DIMES: A Differentiable Meta Solver for Combinatorial Optimization Problems
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Abstract
Recently, deep reinforcement learning (DRL) models have shown promising resultsin solving NP-hard Combinatorial Optimization (CO) problems.
However, most DRL solvers can only scale to a few hundreds of nodes for combinatorial optimization problems on graphs, such as the Traveling Salesman Problem (TSP).
Thispaper addresses the scalability challenge in large-scale combinatorial optimizationby proposing a novel approach, namely, DIMES.
Unlike previous DRL methodswhich suffer from costly autoregressive decoding or iterative refinements of discretesolutions, DIMES introduces a compact continuous space for parameterizing theunderlying distribution of candidate solutions.
Such a continuous space allowsstable REINFORCE-based training and fine-tuning via massively parallel sampling.
We further propose a meta-learning framework to enable effective initializationof model parameters in the fine-tuning stage.
Extensive experiments show that
DIMES outperforms recent DRL-based methods on large benchmark datasets for
Traveling Salesman Problems and Maximal Independent Set problems.
Introduction
Combinatorial Optimization (CO) is a fundamental problem in computer science.
It has importantreal-world applications such as shipment planning, transportation, robots routing, biology, circuitdesign, and more [53].
However, due to the NP-hard complexity, a significant portion of the COproblems suffer from an exponential computational cost when using traditional algorithms.
As awell-known example, the Traveling Salesman Problem (TSP) has been intensively studied [29, 47]for finding the most cost-effective tour over an input graph where each node is visited once and onlyonce before finally returning to the starting node.
Over the past decades, significant efforts has beenmade for designing more efficient heuristic solvers [5, 18] for approximating the solution in a reducedsearch space.
Recent development in deep reinforcement learning (DRL) have shown promises in solving COproblems without manual injection of domain-specific expert knowledge [7, 34, 37].
The appeal ofneural methods is that they can learn useful patterns (such as graph motifs) from data that might bedifficult to discover by hand.
A typical category of DRL solvers, namely the construction heuristicslearners , [7, 34] use a Markov decision process (MDP) to grow partial solutions by adding onenew node per step, with a trained strategy which gives good solutions higher probabilities.
Anothercategory of DRL-based solvers, namely the improvement heuristics learners [10, 58], iterativelyrefine a feasible solution with a neural network-guided local Operations Research (OR) operations[50].
A major limitation of these DRL solvers is in their scalability on large CO problems.
Forexample, current DRL solvers for TSP can only scale to graphs with up to hundreds of nodes.
The bad scalability of these DRL methods lies in the fact that they suffer from costly decoding of
CO solutions, which is typically linear in the number of nodes in the input graph.
Since the reward of reinforcement learning is determined after decoding a complete solution (with either a chain rulefactorization or iterative refinements), either construction or improvement heuristic learners wouldencounter the sparse reward problem when dealing with large graphs [34, 27, 31].
While such anoverhead can be partially alleviated by constructing several parts of the solution in parallel [1] forlocally decomposable CO problems 1 , such as for maximum independent set (MIS) problems [44],how to scale up neural solvers for CO problems in general, including the locally non-decomposableones (such as TSP) is still an open challenge.
In this paper, we address this challenge by proposing a novel framework, namely DIMES (DIfferentiable MEta Solver) for solving combinatorial optimization problems.
Instead of previous DRL-based
CO solves that rely on construction or improvement heuristics, we introduce a compact continuousspace to parameterize the underlying distribution of candidate solutions which allows massivelyparallel on-policy sampling without the costly decoding process, and effectively reduces the varianceof the gradients by the REINFORCE algorithm [57] during both training and fine-tuning phases.
Wefurther propose a meta-learning framework to enable effective initialization of model parameters inthe fine-tuning stage.
To our knowledge, we are the first to apply meta-learning over a collection of
CO problem instances, where each instance graph is treated as one of a collection tasks in a unifiedframework.
We need to point out that the idea of designing a continuous space for combinatorial optimizationproblems has been tried by the heatmaps approaches in the literature [40, 25, 17, 36].
However, thereare several major distinctions between the existing methods and our proposed one.
Previous workgenerates heatmaps based on supervised signals (each training graph is paired with its best solution)[4, 19], which are costly to obtain.
DIMES is directly optimized with gradients estimated by the
REINFORCE algorithm, which do not require supervised signals.
