scholarly journals An Edmonds-Gallai-Type Decomposition for the j-Restricted k-Matching Problem

10.37236/7837 ◽  
2020 ◽  
Vol 27 (1) ◽  
Author(s):  
Yanjun Li ◽  
Jácint Szabó
Keyword(s):  
Positive Integer ◽  
Undirected Graph ◽  
Decomposition Theorem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Alternative Proof ◽  
Connected Components ◽  
Matching Problem ◽  
Np Hardness ◽  
Type Decomposition

Given a non-negative integer $j$ and a positive integer $k$, a $j$-restricted $k$-matching in a simple undirected graph is a $k$-matching, so that each of its connected components has at least $j+1$ edges. The maximum non-negative node weighted $j$-restricted $k$-matching problem was recently studied by Li who gave a polynomial-time algorithm and a min-max theorem for $0 \leqslant j < k$, and also proved the NP-hardness of the problem with unit node weights and $2 \leqslant k \leqslant j$. In this paper we derive an Edmonds–Gallai-type decomposition theorem for the $j$-restricted $k$-matching problem with $0 \leqslant j < k$, using the analogous decomposition for $k$-piece packings given by Janata, Loebl and Szabó, and give an alternative proof to the min-max theorem of Li.

COMPUTATIONAL COMPLEXITY OF THE PERFECT MATCHING PROBLEM IN HYPERGRAPHS WITH SUBCRITICAL DENSITY

2010 ◽  
Vol 21 (06) ◽  
pp. 905-924 ◽  
Author(s):  
MAREK KARPIŃSKI ◽  
ANDRZEJ RUCIŃSKI ◽  
EDYTA SZYMAŃSKA
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Perfect Matching ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Existence Problem ◽  
Matching Problem ◽  
Uniform Hypergraphs ◽  
The Status ◽  
Np Complete

In this paper we consider the computational complexity of deciding the existence of a perfect matching in certain classes of dense k-uniform hypergraphs. It has been known that the perfect matching problem for the classes of hypergraphs H with minimum ((k - 1)–wise) vertex degreeδ(H) at least c|V(H)| is NP-complete for [Formula: see text] and trivial for c ≥ ½, leaving the status of the problem with c in the interval [Formula: see text] widely open. In this paper we show, somehow surprisingly, that ½ is not the threshold for tractability of the perfect matching problem, and prove the existence of an ε > 0 such that the perfect matching problem for the class of hypergraphs H with δ(H) ≥ (½ - ε)|V(H)| is solvable in polynomial time. This seems to be the first polynomial time algorithm for the perfect matching problem on hypergraphs for which the existence problem is nontrivial. In addition, we consider parallel complexity of the problem, which could be also of independent interest.

scholarly journals A polynomial‐time algorithm for simple undirected graph isomorphism

2021 ◽  
Author(s):  
Jing He ◽  
Guangyan Huang ◽  
Jie Cao ◽  
Zhiwang Zhang ◽  
Hui Zheng ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Undirected Graph ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

scholarly journals The maximum weight spanning star forest problem on cactus graphs

2015 ◽  
Vol 07 (02) ◽  
pp. 1550018 ◽  
Author(s):  
Viet Hung Nguyen
Keyword(s):  
Connected Graph ◽  
Genomic Sequence ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Connected Component ◽  
Maximum Weight ◽  
Cactus Graphs ◽  
Np Hardness

A star is a graph in which some node is incident with every edge of the graph, i.e., a graph of diameter at most 2. A star forest is a graph in which each connected component is a star. Given a connected graph G in which the edges may be weighted positively. A spanning star forest of G is a subgraph of G which is a star forest spanning the nodes of G. The size of a spanning star forest F of G is defined to be the number of edges of F if G is unweighted and the total weight of all edges of F if G is weighted. We are interested in the problem of finding a Maximum Weight spanning Star Forest (MWSFP) in G. In [C. T. Nguyen, J. Shen, M. Hou, L. Sheng, W. Miller and L. Zhang, Approximating the spanning star forest problem and its applications to genomic sequence alignment, SIAM J. Comput. 38(3) (2008) 946–962], the authors introduced the MWSFP and proved its NP-hardness. They also gave a polynomial time algorithm for the MWSF problem when G is a tree. In this paper, we present a linear time algorithm that solves the MSWF problem when G is a cactus.

