Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

In the paper we consider the problem of scheduling $n$ identical jobs on 4 uniform machines with speeds $s_1 \geq s_2 \geq s_3 \geq s_4,$ respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree $\Delta$, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if $s_1=s_2=s_3$. If, however, $\Delta \leq 4$ and $s_1 \geq 12 s_2$, $s_2=s_3=s_4$, then the problem can be solved to optimality in time $O(n^{1.5})$. The same algorithm returns a solution of value at most 2 times optimal provided that $s_1 \geq 2s_2$. Finally, we study the case $s_1 \geq s_2 \geq s_3=s_4$ and give an $O(n^{1.5})$-time $32/15$-approximation algorithm in all such situations

Scheduling of Identical Jobs with Bipartite Incompatibility Graphs on Uniform Machines. Computational Experiments

Abstract. In the paper we consider the problem of scheduling of unit-length jobs on 3 or 4 uniform parallel machines to minimize schedule length or total completion time. We assume that jobs are subject to some kind of mutual exclusion constraints, modeled by a bipartite graph of bounded degree. The edges of the graph correspond to pairs of jobs that cannot be processed on the same machine. Although the problem is generally NP-hard, we show that under some conditions imposed on machine speeds and the structure of  incompatibility graph our problem can be solved to optimality in polynomial time. Theoretical considerations are accompanied by computer experiments  with some particular model of scheduling

Total Completion Time Minimization for Scheduling with Incompatibility Cliques

This paper considers parallel machine scheduling with incompatibilities between jobs. The jobs form a graph and no two jobs connected by an edge are allowed to be assigned to the same machine. In particular, we study the case where the graph is a collection of disjoint cliques. Scheduling with incompatibilities between jobs represents a well-established line of research in scheduling theory and the case of disjoint cliques has received increasing attention in recent years. While the research up to this point has been focused on the makespan objective, we broaden the scope and study the classical total completion time criterion. In the setting without incompatibilities, this objective is well known to admit polynomial time algorithms even for unrelated machines via matching techniques. We show that the introduction of incompatibility cliques results in a richer, more interesting picture. Scheduling on identical machines remains solvable in polynomial time, while scheduling on unrelated machines becomes APX-hard. Furthermore, we study the problem under the paradigm of fixed-parameter tractable algorithms (FPT). In particular, we consider a problem variant with assignment restrictions for the cliques rather than the jobs. We prove that it is NP-hard and can be solved in FPT time with respect to the number of cliques. Moreover, we show that the problem on unrelated machines can be solved in FPT time for reasonable parameters, e.g., the parameter pair: number of machines and maximum processing time. The latter result is a natural extension of known results for the case without incompatibilities and can even be extended to the case of total weighted completion time. All of the FPT results make use of n-fold Integer Programs that recently have received great attention by proving their usefulness for scheduling problems

Approximation Algorithms for Problems in Makespan Minimization on Unrelated Parallel Machines

A fundamental problem in scheduling is makespan minimization on unrelated parallel machines (R||Cmax). Let there be a set J of jobs and a set M of parallel machines, where every job Jj ∈ J has processing time or length pi,j ∈ ℚ+ on machine Mi ∈ M. The goal in R||Cmax is to schedule the jobs non-preemptively on the machines so as to minimize the length of the schedule, the makespan. A ρ-approximation algorithm produces in polynomial time a feasible solution such that its objective value is within a multiplicative factor ρ of the optimum, where ρ is called its approximation ratio. The best-known approximation algorithms for R||Cmax have approximation ratio 2, but there is no ρ-approximation algorithm with ρ \u3c 3/2 for R||Cmax unless P=NP. A longstanding open problem in approximation algorithms is to reconcile this hardness gap. We take a two-pronged approach to learn more about the hardness gap of R||Cmax: (1) find approximation algorithms for special cases of R||Cmax whose approximation ratios are tight (unless P=NP); (2) identify special cases of R||Cmax that have the same 3/2-hardness bound of R||Cmax, but where the approximation barrier of 2 can be broken. This thesis is divided into four parts. The first two parts investigate a special case of R||Cmax called the graph balancing problem when every job has one of two lengths and the machines may have one of two speeds. First, we present 3/2-approximation algorithms for the graph balancing problem with one speed and two job lengths. In the second part of this thesis we give an approximation algorithm for the graph balancing problem with two speeds and two job lengths with approximation ratio (√65+7)/8 ≈ 1.88278. In the third part of the thesis we present approximation algorithms and hardness of approximation results for two problems called R||Cmax with simple job-intersection structure and R||Cmax with bounded job assignments. We conclude this thesis by presenting algorithmic and computational complexity results for a generalization of R||Cmax where J is partitioned into sets called bags, and it must be that no two jobs belonging to the same bag are scheduled on the same machine

