Many real discrete optimization problems (DOPs) are $NP$-hard and contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. Large DOPs can be solved due to their special tructure using decomposition approaches. An important example of decomposition approaches is tree decomposition with local decomposition algorithms using the special block matrix structure of constraints which can exploit sparsity in the interaction graph of a discrete optimization problem. In this paper, discrete optimization problems with a tree structural graph are solved by local decomposition algorithms. Local decomposition algorithms generate a family of related DO problems which have the same structure but differ in the right-hand sides. Due to this fact, postoptimality techniques in DO are applied.
Deep Dive into Tree decomposition and postoptimality analysis in discrete optimization.
Many real discrete optimization problems (DOPs) are $NP$-hard and contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. Large DOPs can be solved due to their special tructure using decomposition approaches. An important example of decomposition approaches is tree decomposition with local decomposition algorithms using the special block matrix structure of constraints which can exploit sparsity in the interaction graph of a discrete optimization problem. In this paper, discrete optimization problems with a tree structural graph are solved by local decomposition algorithms. Local decomposition algorithms generate a family of related DO problems which have the same structure but differ in the right-hand sides. Due to this fact, postoptimality techniques in DO are applied.
Discrete optimization (DO) problems arise in various application areas such as planning, economical allocation, logistics, scheduling, computer aided design, and robotics. Application areas od discrete optimization models of OR also include supply chain design and management, network optimization, telecommunications, VLSI routing, manufacturing, transportation, scheduling, and finance. The tremendous attention that DO particularly has received in the literature gives some indication of its importance in many research areas. Unfortunately, most of the interesting problems are in the complexity class N P -hard and may require searching a tree of exponential size (if P = N P ) in the worst case. Many real DO problems contain a huge number of variables and/or constraints that make the models intractable for currently available DO solvers.
One of the promising approaches to cope with NP-hardness in solving DO problems is the construction of decomposition methods [34]. Decomposition techniques usually determine subproblems the solutions of which can be combined to create a solution of the initial DO problem. Usually, DO problems from applications have a special structure, and the matrices of constraints for large-scale problems have a lot of zero elements (sparse matrices). The nonzero elements of the matrix often fall into a limited number of blocks. The block form of many DO problems is usually caused by the weak connectedness of subsystems of real-world systems.
The search for graph structures appropriate for the application of dynamic programming caused a series of papers dedicated to tree decomposition research ( [35], [3], [6], [28], [14], [7], [9], [22]). Tree decomposition methods aim to merge variables such that the meta-graph is a tree of meta-nodes. Tree decomposition and the related notion of a treewidth (Robertson, Seymour [35]) play a very important role in algorithms, for many N P -complete problems on graphs that are otherwise intractable become polynomial time solvable when these graphs have a tree decomposition with restricted maximal size of cliques (or have a bounded treewidth [7], [9]).
Most of the works based on tree decomposition approach only present theoretical results [27], see the recent survey by Hicks et al. [24]. But only few papers on applications of this powerful tool in the area of DO exist [29], [24].
The algorithmic importance of the tree decomposition was caused by results of Courcelle [12] and Arnborg et al. [2] which showed that several N P -hard problems posed in monadic second-order logic can be solved in polynomial time using dynamic programming techniques on input graphs with bounded treewidth Tree decomposition based algorithms demonstrated their efficiency on solving frequency assignment problem Koster et al. [29], ring-routing problems [10], and traveling salesman problem [11].
Efficiency of tree decomposition based algorithms crucially depends on interaction graph structure of the DO problem, so that it has a time complexity O(n • 2 tw+1 ), where tw is the treewidth of the graph. If the interaction graph is rather sparse or, in other words, it has a relatively small treewidth, then complexity of the tree decomposition algorithm is reasonable.
Necessity of reduction of enumeration while solving problems corresponding to meta-nodes of the tree decomposition causes expediency and an urgency of development of tools that could help to cope with this difficulty.
In this paper, discrete optimization problems with a tree structural graph are solved by local decomposition algorithms that belong to dynamic programming paradigm. Local decomposition algorithm generates a family of related DO problems which have the same structure but differ in the right-hand sides. Due to this fact, postoptimality techniques in DO are applied.
Consider a DOP with constraints:
subject to
where X = {x 1 , . . . , x n } is a set of discrete variables, functions f i (X i ) are called components of the objective function and can be defined in tabular form, X k ⊂ X, k ∈ K = {1, 2, . . . , t} , t is the number of components of objective function, K is a set of indices of components;
We shall consider further a linear objective function (5):
Definition 1. [5]. Variables x ∈ X and y ∈ Y interact in DOP with constraints if they both appear either in the same component of objective function, or in the same constraint (in other words, if variables are both either in a set X k , or in a set X Si ).
Graph representation of a DOP structure may be done with various detailization. Structural graph of a DOP defines which variables are in which constraints.
An interaction graph [5] represents a structure of the DOP in a natural way. Definition 2. [5]. The interaction graph of a DOP is an undirected graph G = (X, E), such that 1. Vertices X of G correspond to variables of DOP; 2. Two vertices of G are adjacent iff corresponding variables interact.
Further, we shall use the notion of vertices that corr
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