Discrete Applied Mathematics最新文献

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Multiplicity of signless Laplacian eigenvalue 2 of a connected graph with a perfect matching 具有完美匹配的连通图的无符号拉普拉奇特征值 2 的倍率

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1016/j.dam.2024.11.003

Jinxing Zhao , Xiaoxiang Yu

The signless Laplacian matrix of a graph

G

is defined as

Q (G) = D (G) + A (G)

, where

D (G)

is the degree-diagonal matrix of

G

and

A (G)

is the adjacency matrix of

G

. The multiplicity of an eigenvalue

μ

Q (G)

is denoted by

m_{Q} (G, μ)

. Let

G = C (T_{1}, \dots, T_{g})

be a unicyclic graph with a perfect matching, where

T_{i}

is a rooted tree attached at the vertex

v_{i}

of the cycle

C_{g}

. It is proved that

Q (G)

has 2 as an eigenvalue if and only if

g + t

is divisible by 4, where

t

is the number of

T_{i}

of even orders. Another main result of this article gives a characterization for a connected graph

G

with a perfect matching such that

m_{Q} (G, 2) = θ (G) + 1

, where

θ (G) = | E (G) | - | V (G) | + 1

is the cyclomatic number of

G

图 G 的无符号拉普拉斯矩阵定义为 Q(G)=D(G)+A(G)，其中 D(G) 是 G 的阶对角矩阵，A(G) 是 G 的邻接矩阵。Q(G) 的特征值 μ 的多重性用 mQ(G,μ) 表示。设 G=C(T1,...Tg) 是一个完美匹配的单环图，其中 Ti 是连接在循环 Cg 的顶点 vi 上的有根树。本文证明，当且仅当 g+t 能被 4 整除时，Q(G) 的特征值为 2，其中 t 是偶数阶 Ti 的个数。本文的另一个主要结果给出了具有完美匹配的连通图 G 的特征，即 mQ(G,2)=θ(G)+1，其中θ(G)=|E(G)|-|V(G)|+1 是 G 的循环数。

引用次数: 0

Rainbow short linear forests in edge-colored complete graph 边色完整图中的彩虹短线性森林

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1016/j.dam.2024.11.002

Menglu He, Zemin Jin

An edge-colored graph

G

is called rainbow if no two edges of

G

have the same color. For a graph

G

and a subgraph

H \subseteq G

, the anti-Ramsey number

A R (G, H)

is the maximum number of colors in an edge-coloring of

G

such that

G

contains no rainbow copy of

H

. Recently, the anti-Ramsey problem for disjoint union of graphs received much attention. In particular, several researchers focused on the problem for graphs consisting of small components. In this paper, we continue the work in this direction. We refine the bound and obtain the precise value of

A R (K_{n}, P_{3} \cup t P_{2})

for all

n \geq 2 t + 3

. Additionally, we determine the value of

A R (K_{n}, 2 P_{3} \cup t P_{2})

for any integers

t \geq 1

and

n \geq 2 t + 7

如果一个边着色的图 G 没有两条边的颜色相同，则称其为彩虹图。对于图 G 和子图 H⊆G，反拉姆齐数 AR(G,H) 是指在 G 的边染色中，G 不包含 H 的彩虹副本的最大颜色数。特别是，一些研究人员重点研究了由小分量组成的图的反拉姆齐问题。在本文中，我们将继续这一方向的研究。我们完善了边界，并得到了所有 n≥2t+3 时 AR(Kn,P3∪tP2) 的精确值。此外，我们还确定了任意整数 t≥1 和 n≥2t+7 时的 AR(Kn,2P3∪tP2) 值。

引用次数: 0

Resistance distances in generalized join graphs 广义连接图中的电阻距离

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1016/j.dam.2024.11.013

Shaohan Xu, Kexiang Xu

<div><div>Let <math><mi>H</mi></math> be a graph with vertex set <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow></mrow></math>. The generalized join graph <math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math> is obtained from <math><mi>H</mi></math> by replacing each vertex <math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math> with a graph <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math> and joining each vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math> with each vertex in <math><msub><mrow><mi>G</mi></mrow><mrow><mi>j</mi></mrow></msub></math> provided <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math>. If every <math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math> is an independent set of <math><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math> vertices, then we write <math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math> as <math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math>, which is called the blow-up of <math><mi>H</mi></math>. In this paper we introduce the local complement transformation in electrical networks and obtain an electrically equivalent graph of <math><mrow><mi>H</mi><mrow><mo>[</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow></mrow></math>. As their applications, we obtain formulae for resistance distances of some vertex-weighted graphs and give a unified technique to compute resistance distances in <math><mrow><mi>H</mi>