As a result, DIMES can scale tolarge graphs with up to tens of thousands of nodes, and predict (nearly) optimal solutions withoutthe need for costly generation of supervised training data or human specification of problem-specificheuristics.
In our experiments, we show that DIMES outperforms strong baselines among DRL-based solvers on TSP benchmark datasets, and can be successfully scaled up to graphs with tens of thousands of nodes.
As a sanity check, we also evaluate our framework on locally decomposable combinatorial optimization problems, including Maximal Independent Set (MIS) problem for synthetic graphs and graphsreduced from satisfiability (SAT) problems.
Our experimental results show that DIMES achievecompetitive performance compared to neural solvers specially designed for locally decomposable CO problems.
Related Work
2.1 DRL-based Construction Heuristics Learners
Construction heuristics methods create a solution of CO problem instance in one shot without furthermodifications.
Bello et al.
[7] are the first to tackle combinatorial optimization problems usingneural networks and reinforcement learning.
They used a Pointer Network (PtrNet) [54] as the policynetwork and used the actor-critic algorithm [33] for training on TSP and KnapSack instances.
Furtherimproved models have been developed afterwards [13, 34, 49, 14, 38], such as attention models[52], better DRL algorithms [30, 42, 35, 37, 46, 59, 55], for an extended scope of CO problems suchas Capacitated Vehicle Routing Problem (CVRP) [45], Job Shop Scheduling Problem (JSSP) [62],
Maximal Independent Set (MIS) problem [30, 1], and boolean satisfiability problem (SAT) [61].
Our proposed method in this paper belongs to the category of construction heuristics leaners in thesense of producing a one-shot solution per problem instance.
But unlike previous methods whichgenerate the solutions via a constructive Markov decision process (MDP) with rather costly decodingsteps (adding one un-visited node per step to a partial solution), we introduce a compact continuesspace to parameterize the underlying distribution of discrete candidate solutions, and to allow efficientsampling from such distribution without costly neural network-involved decoding.
2.2 DRL-based Improvement Heuristics Learners
Contrast to construction heuristics, DRL-based improvement heuristics methods train a neural networkto iteratively correct the mistakes locally for improving the quality of the current solution until thetime budget runs out.
Such DRL-based improvement heuristics methods are usually inspired byclassical local search algorithms such as 2-opt [11] and the large neighborhood search (LNS)
[51],and have been demonstrated with outstanding results by many [58, 41, 56, 12, 10, 22, 60, 43, 24, 31].
Improvement heuristics methods generally show better performance than construction heuristics butare slower in computation in return.
2.3 Supervised Learners for CO Problems
Vinyals et al.
[54] trained a Pointer Network to predict a TSP solution based on supervised signalsfrom the Held-Karp algorithm [6] or approximate algorithms.
Li et al.
[40] and Joshi et al.
[25]trained a graph convolutional network to predict the possibility of each node or edge to be includedin the optimal solutions of MIS and TSP problems, respectively.
Recently, Joshi et al.
[26] showedthat unsupervised reinforcement learning leads to better emergent generalization over various sizedgraphs than supervised learning.
Our work in this paper provides further evidence for the benefits ofthe unsupervised training, or more specifically, unsupervised generation of heatmaps [40, 25, 17, 36],for combinatorial optimization problems.
Proposed Method
3.1 Formal Definitions
Following a conventional notation [48] we define F s as the set of discrete feasible solutions for a COproblem instance s , and c s : F s → R as the cost function over any element f ∈ F s .
The objective isto find the optimal solution as:
[Equation]
For the Traveling Salesman Problem (TSP), F s is the set of all the routes that visits each node exactlyonce and returns to the starting node by the end, and c s calculates the cost for each route f ∈ F s bysumming up the edge weights in the route.
The size of F s for TSP is n !
for a graph with n nodes.
For the Maximal Independent Set (MIS) problem, F s is a subset of the power set S s = { 0 , 1 } n andconsists of all the independent subsets where each node a subset has no connection to any other nodein the same subset, and c s calculates the negation of the size of each independent subset.
We parameterize the solution space with a continuous and differentiable vector θ ∈ R |V s | , where V sdenotes the variables in the problem instance s (e.g., edges in TSP and nodes in MIS), and estimatesthe probability of each feasible solution f as:
[Equation]
where p θ is an energy function over the discrete feasible solution space, f is a |V s | -dimensionalvector with element f i ∈ { 0 , 1 } indicating whether the i th variable is included in feasible solution f ,and the higher value of θ i means a higher probability for the i th variable produced by p θ ( f | s ) .