How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach

In this paper we study a resource allocation problem that encodes correlation between items in terms of \conflict and maximizes the minimum utility of the agents under a conflict free allocation. Admittedly, the problem is computationally hard even under stringent restrictions because it encodes a variant of the {\sc Maximum Weight Independent Set} problem which is one of the canonical hard problems in both classical and parameterized complexity. Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from the classical complexity perspective to draw the boundary between {\sf NP}-hardness and tractability for a constant number of agents. The problem was shown to be hard even for small constant number of agents and various other restrictions on the underlying graph. Notwithstanding this computational barrier, we notice that there are several parameters that are worth studying: number of agents, number of items, combinatorial structure that defines the conflict among the items, all of which could well be small under specific circumstancs. Our search rules out several parameters (even when taken together) and takes us towards a characterization of families of input instances that are amenable to polynomial time algorithms when the parameters are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)} algorithm for our problem where $m$ denotes the number of items that significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using ideas from FFT based fast polynomial multiplication; and we identify simple graph classes relevant to our problem's motivation that admit efficient algorithms

Graph theoretic and algorithmic aspect of the equitable coloring problem in block graphs

An equitable coloring of a graph $G=(V,E)$ is a (proper) vertex-coloring of $G$, such that the sizes of any two color classes differ by at most one. In this paper, we consider the equitable coloring problem in block graphs. Recall that the latter are graphs in which each 2-connected component is a complete graph. The problem remains hard in the class of block graphs. In this paper, we present some graph theoretic results relating various parameters. Then we use them in order to trace some algorithmic implications, mainly dealing with the fixed-parameter tractability of the problem.Comment: 21 pages, 2 figure

Parametrisierte Algorithmen für Ganzzahlige Lineare Programme und deren Anwendungen für Zuweisungsprobleme

This thesis is concerned with solving NP-hard problems. We consider two prominent strategies of coping with such computationally hard questions efficiently. The first approach aims to design approximation algorithms, that is, we are content to find good, but non-optimal solutions in polynomial time. The second strategy is called Fixed-Parameter Tractability (FPT) and considers parameters of the instance to capture the hardness of the problem and by that, obtain efficient algorithms with respect to the remaining input. This thesis employs both strategies jointly to develop efficient approximation and exact algorithms using parameterization and modeling the problem as structured integer linear programs (ILPs), which can be solved in FPT. In the first part of this work, we concentrate on these well-structured ILPs. On the one hand, we develop an efficient algorithm for block-structured integer linear programs called n-fold ILPs. On the other hand, we investigate the similarly block-structured 2-stage stochastic ILPs and prove conditional lower bounds regarding the running time of any algorithm solving them that match the best known upper bounds. We also prove the tightness of certain structural parameters called sensitivity and proximity for ILPs which arise from combinatorial questions such as allocation problems. The second part utilizes n-fold ILPs and structural properties to add to and improve upon known results for Scheduling and Bin Packing problems. We design exact FPT algorithms for the Scheduling With Clique Incompatibilities, Bin Packing, and Multiple Knapsack problems. Further, we provide constant-factor approximation algorithms and polynomial time approximation schemes (PTAS) for the Class Constraint Scheduling problems. Broadening our scope, we also investigate this problem and the closely related Cardinality Constraint Scheduling problem in the online setting and derive lower bounds for the approximation ratios as well as a PTAS for them. Altogether, this thesis contributes to the knowledge about structured ILPs, proves their limits and reaffirms their usefulness for a plethora of allocation problems. In doing so, various new and improved algorithms with respect to the running time or approximation quality emerge