设 H 是顶点集 V(H)={v1,v2,...,vk} 的图。将每个顶点 vi 替换为一个图 Gi，并将 Gi 中的每个顶点与 Gj 中的每个顶点连接（条件是 vivj∈E(H)），就得到广义连接图 H[G1,G2,...,Gk]。如果每个 Gi 都是 ni 个顶点的独立集合，那么我们就把 H[G1,G2,...,Gk] 写成 H[n1,n2,...,nk]，这就是所谓的 H 放大图。作为其应用，我们得到了一些顶点加权图的电阻距离公式，并给出了一种统一的技术来计算 H[G1,G2,...Gk] 中的电阻距离，当 1≤i≤k 的每个 Gi 都是由匹配顶点和孤立顶点组成的图时，从而得到了当 H 是一些给定图时 H[G1,G2,...Gk] 的电阻距离的封闭公式。

引用次数: 0

Partitions of Zm with identical representation functions 具有相同表示函数的 Zm 分区

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-16 DOI: 10.1016/j.dam.2024.11.010

Cui-Fang Sun, Zhi Cheng

<div><div>For any positive integer <math><mi>m</mi></math>, let <math><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></math> be the set of residue classes modulo <math><mi>m</mi></math>. For <math><mrow><mi>A</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math> and <math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math>, let the representation function <math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math> denote the number of solutions of the equation <math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>=</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>+</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover></mrow></math> with unordered pairs <math><mrow><mrow><mo>(</mo><mover><mrow><mi>a</mi></mrow><mo>¯</mo></mover><mo>,</mo><mover><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>∈</mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow></math>. Let <math><mrow><mi>m</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>α</mi></mrow></msup><mi>M</mi></mrow></math>, where <math><mi>α</mi></math> is a nonnegative integer and <math><mi>M</mi></math> is a positive odd integer. In this paper, we prove that if <math><mrow><mi>M</mi><mo>=</mo><mn>1</mn></mrow></math> and <math><mrow><mn>2</mn><mo>∤</mo><mi>α</mi></mrow></math>, then there exist two distinct sets <math><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊆</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math> with <math><mrow><mi>A</mi><mo>∪</mo><mi>B</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>∖</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mrow><mo>{</mo><mover><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>¯</mo></mover><mo>}</mo></mrow></mrow></math> such that <math><mrow><msub><mrow><mi>R</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>B</mi></mrow></msub><mrow><mo>(</mo><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>)</mo></mrow></mrow></math> for all <math><mrow><mover><mrow><mi>n</mi></mrow><mo>¯</mo></mover><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></math>. We also prove that if <math><mrow><mi>M</mi><mo>≥</mo><mn>3</mn></mrow></math>

对于任意正整数 m，让 Zm 成为残差类 modulo m 的集合。对于 A⊆Zm 和 n¯∈Zm，让表示函数 RA(n¯) 表示方程 n¯=a¯+a′¯ 的解的数目，其中无序对 (a¯,a′¯)∈A×A.设 m=2αM，其中 α 为非负整数，M 为正奇数。本文将证明，若 M=1 且 2∤α，则存在两个不同的集合 A,B⊆Zm，其中 A∪B=Zm∖{r1¯},A∩B={r2¯}，从而对于所有 n¯∈Zm，RA(n¯)=RB(n¯)。我们还证明，如果 M≥3 或 M=1 且 2∣α，则不存在两个不同的集合 A,B⊆Zm 且 A∪B=Zm∖{r1¯} 和 A∩B={r2¯} 使得 RA(n¯)=RB(n¯) for all n¯∈Zm

引用次数: 0

Complexity of Maker–Breaker games on edge sets of graphs 图边缘集上的闯关游戏的复杂性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-15 DOI: 10.1016/j.dam.2024.11.012

Eric Duchêne , Valentin Gledel , Fionn Mc Inerney , Nicolas Nisse , Nacim Oijid , Aline Parreau , Miloš Stojaković

We study the algorithmic complexity of Maker–Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the

H

-game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph

H

in the second. We prove that deciding who wins the perfect matching game and the

H

-game is

PSPACE

-complete, even for the latter in small-diameter graphs if

H

is a tree. Toward finding the smallest graph

H

for which the

H

-game is

PSPACE

-complete, we also prove that such an

H

of order 51 and size 57 exists.

We then give several positive results for the

H

-game. As the

H

-game is already

PSPACE

-complete when

H

is a tree, we mainly consider the case where

H

belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the

P_{4}

-game in general graphs and the

K_{1, ℓ}

-game in trees. Then, we prove that the

K_{1, ℓ}

-game in any graph, and the

H

-game in trees are both

FPT

parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest.

Another natural direction to take is to consider the

H

-game when

H

is a cycle. While we were unable to resolve this case, we prove that the related arboricity-

k

game is polynomial-time solvable. In particular, when

k = 2

, Maker wins this game if she claims the edges of any cycle.