3.2 Gradient-based Optimization
When the combinatorial problem is locally decomposable, such a MIS, a penalty loss [28, 2] can beadded to suppress the unfeasible solutions as:
[Equation]
(3) where β > max f ∈F s c s ( f ) .
The objective function ℓ Erdos can thus be calculated analytically andenable end-to-end training.
However, this is not possible for more structured combinatorial problems such as TSP 2 .
Therefore, we propose to directly optimize the expected cost over the underlyingpopulation of feasible solutions, which is defined as:
[Equation]
Optimizing this objective requires efficient sampling, with which REINFORCE-based [57] gradientestimation can be calculate.
Nevertheless, a common practice to sample from the energy p θ functionsrequires MCMC [39], which is not efficient enough.
So we propose to design an auxiliary distributionq θ over the feasible solutions F s , such that the following conditions hold: 1) sampling from q θ isefficient, and 2) q θ and p θ should convergence to the same optimal θ ∗ .
Then, we can replace p θ byq θ in our objective function as:
[Equation]
and get the REINFORCE-based update rule as:
[Equation]
where b ( s ) denotes a baseline function that does not depend on f and estimates the expected costto reduce the variance of the gradients.
In this paper, we use a sampling-based baseline functionproposed by Kool et al.
Next, we specify the auxiliary distributions for TSP and MIS, respectively.
For brevity, we omit theconditional notations of s for all probability formulas in the rest of the paper.
3.2.1 Auxiliary Distribution for TSP
For the TSP problem for an n -node graph, each feasible solution f consists of n edges, which canbe specified as a permutation of n nodes denoted as π f , where π f (0) =
π f ( n ) , and π f ( i ) ̸
= π f ( j ) ,∀ i > 0 , j > 0 , i ̸
= j .
Assuming the starting node is j we have:
[Equation]
To make q TSP θ serve as an auxiliary distribution, we define:
[Equation]
Here the higher valued θ i,j means the higher probability for the edge from node i to node j to besampled.
More importantly, notice that we use matrix θ ∈ R n × n to parameterize the probabilisticdistribution of n !
discrete feasible solutions.
The compact, continuous and differentiable spaceof θ allows us to leverage gradient-based optimization without costly MDP-based construction offeasible solutions, which has been a bottleneck for scaling up in representative DRL solvers so far.
Inother words, we also no longer need costly MCMC-based sampling for optimizing our model dueto the chain-rule decomposition.
Instead, we use autoregressive factorization for sampling from theauxiliary distribution, which is faster than sampling with MCMC from the distribution defined by theenergy function.
The last question is how to choose the first node in a tour.
In this paper, we choose the starting noderandomly with a uniform distribution, which yields:
[Equation]
3.2.2 Auxiliary Distribution for MIS
For the Maximal Independent Set (MIS) problem, the feasible solution is a set of independent nodes,which means that none of the node has any link to any other node in the same set.
To ease the analysis,we further impose a constraint to the MIS solutions such that each set is not a proper subset of anyother independent set in the feasible domain.
To enable the chain-rule decomposition in probability estimation, we introduce a as an ordering ofthe independent nodes in solution f , and { a } f as the set of all possible orderings of the nodes in f .
The chain rule applied to a can thus be defined as:
[Equation]
where G ( a <i ) denotes the set of available nodes for growing partial solution ( a 1 , . .
, a i − 1 ) , i.e., thenodes that are not connected to any nodes in { a 1 , . .
, a i − 1 } .
Notice again that the parameterizationspace for MIS θ ∈ R n (where n denotes the number of nodes in the graph) is compact, continuousand differentiable, which allows efficient gradient-driven optimization.
Due to the space limit, we leave the proof of the convergence between p θ and q θ (i.e., q TSP θ andq MIS θ ) to the appendix.
3.3 Meta-Learning Framework
Model-Agnostic Meta-Learning (MAML) [16] is originally proposed for few-shot learning. In the MAML framework, a model is first trained on a collection of tasks simultaneously, and then adaptsits model parameters to each task.
