Approximation algorithms for job scheduling with block-type conflict graphs

TL;DR

The problem of scheduling jobs on parallel machines, under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered and an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines is presented.

Abstract

The problem of scheduling jobs on parallel machines (identical, uniform, or unrelated), under incompatibility relation modeled as a block graph, under the makespan optimality criterion, is considered in this paper. No two jobs that are in the relation (equivalently in the same block) may be scheduled on the same machine in this model. The presented model stems from a well-established line of research combining scheduling theory with methods relevant to graph coloring. Recently, cluster graphs and their extensions like block graphs were given additional attention. We complement hardness results provided by other researchers for block graphs by providing approximation algorithms. In particular, we provide a $2$-approximation algorithm for $P|G = block\ graph|C_{max}$ and a PTAS for the case when the jobs are unit time in addition. In the case of uniform machines, we analyze two cases. The first one is when the number of blocks is bounded, i.e. $Q|G = k-block\ graph|C_{max}$. For this case, we provide a PTAS, improving upon results presented by D. Page and R. Solis-Oba. The improvement is two-fold: we allow richer graph structure, and we allow the number of machine speeds to be part of the input. Due to strong NP-hardness of $Q|G = 2-clique\ graph|C_{max}$, the result establishes the approximation status of $Q|G = k-block\ graph|C_{max}$. The PTAS might be of independent interest because the problem is tightly related to the NUMERICAL k-DIMENSIONAL MATCHING WITH TARGET SUMS problem. The second case that we analyze is when the number of blocks is arbitrary, but the number of cut-vertices is bounded and jobs are of unit time. In this case, we present an exact algorithm. In addition, we present an FPTAS for graphs with bounded treewidth and a bounded number of unrelated machines.

Figures (6)

• Figure 1: A block graph $G$ (left) and its block-cut forest $T_G$ (right). The respective blocks are marked with the same line type.
• Figure 2: The figures present the order in which the knowledge about the colorings of subgraphs is constructed. In consecutive figures the lines represent subgraphs corresponding to the sets of distinct colorings that are consecutively constructed. Observe also that the calls of the algorithms give the intuitive meaning of the presented algorithms.
• Figure 3: The sample block graph with sample colorings represented by $(0, 0, 0, 1; 0, 0, 0, 2)$. The meaning is that the color assigned to $J_1$ is of cardinality $3$ and there are $2$ other colors of cardinality $3$ each.
• Figure 4: An example graph $G$ with $2$ blocks and $f(J_4) = M_2$, and its flow network $F(G, C, f)$.
• Figure 5: A high-level overview of the conversion of configuration sets $U_{i-1}$ to $U_i$ in \ref{['alg:ptas_uniform_block']} for some block $V_{k'}$ is illustrated. Keep in mind that when transforming the configurations, the jobs that have processing requirement $c_i \ge p_j > c_{i-1}$ and are not cut-vertices are added. Finally, if the difference between the capacities of the machines is big (and $shift \ge \tau$) then it might be the case that some new jobs will be added to $n_{0,k'}$; the dashed line is added to emphasize that it is the only case when the same field is filled with values from both: the old vector and the set of jobs.

Theorems & Definitions (36)

• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• Lemma 2.5
• proof
• Lemma 2.6