In the Multiterminal Cut problem we are given an edge-weighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, max-flow problem, and can be solved in polynomial time. We show that the problem becomes NP-hard as soon as k = 3, but ca...

Today we discuss two NP-hard problems, Max-Cut and minimum bisection. As these problems are NP-hard, unless the widely believed P 6= NP conjecture is false, these problem cannot be solved in polynomial time for every instance. These leaves us with a couple of alternatives: either we look for algorithms that approximate the solution, or consider algorithms that work for “typical” instances, but ...

We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP–hard problems are easier to solve. In particular, whether there exist algorithms that solve correctly and in polynomial time all sufficiently stable instances of s...

For a graph G and a collection of vertex pairs {(s1, t1), . . . , (sk, tk)}, the k disjoint paths problem is to find k vertex-disjoint paths P1, . . . , Pk, where Pi is a path from si to ti for each i = 1, . . . , k. In the corresponding optimization problem, the shortest disjoint paths problem, the vertex-disjoint paths Pi have to be chosen such that a given objective function is minimized. We...

The simple max-cut problem is as follows: given a graph, find a partition of its vertex set into two disjoint sets, such that the number of edges having one endpoint in each set is as large as possible. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The simple max-cut decision problem is known to be NP-complete for split graphs. An indifference grap...

This note confirms a conjecture of (Bramoullé in Games Econ Behav 58:30–49, 2007). The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than n1− , where n i...

In the maximum constraint satisfaction problem (Max CSP), one is given a finite collection of positive-weight constraints on overlapping sets of variables, and the goal is to assign values from a given domain to the variables so that the total weight of satisfied constraints is maximized. We consider this problem and its variant Max AW CSP where the weights are allowed to be both positive and n...

A k-coloring of G = (V,E) is a partition S = (S1, . . . , Sk) of the node set V of G into stable sets Si (a stable set is a set of pairwise non adjacent nodes). In the usual case, the objective is to determine a node coloring minimizing k. A natural generalization of this problem is obtained by assigning a strictly positive integer weight w(v) for any node v ∈ V , and defining the weight of sta...

In this paper, we consider the generalized min-sum set cover problem, introduced by Azar, Gamzu, and Yin [1]. Bansal, Gupta, and Krishnaswamy [2] give a 485approximation algorithm for the problem. We are able to alter their algorithm and analysis to obtain a 28-approximation algorithm, improving the performance guarantee by an order of magnitude. We use concepts from α-point scheduling to obtai...