The standard MAML uses second-order derivatives in training,which are costly to compute. To reduce computation burden, the authors also propose first-orderapproximation that does not require second-order derivatives.
Inspired by MAML, we train a graph neural network (GNN) over a collection of CO problems in away that the it can capture the common nature across all the instances, and adapt its model parameterseffectively to each problem instance based on the features/structure of each input graph.
Let F Φbe the graph neural network with parameter Φ , and denote by κ s the input features of an instancegraph s in collection C , by A s the adjacency matrix of the input graph, and by θ s :=
F Φ ( κ s , A s )the instance-specific initialization of model parameters.
The vanilla loss function is defined as theexpected cost of the solution for any graph in the collection as:
[Equation]
The gradient-based updates can thus be written as:
[Equation]
12) where ∇ θ s ℓ q ( θ s ) is estimated using the REINFORCE algorithm (Equation 6).
Since ℓ q does notdepend on the ground-truth labels, we can further fine-tune neural network parameters on each singletest instance with REINFORCE-based updates, which is referred to as active search [7, 23].
Specifically, the fine-tuned parameters Φ ( T ) s is computed using one or more gradient updates for eachgraph instance s .
For example, when adapting to a problem instance s using T gradient updates withlearning rate α , we have:
[Equation] [Equation]
Next, we optimize the performance of the graph neural network with updated parameters (i.e., Φ ( T ) s )with respect to Φ , with a meta objective:
[Equation]
15) and calculate the meta-updates as:
[Equation]
Notice that we adopt the first-order approximation to optimize this objective, which ignores theupdate via the gradient term of ∇ Φ L ( Φ | { s } ) .
We defer the derivation of the approximation formulato the appendix.
Alg. 1 in the appendix illustrates the full training process of the meta-learning.
3.4 Per-Instance Search
Given a fine-tuned (i.e., after active search) continuous parameterization of the solution space θ ( T ) s ,the per-instance search decoding aims to search for a feasible solution that minimizes the cost functionc .
In this paper, we adotp three decoding strategies, i.e., greedy decoding, sampling, and Monte Carlotree search.
Due to the space limit, the detailed description of three decoding strategies can be foundin the appendix.
3.5 Graph Neural Networks
Based on the shape of the differentiable variable θ required by each problem (i.e., R n × n for TSP and 202 R n for MIS), we use Anisotropic Graph Neural Networks [9] and Graph Convolutional Networks[32] as the backbone network for TSP and MIS tasks, respectively.
Due to the space limit, the detailedneural architecture design can be found in the appendix.
Experiments
[Figure]
4.1 Experiments for Traveling Salesman Problem
[Figure]
4.1.1 Experimental Settings
Data Sets The training instances are generated on the fly.
We closely follow the data generationprocedure of previous works, e.g., [34].
We generate 2-D Euclidean TSP instances by sampling eachnode independently from a uniform distribution over the unit square.
The TSP problems of differentscales are named TSP-500/1000/10000, respectively, where TSP- N indicates the TSP instance on Nnodes.
For testing, we use the test instances generated by Fu et al.
There are 128 test instancesin each of TSP-500/1000, and 16 test instances in TSP-10000.
Evaluation Metrics For model comparison, we report the average length (Length), average performance drop (Drop) and averaged inference latency time (Time), respectively, where Length (theshorter, the better) is the average length of the system-predicted tour for each test-set graph, Drop(the smaller, the better) is the average of relative performance drop in terms of the solution lengthcompared to a baseline method, and Time (the smaller, the better) is the total clock time for generatingsolutions for all test instance, in seconds (s), minutes (m), or hours (h).
Training and Hardware Due to the space limit, please refer to the appendix.
4.1.2 Main Results
Our main results are summarized in Table 1, with T = 15 for TSP-500, T = 14 for TSP-1000, and
T = 12 for TSP-10000.
For the evaluation of DIMES, we fine-tune the DIMES on each instancefor 100 steps (TSP-500 & TSP-1000) or for 50 steps (TSP-10000). For the sampling in DIMES, weuse the temperature parameter τ = 0 . 01 for DIMES+S and τ =
for DIMES +AS +S
We compare
DIMES with 14 other TSP solvers on the same test sets We divide those 14 methods into twocategories traditional
OR methods and 8 learning -based methods.
• Traditional operations research methods include two exact solvers, i
e., Concorde [4] and Gurobi[19], and a strong heuristic solver named LKH-3 [20].
