A heuristic method for solving the problem of partitioning graphs with supply and demand

Abstract

In this paper we present a greedy algorithm for solving the problem of the maximum partitioning of graphs with supply and demand (MPGSD). The goal of the method is to solve the MPGSD for large graphs in a reasonable time limit. This is done by using a two stage greedy algorithm, with two corresponding types of heuristics. The solutions acquired in this way are improved by applying a computationally inexpensive, hill climbing like, greedy correction procedure. In our numeric experiments we analyze different heuristic functions for each stage of the greedy algorithm, and show that their performance is highly dependent on the properties of the specific instance. Our tests show that by exploring a relatively small number of solutions generated by combining different heuristic functions, and applying the proposed correction procedure we can find solutions within only a few percent of the optimal ones.

Partitioning of supply/demand graphs with capacity limitations: an ant colony approach

“…Using a second heuristic h n the best v ∈ N k i is selected and added to S k i . An in-depth analysis of potential heuristics is given in our previous work (Jovanovic and Bousselham 2014;Jovanovic et al 2015). This procedure will be repeated until it is not possible to expand any of the subgraphs.…”

“…There is a wide range of potential heuristics h se and h e for selecting the subgraph S k i that is to be expanded, and the subgraph S k j that will be merged with S k i . Such heuristics would follow a similar logic to the ones presented in articles by Jovanovic and Bousselham (2014) and Jovanovic et al (2015). Due to the fact that we are using these heuristics only as a basis for an ACO algorithm we will use two basic ones based on the total supply of subgraphs.…”

“…The final step is just selecting the N subgraphs that cover the most demand. It is important to point out that for the proposed algorithm to be computationally effective it is necessary to use some auxiliary structures similar to the ones used in Jovanovic and Bousselham (2014); Jovanovic et al (2015). An illustration of the algorithm is given in Fig.…”

“…Some examples are the use of a balanced partitioning (Andreev and Räcke 2004), minimizing the number or weight of cuts (Reinelt et al 2008; Barnes et al 1988), or by limiting the number of cuts (Reinelt and Wenger 2010). The problem of finding the maximal partitioning of graphs with supply and demand (MPGSD) has shown to be closely related to the maximization of selfadequacy of interconnected microgrids (Jovanovic and Bousselham 2014;Jovanovic et al 2015). Focus of the research for MPGSD, to a large extent, has been dedicated to the theoretical aspects of this problem (Ito et al 2008;Narayanaswamy and Ramakrishna 2012;Ito et al 2005;Kawabata and Nishizeki 2013).…”

“…The ant colony optimization (ACO) (Dorigo and Blum 2005) metaheuristic has shown to be very suitable for such a task in case of graph problems. In the context of the problem of interest, ACO has formerly proven to be an effective approach for solving various graph partitioning problems like the multiway (Tashkova et al 2011) and balanced (Comellas and Sapena 2006) ones, and also for the closely related graph cutting (Hinne and Marchiori 2011) and covering (Jovanovic andTuba 2011, 2013) problems. In our previous work, the ACO method has been successfully applied to the MPGSD .…”