# Finding minimum-quotient cuts in planar graphs

## Abstract

Given a graph G = (V, E) where each vertex v {element_of} V is assigned a weight w(v) and each edge e {element_of} E is assigned a cost c(e), the quotient of a cut partitioning the vertices of V into sets S and {bar S} is c(S, {bar S})/min{l_brace}w(S), w(S){r_brace}, where c(S, {bar S}) is the sum of the costs of the edges crossing the cut and w(S) and w({bar S}) are the sum of the weights of the vertices in S and {bar S}, respectively. The problem of finding a cut whose quotient is minimum for a graph has in recent years attracted considerable attention, due in large part to the work of Rao and Leighton and Rao. They have shown that an algorithm (exact or approximation) for the minimum-quotient-cut problem can be used to obtain an approximation algorithm for the more famous minimumb-balanced-cut problem, which requires finding a cut (S,{bar S}) minimizing c(S,{bar S}) subject to the constraint bW {le} w(S) {le} (1 {minus} b)W, where W is the total vertex weight and b is some fixed balance in the range 0 < b {le} {1/2}. Unfortunately, the minimum-quotient-cut problem is strongly NP-hard for general graphs, and themore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Labs., Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 10150229

- Report Number(s):
- SAND-92-2585C; CONF-9305153-3

ON: DE93011675

- DOE Contract Number:
- AC04-76DP00789

- Resource Type:
- Conference

- Resource Relation:
- Conference: Association for Computing Machinery (ACM) symposium on the theory of computing,San Diego, CA (United States),May 1993; Other Information: PBD: 18 Nov 1992

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; DIAGRAMS; OPTIMIZATION; ALGORITHMS; 990200; MATHEMATICS AND COMPUTERS

### Citation Formats

```
Park, J K, and Phillips, C A.
```*Finding minimum-quotient cuts in planar graphs*. United States: N. p., 1992.
Web.

```
Park, J K, & Phillips, C A.
```*Finding minimum-quotient cuts in planar graphs*. United States.

```
Park, J K, and Phillips, C A. Wed .
"Finding minimum-quotient cuts in planar graphs". United States.
```

```
@article{osti_10150229,
```

title = {Finding minimum-quotient cuts in planar graphs},

author = {Park, J K and Phillips, C A},

abstractNote = {Given a graph G = (V, E) where each vertex v {element_of} V is assigned a weight w(v) and each edge e {element_of} E is assigned a cost c(e), the quotient of a cut partitioning the vertices of V into sets S and {bar S} is c(S, {bar S})/min{l_brace}w(S), w(S){r_brace}, where c(S, {bar S}) is the sum of the costs of the edges crossing the cut and w(S) and w({bar S}) are the sum of the weights of the vertices in S and {bar S}, respectively. The problem of finding a cut whose quotient is minimum for a graph has in recent years attracted considerable attention, due in large part to the work of Rao and Leighton and Rao. They have shown that an algorithm (exact or approximation) for the minimum-quotient-cut problem can be used to obtain an approximation algorithm for the more famous minimumb-balanced-cut problem, which requires finding a cut (S,{bar S}) minimizing c(S,{bar S}) subject to the constraint bW {le} w(S) {le} (1 {minus} b)W, where W is the total vertex weight and b is some fixed balance in the range 0 < b {le} {1/2}. Unfortunately, the minimum-quotient-cut problem is strongly NP-hard for general graphs, and the best polynomial-time approximation algorithm known for the general problem guarantees only a cut whose quotient is at mostO(lg n) times optimal, where n is the size of the graph. However, for planar graphs, the minimum-quotient-cut problem appears more tractable, as Rao has developed several efficient approximation algorithms for the planar version of the problem capable of finding a cut whose quotient is at most some constant times optimal. In this paper, we improve Rao`s algorithms, both in terms of accuracy and speed. As our first result, we present two pseudopolynomial-time exact algorithms for the planar minimum-quotient-cut problem. As Rao`s most accurate approximation algorithm for the problem -- also a pseudopolynomial-time algorithm -- guarantees only a 1.5-times-optimal cut, our algorithms represent a significant advance.},

doi = {},

url = {https://www.osti.gov/biblio/10150229},
journal = {},

number = ,

volume = ,

place = {United States},

year = {1992},

month = {11}

}