我们研究了在一般图的边缘集上进行的制造者-破坏者博弈的算法复杂性。我们主要考虑完全匹配博弈和 H 博弈。在第一个博弈中，如果制造者要求得到完美匹配的边，而在第二个博弈中要求得到固定图 H 的副本，那么制造者就赢了。我们证明，决定谁赢得完美匹配博弈和 H-博弈是 PSPACE-complete的，即使后者是在小直径图中，如果 H 是一棵树的话。为了找到H-博弈PSPACE-complete的最小图H，我们还证明了这样一个阶数为51、大小为57的图H的存在。由于当 H 是树时，H-game 已经是 PSPACE-complete，所以我们主要考虑 H 属于树的子类的情况。具体来说，我们设计了两种线性时间算法，它们都基于结构特征，用于决定一般图中 P4 对弈和树中 K1,ℓ 对弈的胜者。然后，我们证明了任意图中的 K1,ℓ 对局和树中的 H 对局都是以对局长度为参数的 FPT，这显著地增加了具有这一性质的对局的简短列表，这也是我们的兴趣所在。虽然我们无法解决这种情况，但我们证明了相关的arboricity-k博弈是多项式时间可解的。特别是，当 k=2 时，如果制造者要求任何循环的边，她就会赢得这个博弈。

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引用次数: 0

Minimal trees with respect to exponential Zagreb indices 与指数萨格勒布指数有关的最小树

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-15 DOI: 10.1016/j.dam.2024.11.007

Nasrin Dehgardi , Mahdieh Azari

The concept of the exponential of vertex-degree-based topological indices was raised by Rada (2019) with the aim of studying the discrimination ability of this kind of invariants. In this paper, the minimum values of the exponential of the second Zagreb index and two variants of the first Zagreb index among all

n

-vertex trees with maximum degree

Δ

are characterized and the corresponding minimal trees are introduced. In addition, results are extended to the class of connected graphs with

n

vertices and maximum degree

Δ

拉达（2019）提出了基于顶点度的拓扑指数指数的概念，目的是研究这类不变式的判别能力。本文表征了具有最大度 Δ 的所有 n 个顶点树中第二个萨格勒布指数的指数值和第一个萨格勒布指数的两个变体的最小值，并介绍了相应的最小树。此外，还将结果扩展到具有 n 个顶点和最大度数 Δ 的连通图类。

引用次数: 0

Packing 2- and 3-stars into (2,3)-regular graphs 将 2 星和 3 星打包到 (2,3) 不规则图中

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-14 DOI: 10.1016/j.dam.2024.10.022

Wenying Xi , Wensong Lin , Yuquan Lin

Let

i

be a positive integer, an

i

-star denoted by

S_{i}

is a complete bipartite graph

K_{1, i}

. An

{S_{2}, S_{3}}

-packing of a graph

G

is a collection of vertex-disjoint subgraphs of

G

in which each subgraph is a 2-star or a 3-star. The maximum

{S_{2}, S_{3}}

-packing problem is to find an

{S_{2}, S_{3}}

-packing of a given graph containing the maximum number of vertices. The

{S_{2}, S_{3}}

-factor problem is to answer whether there is an

{S_{2}, S_{3}}

-packing containing all vertices of the given graph. The

{S_{2}, S_{3}}

-factor problem is NP-complete in cubic graphs. In this paper we design a quadratic-time algorithm for finding an

{S_{2}, S_{3}}

-packing of

G

that covers at least thirteen-sixteenths of its vertices with only a few exceptions. We also present some

(2, 3)

-regular graphs with their maximum

{S_{2}, S_{3}}

-packings covering exactly thirteen-sixteenths of their vertices.

让 i 为正整数，用 Si 表示的 i-star 是一个完整的双方形图 K1,i。图 G 的 {S2,S3} 组合是 G 的顶点相交子图的集合，其中每个子图都是 2-star 或 3-star。最大{S2,S3}堆积问题是指找到一个包含最多顶点数的给定图的{S2,S3}堆积。{S2,S3}因子问题是回答是否存在包含给定图形所有顶点的{S2,S3}堆积。在立方图中，{S2,S3} 因子问题是 NP-完全的。在本文中，我们设计了一种二次方时间算法，用于找到 G 的 {S2,S3} 组合，该组合至少覆盖了其十六分之三的顶点，只有少数例外。我们还介绍了一些 (2,3) 不规则图，它们的最大 {S2,S3} 组合正好覆盖了其十六分之十三的顶点。

引用次数: 0

On the minimum spectral radius of graphs with given order and dissociation number 关于给定阶数和解离数的图形的最小谱半径

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-14 DOI: 10.1016/j.dam.2024.11.014

Jing Zhao , Huiqing Liu , Jin Xiong

A dissociation set in a graph

G

is a set of vertices that induces a subgraph of maximum degree at most 1. The cardinality of a maximum dissociation set in