Algorithms and Bounds for L-Drawings of Directed Graphs

2018 ◽  
Vol 29 (04) ◽  
pp. 461-480
Author(s):  
Patrizio Angelini ◽  
Giordano Da Lozzo ◽  
Marco Di Bartolomeo ◽  
Valentino Di Donato ◽  
Maurizio Patrignani ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Adjacency Matrix ◽  
Experimental Analysis ◽  
Directed Graphs ◽  
Expressive Power ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Matrix Representations ◽  
Np Hardness

We introduce L-drawings, a novel paradigm for representing directed graphs aiming at combining the readability features of orthogonal drawings with the expressive power of adjacency matrix representations. In an L-drawing, vertices have exclusive [Formula: see text]- and [Formula: see text]-coordinates and edges consist of two segments, one exiting the source vertically and one entering the destination horizontally. We study the problem of computing L-drawings using minimum ink. We prove its NP-hardness and provide a heuristic based on a polynomial-time algorithm that adds a vertex to a drawing using the minimum additional ink. We performed an experimental analysis of the heuristic which confirms its effectiveness.

The Harmonious Chromatic Number of Bounded Degree Trees

1996 ◽  
Vol 5 (1) ◽  
pp. 15-28 ◽  
Author(s):  
Keith Edwards
Keyword(s):  
Positive Integer ◽  
Chromatic Number ◽  
Maximum Degree ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Simple Graph ◽  
Bounded Degree ◽  
Bounded Degree Trees ◽  
Fixed Positive Integer ◽  
Proper Vertex

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring. Let d be a fixed positive integer. We show that there is a natural number N(d) such that if T is any tree with m ≥ N(d) edges and maximum degree at most d, then the harmonious chromatic number h(T) is k or k + 1, where k is the least positive integer such that . We also give a polynomial time algorithm for determining the harmonious chromatic number of a tree with maximum degree at most d.

scholarly journals A Short Proof of the Hajnal–Szemerédi Theorem on Equitable Colouring

2008 ◽  
Vol 17 (2) ◽  
pp. 265-270 ◽  
Author(s):  
H. A. KIERSTEAD ◽  
A. V. KOSTOCHKA
Keyword(s):  
Positive Integer ◽  
Polynomial Time ◽  
Maximum Degree ◽  
Short Proof ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Proper Vertex ◽  
Colour Classes ◽  
Vertex Colouring

A proper vertex colouring of a graph is equitable if the sizes of colour classes differ by at most one. We present a new shorter proof of the celebrated Hajnal–Szemerédi theorem: for every positive integer r, every graph with maximum degree at most r has an equitable colouring with r+1 colours. The proof yields a polynomial time algorithm for such colourings.

THE BENFORD-NEWCOMB DISTRIBUTION AND UNAMBIGUOUS CONTEXT-FREE LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 717-727
Author(s):  
BALA RAVIKUMAR
Keyword(s):  
Positive Integer ◽  
Polynomial Time ◽  
Formal Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Language L ◽  
Combinatorial Technique ◽  
Pumping Lemma ◽  
Context Free

For a string w ∈ {0,1, 2,…, d-1}*, let vald(w) denote the integer whose base d representation is the string w and let MSDd(x) denote the most significant or the leading digit of a positive integer x when x is written as a base d integer. Hirvensalo and Karhumäki [9] studied the problem of computing the leading digit in the ternary representation of 2x ans showed that this problem has a polynomial time algorithm. In [16], some applications are presented for the problem of computing the leading digit of AB for given inputs A and B. In this paper, we study this problem from a formal language point of view. Formally, we consider the language Lb,d,j = {w|w ∈ {0,1, 2,…, d-1}*, 1 ≤ j ≤ 9, MSDb(dvalb(w))) = j} (and some related classes of languages) and address the question of whether this and some related languages are context-free. Standard pumping lemma proofs seem to be very difficult for these languages. We present a unified and simple combinatorial technique that shows that these languages are not unambiguous context-free languages. The Benford-Newcomb distribution plays a central role in our proofs.

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