Adam Kurpisz

Approximating a two-machine flow shop scheduling under discrete scenario uncertainty

This paper deals with the two machine permutation flow shop problem with uncertain data, whose deterministic counterpart is known to be polynomially solvable. In this paper, it is assumed that job processing times are uncertain and they are specified as a discrete scenario set. For this uncertainty representation, the min–max and min–max regret criteria are adopted. The min–max regret version of the problem is known to be weakly NP-hard even for two processing time scenarios. In this paper, it is shown that the min–max and min–max regret versions of the problem are strongly NP-hard even for two scenarios. Furthermore, the min–max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not (4/3−ϵ)-approximable for any ϵ>0 unless P=NP if the number of scenarios is a part of the input. On the other hand, the min–max regret version is not at all approximable even for two scenarios.

Approximating the min-max (regret) selecting items problem

In this paper the problem of selecting p items out of n available to minimize the total cost is discussed. This problem is a special case of many important combinatorial optimization problems such as 0-1 knapsack, minimum assignment, single machine scheduling, minimum matroid base or resource allocation. It is assumed that the item costs are uncertain and they are specified as a scenario set containing K distinct cost scenarios. In order to choose a solution the min-max and min-max regret criteria are applied. It is shown that both min-max and min-max regret problems are not approximable within any constant factor unless P=NP, which strengthens the results known up to date. In this paper a deterministic approximation algorithm with performance ratio of O(lnK) for the min-max version of the problem is also proposed.

On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n-1. These problems are the hardest for the Lasserre hierarchy in this sense.

Recoverable Robust Combinatorial Optimization Problems

This paper deals with two Recoverable Robust (RR) models for combinatorial optimization problems with uncertain costs. These models were originally proposed by Busing (2012) for the shortest path problem with uncertain costs. In this paper, we generalize the RR models to a class of combinatorial optimization problems with uncertain costs and provide new positive and negative complexity results in this area.

Improved Approximation for the Maximum Duo-Preservation String Mapping Problem

In this paper we present improved approximation results for the max duo-preservation string mapping problem (MPSM) introduced in [Chen et al., Theoretical Computer Science, 2014] that is complementary to the well-studied min common string partition problem (MCSP). When each letter occurs at most k times in each string the problem is denoted by k-MPSM. First, we prove that k-MPSM is APX-Hard even when k = 2. Then, we improve on the previous results by devising two distinct algorithms: the first ensures approximation ratio 8/5 for k = 2 and ratio 3 for k = 3, while the second guarantees approximation ratio 4 for any bigger value of k. Finally, we address the approximation of constrained maximum induced subgraph (CMIS, a generalization of MPSM, also introduced in [Chen et al., Theoretical Computer Science, 2014]), and improve the best known 9-approximation for 3-CMIS to a 6-approximation, by using a configuration LP to get a better linear relaxation. We also prove that such a linear program has an integrality gap of k, which suggests that no constant approximation (i.e. independent of k) can be achieved through rounding techniques.

Competitive-Ratio Approximation Schemes for Makespan Scheduling Problems

The concept of competitive-ratio approximation scheme was recently proposed in [7]. Such a scheme algorithmically constructs an online algorithm with a competitive ratio arbitrarily close to the best possible competitive ratio for a given online problem. In this paper we continue this line of research by addressing several makespan scheduling problems and introducing new ideas: we combine the classical technique of structuring and simplifying the input instance for approximation schemes, with the new technique of guessing the end of the schedule (time after which no job is processed and released), which allows us to reduce the infinite-size set of on-line algorithms to a relevant set of finite size. This is the key idea for eventually allowing an enumeration scheme that finds a near optimal on-line algorithm. We demonstrate how this technique can be successfully applied to three basic makespan online over time scheduling problems: scheduling on unrelated parallel machines, job shop scheduling and single machine scheduling with delivery times.

Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree

A Lasserre Lower Bound for the Min-Sum Single Machine Scheduling Problem

2d

An unbounded Sum-of-Squares hierarchy integrality gap for a polynomially solvable problem

and an odd number of variables

Approximability of the robust representatives selection problem

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