G

is called the dissociation number of

G

. Huang et al. determined the graphs with the minimum spectral radius among all connected graphs with given order

n

and dissociation number

2, ⌈ \frac{2 n}{3} ⌉, n - 2, n - 1

, respectively. In this paper, we characterize the graphs that attain the minimum spectral radius among all connected graphs with given order

n

and dissociation number

⌈ \frac{2 n}{3} ⌉ - 1

图 G 中的解离集是引起最大阶数至多为 1 的子图的顶点集合。在本文中，我们将描述在给定阶数 n 和解离数⌈2n3⌉-1 的所有连通图中达到最小谱半径的图的特征。

引用次数: 0

On vertices of outdegree k in minimally k-arc-connected digraphs 论最小k弧连接数图中缺度为k的顶点

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-13 DOI: 10.1016/j.dam.2024.11.009

Jun Fan, Xiaomin Hu, Weihua Yang, Shuang Zhao

Let

k

be a positive integer, and

D = (V (D), E (D))

be a minimally

k

-arc-connected simple digraph. Mader conjectured (Combinatorics 2 (1996) 423-449) that there are at least

k + 1

vertices of outdegree

k

D

. In this paper we prove that there are at least four vertices of outdegree

k

for

k \geq 3

设 k 为正整数，D=(V(D),E(D)) 为最小 k 弧连接的简单图。Mader 猜想（《组合论》2 (1996) 423-449）D 中至少有 k+1 个缺度为 k 的顶点。

引用次数: 0

Tight bounds for budgeted maximum weight independent set in bipartite and perfect graphs 二方图和完美图中预算最大权重独立集的严格界限

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-11-13 DOI: 10.1016/j.dam.2024.10.023

Ilan Doron-Arad, Hadas Shachnai

We consider the classic budgeted maximum weight independent set (BMWIS) problem. The input is a graph

G = (V, E)

, where each vertex

v \in V

has a weight

w (v)

and a cost

c (v)

, and a budget

B

. The goal is to find an independent set

S \subseteq V

G

such that

\sum_{v \in S} c (v) \leq B

, which maximizes the total weight

\sum_{v \in S} w (v)

. Since the problem on general graphs cannot be approximated within ratio

{| V |}^{1 - ɛ}

for any

ɛ > 0

, BMWIS has attracted significant attention on graph families for which a maximum weight independent set can be computed in polynomial time. Two notable such graph families are bipartite and perfect graphs. BMWIS is known to be NP-hard on both of these graph families; however, prior to this work, the best possible approximation guarantees for these graphs were wide open.

In this paper, we give a tight 2-approximation for BMWIS on perfect graphs and bipartite graphs. In particular, we give a

(2 - ɛ)

lower bound for BMWIS on bipartite graphs, already for the special case where the budget is replaced by a cardinality constraint, based on the Small Set Expansion Hypothesis (SSEH). For the upper bound, we design a 2-approximation for BMWIS on perfect graphs using a Lagrangian relaxation based technique. Finally, we obtain a tight lower bound for the capacitated maximum weight independent set (CMWIS) problem, the special case of BMWIS where

w (v) = c (v) \forall v \in V

. We show that CMWIS on bipartite and perfect graphs is unlikely to admit an efficient polynomial-time approximation scheme (EPTAS). Thus, the existing PTAS for CMWIS is essentially the best we can expect.

我们考虑的是经典的预算最大权重独立集（BMWIS）问题。输入是一个图 G=(V,E)，其中每个顶点 v∈V 都有权重 w(v)、成本 c(v)和预算 B。目标是在 G 中找到一个独立集 S⊆V，使得∑v∈Sc(v)≤B，从而最大化总权重∑v∈Sw(v)。由于对于任意ɛ>0，一般图上的问题无法在比率 |V|1-ɛ 内近似解决，因此 BMWIS 在可以在多项式时间内计算最大权重独立集的图族中引起了极大关注。其中两个著名的图族是二方图和完美图。众所周知，BMWIS 在这两个图族上都是 NP-hard；然而，在这项工作之前，这些图的最佳近似保证还没有定论。特别是，我们根据小集扩展假说 (SSEH)，给出了双方形图上 BMWIS 的 (2-ɛ) 下限，这已经适用于预算被万有引力约束取代的特殊情况。对于上界，我们使用基于拉格朗日松弛的技术，为完美图上的 BMWIS 设计了一个 2 近似值。最后，我们得到了有容最大权重独立集（CMWIS）问题的紧密下界，该问题是 BMWIS 的特例，其中 w(v)=c(v)∀v∈V 。我们的研究表明，双方图和完美图上的 CMWIS 不可能有高效的多项式时间逼近方案（EPTAS）。因此，针对 CMWIS 的现有 PTAS 基本上是我们所能期待的最佳方案。

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