Integral Biflow Maximization
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913445107990528 |
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| author | Ding, Guoli Tao, Rongchuan Yang, Mengxi Zang, Wenan |
| author_facet | Ding, Guoli Tao, Rongchuan Yang, Mengxi Zang, Wenan |
| contents | Let $G=(V,E)$ be a graph with four distinguished vertices, two sources $s_1, s_2$ and two sinks $t_1,t_2$, let $c:\, E \rightarrow \mathbb Z_+$ be a capacity function, and let ${\cal P}$ be the set of all simple paths in $G$ from $s_1$ to $t_1$ or from $s_2$ to $t_2$. A biflow (or $2$-commodity flow) in $G$ is an assignment $f:\, {\cal P}\rightarrow \mathbb R_+$ such that $\sum_{e \in Q \in {\cal P}}\, f(Q) \le c(e)$ for all $e \in E$, whose value is defined to be $\sum_{Q \in {\cal P}}\, f(Q)$. A bicut in $G$ is a subset $K$ of $E$ that contains at least one edge from each member of ${\cal P}$, whose capacity is $\sum_{e\in K}\, c(e)$. In 1977 Seymour characterized, in terms of forbidden structures, all graphs $G$ for which the max-biflow (integral) min-bicut theorem holds true (that is, the maximum value of an integral biflow is equal to the minimum capacity of a bicut for every capacity function $c$); such a graph $G$ is referred to as a Seymour graph. Nevertheless, his proof is not algorithmic in nature. In this paper we present a combinatorial polynomial-time algorithm for finding maximum integral biflows in Seymour graphs, which relies heavily on a structural description of such graphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17821 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integral Biflow Maximization Ding, Guoli Tao, Rongchuan Yang, Mengxi Zang, Wenan Combinatorics 90C27, 05C85, 68Q25 Let $G=(V,E)$ be a graph with four distinguished vertices, two sources $s_1, s_2$ and two sinks $t_1,t_2$, let $c:\, E \rightarrow \mathbb Z_+$ be a capacity function, and let ${\cal P}$ be the set of all simple paths in $G$ from $s_1$ to $t_1$ or from $s_2$ to $t_2$. A biflow (or $2$-commodity flow) in $G$ is an assignment $f:\, {\cal P}\rightarrow \mathbb R_+$ such that $\sum_{e \in Q \in {\cal P}}\, f(Q) \le c(e)$ for all $e \in E$, whose value is defined to be $\sum_{Q \in {\cal P}}\, f(Q)$. A bicut in $G$ is a subset $K$ of $E$ that contains at least one edge from each member of ${\cal P}$, whose capacity is $\sum_{e\in K}\, c(e)$. In 1977 Seymour characterized, in terms of forbidden structures, all graphs $G$ for which the max-biflow (integral) min-bicut theorem holds true (that is, the maximum value of an integral biflow is equal to the minimum capacity of a bicut for every capacity function $c$); such a graph $G$ is referred to as a Seymour graph. Nevertheless, his proof is not algorithmic in nature. In this paper we present a combinatorial polynomial-time algorithm for finding maximum integral biflows in Seymour graphs, which relies heavily on a structural description of such graphs. |
| title | Integral Biflow Maximization |
| topic | Combinatorics 90C27, 05C85, 68Q25 |
| url | https://arxiv.org/abs/2407.17821 |