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We study a scheduling problem where the jobs we have to perform are composed of one or more tasks. If two jobs sharing a non-empty subset of tasks are scheduled on the same machine, then these shared tasks have to be performed only once. This kind of problem is known in the literature under the names of VM-PACKING or PAGINATION. Our objective is to schedule a set of these objects on two parallel identical machines, with the aim of minimizing the makespan. This problem is NP-complete as an extension of the PARTITION problem. In this paper we present three exact algorithms with worst-case time-complexity guarantees, by exploring different branching techniques. Our first algorithm focuses on the relation between jobs sharing one or more symbols in common, whereas the two other algorithms branches on the shared symbols.

The maximum duo-preservation string mapping (Max-Duo) problem is the complement of the well studied minimum common string partition problem, both of which have applications in many fields including text compression and bioinformatics. k-Max-Duo is the restricted version of Max-Duo, where every letter of the alphabet occurs at most k times in each of the strings, which is readily reduced into the well known maximum independent set (MIS) problem on a graph of maximum degree $$\Delta \le 6(k-1)$$ . In particular, 2-Max-Duo can then be approximated arbitrarily close to 1.8 using the state-of-the-art approximation algorithm for the MIS problem on bounded-degree graphs. 2-Max-Duo was proved APX-hard and very recently a $$(1.6 + \epsilon )$$ -approximation algorithm was claimed, for any $$\epsilon > 0$$ . In this paper, we present a vertex-degree reduction technique, based on which, we show that 2-Max-Duo can be approximated arbitrarily close to 1.4.

Given a graph $$G = (V, E)$$ , we seek for a collection of vertex disjoint paths each of order at most 3 that together cover all the vertices of V. The problem is called 3-path partition, and it has close relationships to the well-known path cover problem and the set cover problem. The general k-path partition problem for a constant $$k \ge 3$$ is NP-hard, and it admits a trivial k-approximation. When $$k = 3$$ , the previous best approximation ratio is 1.5 due to Monnot and Toulouse (Oper Res Lett 35:677–684, 2007), based on two maximum matchings. In this paper we first show how to compute in polynomial time a 3-path partition with the least 1-paths, and then apply a greedy approach to merge three 2-paths into two 3-paths whenever possible. Through an amortized analysis, we prove that the proposed algorithm is a 13 / 9-approximation. We also show that the performance ratio 13 / 9 is tight for our algorithm.

Suppose that each vertex of a graph $$G$$ G is either a supply vertex or a demand vertex and is assigned a supply or a demand. All demands and supplies are nonnegative constant numbers in a steady network, while they are functions of a variable $$\lambda $$ ? in a parametric network. Each demand vertex can receive "power" from exactly one supply vertex through edges in $$G$$ G . One thus wishes to partition $$G$$ G to connected components by deleting edges from $$G$$ G so that each component has exactly one supply vertex whose supply is at least the sum of demands in the component. The "partition problem" asks whether $$G$$ G has such a partition. If $$G$$ G has no such partition, one wishes to find the maximum number $$r^*$$ r ? , $$0\le r^*

A partition problem in one-dimensional space is to seek a partition of a set of numbers that maximizes a given objective function. In some partition problems, the partition size, i.e., the number of nonempty parts in a partition, is fixed; while in others, the size can vary arbitrarily. We call the former the size-partition problem and the latter the open-partition problem. In general, it is much harder to solve open problems since the objective functions depend on size. In this paper, we propose a new approach by allowing empty parts and transform the open problem into a size problem allowing empty parts, called a relaxed-size problem. While the sortability theory has been established in the literature as a powerful tool to attack size partition problems, we develop the sortability theory for relaxed-size problems as a medium to solve open problems.

Given a partition of distinct d-dimensional vectors into p parts, the partition sum of the partition is the sum of vectors in each part. The shape of the partition is a p-tuple of the size of each part. A single-shape partition polytope is the convex hull of partition sums of all partitions that have a prescribed shape. A partition is separable if the convex hull of its parts are pairwise disjoint. The separability of a partition is a necessary condition for the associated partition sum to be a vertex of the single-shape partition polytope. It is also a sufficient condition for d=1 or p=2. However, the sufficiency fails to hold for d?3 and p?3. In this paper, we give some geometric sufficient conditions as well as some necessary conditions of vertices in general d and p. Thus, the open case for d=2 and p?3 is resolved.

Given a graph G=(V,E) with edge weights we∈ℝ, the optimum cooperation problem consists in determining a partition of the graph that maximizes the sum of weights of the edges with nodes in the same class plus the number of the classes of the partition. The problem is also known in the literature as the optimum attack problem in networks. Furthermore, a relevant physics application exists.

The three-partition problem is one of the most famous strongly NP-complete combinatorial problems. We introduce properties which, in many cases, can allow either a quick solution of an instance or a reduction of its size. The average effectiveness of the properties proposed is tested through computational experiments.