For LKH-3, we consider two settings: (i) default : following previous work [34], we perform 1 runs with a maximum of 10000 trials (the default configuration of LKH-3); (ii) less trials : we perform 1 run with a maximum of 500 trials for TSP-500/1000 and 250 trials for TSP-10000, so that the running times of LKH-3 match those
of DIMES+MCTS. Besides, we also compare DIMES against simple heuristics, including Nearest,Random, and Farthest Insertion.
• Learning-based methods include 8 variants of the 4 methods with the strongest results in recentbenchmark evaluations, namely EAN [13], AM [34], GCN [25], POMO+EAS [23], and Att-GCN[17], respectively.
Those methods can be further divided into the reinforcement learning (RL)sub-category and the supervised learning (SL) sub-category.
Some reinforcement learning methodscan further adopt an Active Search (AS) stage
to fine-tune on each instance.
The results of thebaselines except the running time of Att-GCN are taken from Fu et al. Note that baselines [Table] [Table]
are trained on small graphs and evaluated on large graphs, while DIMES can be trained directly onlarge graphs.
We re-run the publicly available code of Att-GCN on our hardware to ensure faircomparison of time.
The decoding schemes in each method (if applicable) are further specified as Greedy decoding (G),
Sampling (S), Beam Search (BS), and Monte Carlo Tree Search (MCTS).
The 2-OPT improvements[11] can be optionally used to further improve the neural network-generated solution via heuristiclocal search.
See Section 3.4 for a more detailed descriptions of the various decoding techniques.
As is shown in the table, DIMES significantly outperforms many previous learning-based methods.
Notably, although DIMES is trained without any ground truth solutions, it is able to outperform severalsupervised methods.
DIMES also consistently outperforms simple traditional heuristics.
The bestperformance is achieved by RL+AS+MCTS, which requires considerably more time.
RL+AS+G/Sare faster than RL+AS+MCTS and are competitive to the simple heuristics.
Removing AS in DIMESreduces the running time and leads to only a slight, acceptable performance drop.
Moreover, theyare still better than many previous learning-based methods in terms of solution quality and inferencetime.
[Table]
4.1.3 Ablation Study
On Meta-Learning To study the efficacy of meta-learning, we consider two dimensions of ablations:(i) with or without inner gradient updates: whether to use f Φ ( κ s , A s ) or θ ( T ) s in the objective function;(ii) with or without fine-tuning in the inference phase.
The results on TSP-1000 with training phase T = 10 and greedy decoding are summarized in Table 2a.
Both inner updates and fine-tuning arecrucial to the performance of our method.
That is because meta-learning helps the model generalizeacross problem instances, and fine-tuning helps the trained model adapt to each specific probleminstance.
On Fine-Tuning Parts We study the effect of fine-tuning parts during both training and testing.
In general, the neural architecture we used is a GNN appended with an MLP, whose output iscalled continuous parameterization.
We consider the following fine-tuning parts: (i) the continuousparameterization (Cont. Param.) ; (ii) the parameter of MLP; (iii) the output of GNN and theparameters of MLP; (iv) the parameters of GNN and MLP.
Table 2b summarizes the results withvarious fine-tuning parts for TSP-1000 with training phase T = 5 and greedy decoding.
The resultdemonstrates that (iii) works best.
We conjecture that (iii) makes a nice trade-off between universalityand variance reduction.
On Inner Gradient Update Steps We also study the effect of the number T of inner gradientupdate steps during training.
Table 2c shows the test performance on TSP-1000 by greedy decodingwith various T ’s. As the number of inner gradient updates increases, the test performance improvesaccordingly.
Meanwhile, more inner gradient update steps consumes more training time.
Hence,there is a trade-off between performance and training time in practice.
On Heatmaps for MCTS To study where continuous parameterization of DIMES is essential togood performance in MCTS, we replace it with the following heatmaps: (i) each value is independentlysampled from Unif(0 , 1) ; (ii) 1 / ( r i + 1) , where r i ≥ 1 denotes the rank of the length of the i -th edgeamong those edges that share the source node with it.
This can be regarded as an approximation tothe nearest neighbor heuristics.
We also compare with the Att-GCN heatmap [17].
Comparison ofcontinuous parameterizations for TSP-1000 by MCTS is shown in Table 2d.
