Cut problems in graphs with a budget constraint

Abstract

We study budgeted variants of classical cut problems: the Multiway Cut problem, the Multicut problem, and the k-Cut problem, and provide approximation algorithms for these problems. Specifically, for the budgeted multiway cut and the k-cut problems we provide constant factor approximation algorithms. We show that the budgeted multicut problem is at least as hard to approximate as the sparsest cut problem, and we provide a bi-criteria approximation algorithm for it.

Approximation algorithms for the Weighted t-Uniform Sparsest Cut and some other graph partitioning problems

“…Engelberg et al [11] studied the budgeted variants of some of the well known cut problems. In a Weighted Budgeted Separating Multiway Cut instance an undirected edge-weighted graph, a set of terminal pairs and a budget B are given, where each terminal pair has a specific demand.…”

“…and Vazirani [4]. Using the bicriteria approximation algorithm for the Minimum Graph Bisection problem (also known as the Balanced Cut problem), Andreev and Räcke [12] gave a bicriteria (O ( −2 log n log n), 1 + ) factor approximation algorithm for the Min-Sum k-Partitioning problem for any constant > 0. For the same problem Krauthgamer, Naor and Schwartz [13] gave a bicriteria O ( log k log n, 2) factor approximation algorithm.…”

Approximation Algorithms for the Firefighter Problem: Cuts over Time and Submodularity

“…For example, the work of Hayrapetyan et al [22] and others [3,12] fully utilizes the social-network structure to "cut off" and contain various diffusive processes in a social network. As mentioned earlier, all this work is only concerned with vaccinating a set of nodes before the infection begins, however, and does not have the temporal component of the Firefighter problem.…”

Approximability of the k‐server disconnection problem

“…To do so, consider an s 1 − s 2 cut (N 1 ; N \ N 1 ) and the cut edges F of G. The corresponding edges ofḠ are the edges from F and the edges (i, t l i ) for all is and ls such that i remains connected to s l after the deletion of F due to (7). For example, in Figure 4a, the edges of the s 1 − s 2 cut ({s 1 , 3}; {s 2 , 4, 5}) of G determined by the solution F = {(3, 5)} correspond to the multicut edges (3,5), (3, t 1 3 ), (4, t 2 4 ), (5, t 2 5 ), which are marked "×". We will show that these precisely correspond to the edges of the s 1 − s 2 cut (N 1 ; N \ N 1 ) of G .…”

“…Martel et al also proposed an exact method based on enumeration of "maximal cuts" and the cut-cost submodularity. Among the cut problems with a budget constraint studied by Engelberg et al [5], both the weighted and unweighted version of the budgeted separating multiway cut problem (BSMC), can be considered as special cases of the multiserver extension of the model of Martel et al [9]. They showed that the weighted BSMC is at least as difficult to approximate as the sparsest cut problem, but is approximable within a constant factor on trees.…”