, $\left\vert \vert \{u\in N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert \right\vert\leq 1$. A bipartite graph is \emph{$(a,b)$-biregular} if all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. In this paper we prove that the problem of deciding, if a given graph has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete even for $(3,8)$-biregular bipartite graphs. We also prove that a $(2,2k+1)$-biregular bipartite graph has a locally-balanced $2$-partition with an open neighbourhood if and only if it has no cycle of length $2 \pmod{4}$. Next, we prove that if $G$ is a subcubic bipartite graph that has no cycle of length $2 \pmod{4}$, then $G$ has a locally-balanced $2$-partition with an open neighbourhood. Finally, we show that all doubly convex bipartite graphs have a locally-balanced $2$-partition with an open neighbourhood.

Total vertex-edge domination in graphs: Complexity and algorithms

2021 ◽  
Author(s):  
Nitisha Singhwal ◽  
Palagiri Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Chordal Graphs ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Linear Programming Formulation ◽  
Convex Bipartite Graphs

Let [Formula: see text] be a simple, undirected and connected graph. A vertex [Formula: see text] of a simple, undirected graph [Formula: see text]-dominates all edges incident to at least one vertex in its closed neighborhood [Formula: see text]. A set [Formula: see text] of vertices is a vertex-edge dominating set of [Formula: see text], if every edge of graph [Formula: see text] is [Formula: see text]-dominated by some vertex of [Formula: see text]. A vertex-edge dominating set [Formula: see text] of [Formula: see text] is called a total vertex-edge dominating set if the induced subgraph [Formula: see text] has no isolated vertices. The total vertex-edge domination number [Formula: see text] is the minimum cardinality of a total vertex-edge dominating set of [Formula: see text]. In this paper, we prove that the decision problem corresponding to [Formula: see text] is NP-complete for chordal graphs, star convex bipartite graphs, comb convex bipartite graphs and planar graphs. The problem of determining [Formula: see text] of a graph [Formula: see text] is called the minimum total vertex-edge domination problem (MTVEDP). We prove that MTVEDP is linear time solvable for chain graphs and threshold graphs. We also show that MTVEDP can be approximated within approximation ratio of [Formula: see text]. It is shown that the domination and total vertex-edge domination problems are not equivalent in computational complexity aspects. Finally, an integer linear programming formulation for MTVEDP is presented.

scholarly journals Rejection sampling of bipartite graphs with given degree sequence

2018 ◽  
Vol 10 (2) ◽  
pp. 249-275
Author(s):  
Koko K. Kayibi ◽  
U. Samee ◽  
S. Pirzada ◽  
Mohammad Ali Khan
Keyword(s):  
Markov Chains ◽  
Bipartite Graph ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Sampling Scheme ◽  
Rejection Sampling ◽  
Generation Algorithm ◽  
Running Time ◽  
Edge Swapping ◽  
Simple Realization

Abstract Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where $m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).

ACYCLIC MATCHINGS IN SUBCLASSES OF BIPARTITE GRAPHS

2012 ◽  
Vol 04 (04) ◽  
pp. 1250050 ◽  
Author(s):  
B. S. PANDA ◽  
D. PRADHAN
Keyword(s):  
Time Algorithm ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Chain Graph ◽  
Matching Problem ◽  
Permutation Graph ◽  
Complexity Result ◽  
Np Complete ◽  
A Chain ◽  
Acyclic Matching

A set M ⊆ E is called an acyclic matching of a graph G = (V, E) if no two edges in M are adjacent and the subgraph induced by the set of end vertices of the edges of M is acyclic. Given a positive integer k and a graph G = (V, E), the acyclic matching problem is to decide whether G has an acyclic matching of cardinality at least k. Goddard et al. (Discrete Math.293(1–3) (2005) 129–138) introduced the concept of the acyclic matching problem and proved that the acyclic matching problem is NP-complete for general graphs. In this paper, we propose an O(n + m) time algorithm to find a maximum cardinality acyclic matching in a chain graph having n vertices and m edges and obtain an expression for the number of maximum cardinality acyclic matchings in a chain graph. We also propose a dynamic programming-based O(n + m) time algorithm to find a maximum cardinality acyclic matching in a bipartite permutation graph having n vertices and m edges. Finally, we strengthen the complexity result of the acyclic matching problem by showing that this problem remains NP-complete for perfect elimination bipartite graphs.

Sign in / Sign up

Export Citation Format

Share Document