The result confirms thatthe DIMES continuous parameterization does not simply learn nearest neighbor heuristics, but canidentify non-trivial good candidate edges.
4.2 Experiments For Maximal Independent Set
4.2.1 Experimental Settings
Data Sets We mainly focus on two-types of graphs that recent work [40, 1, 8] shows strugglesagainst, i.e., Erd˝os-Rényi (ER) graphs [15] and SATLIB [21], where the latter is a set of graphsreduced from SAT instances in CNF.
The ER graphs of different scales are named ER-[700-800]and ER-[9000-11000], where ER-[ N - M ] indicates the graph contains N to M nodes.
The pairwiseconnection probability p is set to 0 .
15 and 0 .
02 for ER-[700-800] and ER-[9000-11000], respectively.
The 4,096 training and 5,00 test ER graphs are randomly generated.
For SATLIB, which consists of 40,000 instances, of which we train on 39,500 and test on 500.
Each SAT instance has between403 to 449 clauses.
Since we cannot find the standard train-test splits for both SAT and ER graphsdatasets, we randomly split the datasets and re-run all the baseline methods.
Evaluation Metrics To compare the solving ability of various methods, we report the average sizeof the independent set (Size), average performance drop (Drop) and latency time (Time), respectively,where Size (the larger, the better) is the average size of the system-predicted maximal independent setfor each test-set graph, Drop and Time are defined similarly as in Sec. 4.1.1.
Training and Hardware Due to the space limit, please refer to the appendix.
4.2.2 Main Results
Our main results are summarized in Table 3, where our method (last line) is compared 6 other MISsolvers on the same test sets, including two traditional OR methods (i.e., Gurobi and KaMIS) andfour learning-based methods.
The active search is not used for MIS evaluation since our preliminaryexperiments only show insignificant improvements.
For Gurobi, we formulate the MIS problemas a integer linear program.
For KaMIS, we use the code unmodified from the official repository 3 .
The four learning-based methods can be divided into the reinforcement learning (RL) category, i.e., S2V-DQN [30] and LwD [1]) and the supervised learning (SL) category, i.e., Intel [40] and DGL [8].
We produced the results for all the learning-based methods by running an integrated implementation 4provided by Böther et al.
Notice that as pointed out by Böther et al.
[8], the graph reductionand local 2 -opt search [3] can find near-optimal solutions even starting from a randomly generatedsolution, so we disable the local search or graph reduction techniques during the evaluation for alllearning based methods to show their real CO solving ability.
The methods that cannot produceresults in the 10 × time limit of DIMES are labeled as N/A.
As is shown in Table 3, our DIMES model outperforms previous baseline methods on the mediumscale SATLIB and ER-[700-800] datasets, and significantly outperforms the supervised baseline (i.e.,
Intel) on the large-scale ER-[9000-11000] setting.
This shows that supervised neural CO solverscannot well solve large-scale CO problem due to the expensive annotation problem and generalizationproblem.
In contrast, reinforcement-learning methods are a better choice for large-scale CO problems.
We also find that LwD outperforms DIMES on the large-scale ER-[9000-11000] setting.
We believethis is because LwD is specially designed for locally decomposable CO problems such as MIS andthus can use parallel prediction, but DIMES are designed for general CO problems and only usesautoregressive factorization.
How to better utilize the fact of local decomposability of MIS-likeproblems is one of our future work.
Concluding Remarks
Scalability without significantly scarifying the approximation accuracy is a critical challenge incombinatorial optimization.
In this work we proposed DIMES, a differentiable meta solver that isable to solve large-scale combinatorial optimization problems effectively and efficiently, including TSP and MIS.
The novel parts of DIMES include the compact continuous parameterization and themeta-learning strategy.
Notably, although our method is trained without any ground truth solutions,it is able to outperform several supervised methods.
In comparison with other strong DRL solverson TSP and MIS problems, DIMES can scale up to graphs with over ten thousands nodes while theothers either fail to scale up, or can only produce significantly worse solutions instead in most cases.
One limitation of DIMES is that the continuous parameterization θ is generated in one-shot withoutintermediate steps, which could potentially limit the reasoning power of our method, as shown in the
MIS task. Another limitation is that applying DIMES to a broader ranges of NP-complete problemsthat variables can take multiple values, such as Mixed Integer Programming (MIP), is not trivial andneeds further understanding of the nature of the problems.
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