An iterative ILP approach for constructing a Hamiltonian decomposition of a regular multigraph
aa r X i v : . [ m a t h . C O ] F e b An iterative ILP approach for constructing aHamiltonian decomposition of a regularmultigraph ⋆ Andrey Kostenko [0000 − − − and Andrei Nikolaev [0000 − − − P. G. Demidov Yaroslavl State University, Yaroslavl, Russia, [email protected], [email protected]
Abstract.
A Hamiltonian decomposition of a regular graph is a par-tition of its edge set into Hamiltonian cycles. The problem of findingedge-disjoint Hamiltonian cycles in a given regular graph has many ap-plications in combinatorial optimization and operations research. Ourmotivation for this problem comes from the field of polyhedral combina-torics, as a sufficient condition for vertex nonadjacency in the 1-skeletonof the traveling salesperson polytope can be formulated as the Hamilto-nian decomposition problem in a 4-regular multigraph.In our approach, the algorithm starts by solving the relaxed 2-matchingproblem, then iteratively generates subtour elimination constraints forall subtours in the solution and solves the corresponding ILP-model tooptimality. The procedure is enhanced by the local search heuristic basedon chain edge fixing and cycle merging operations. In the computationalexperiments, the iterative ILP algorithm showed comparable results withthe previously known heuristics on undirected multigraphs and signifi-cantly better performance on directed multigraphs.
Keywords:
Hamiltonian decomposition · Traveling salesperson poly-tope · · Vertex adjacency · Integer linear programming · Sub-tour elimination constraints · Local search
A Hamiltonian decomposition of a regular graph is a partition of its edgeset into Hamiltonian cycles. The problem of finding edge-disjoint Hamiltoniancycles in a given regular graph plays an important role in combinatorial optimiza-tion [23], coding theory [4,5], privacy-preserving distributed mining algorithms[10], analysis of interconnection networks [19] and other areas. See also theoret-ical results on estimating the number of Hamiltonian decompositions of regulargraphs [17]. Our motivation for this problem comes from the field of polyhedralcombinatorics. ⋆ This research was supported by P.G. Demidov Yaroslavl State University ProjectVIP-016 A. Kostenko, A. Nikolaev
We consider a classic traveling salesperson problem: given a complete weightedgraph (or digraph) K n = ( V, E ), it is required to find a Hamiltonian cycle ofminimum weight. We denote by HC n the set of all Hamiltonian cycles in K n .With each Hamiltonian cycle x ∈ HC n we associate a characteristic vector x v ∈ R E by the following rule: x ve = ( , if the cycle x contains an edge e ∈ E, , otherwise.The polytope TSP( n ) = conv { x v | x ∈ HC n } is called the symmetric traveling salesperson polytope . The asymmetric traveling salesperson polytope
ATSP( n ) is defined similarlyas the convex hull of characteristic vectors of all possible Hamiltonian cycles inthe complete digraph K n .The 1- skeleton of a polytope P is the graph whose vertex set is the vertexset of P and edge set is the set of geometric edges or one-dimensional faces of P . The study of 1-skeleton is of interest, since, on the one hand, there are algo-rithms for perfect matching, set covering, independent set, a ranking of objects,problems with fuzzy measures, and many others that are based on the vertexadjacency relation in 1-skeleton and the local search technique (see, for exam-ple, [1,6,11,12,15]). On the other hand, some characteristics of 1-skeleton, suchas the diameter and clique number, estimate the time complexity for differentcomputation models and classes of algorithms [7,9,18].Unfortunately, the classic result by Papadimitriou states that the construc-tion of a 1-skeleton of the traveling salesperson polytope is hard for both directedand undirected graphs. Theorem 1 (Papadimitriou [26]).
The question of whether two vertices ofthe polytopes
TSP( n ) or ATSP( n ) are nonadjacent is NP-complete. Note that the complementary problem of verifying vertex adjacency will beco-NP-complete.
We apply the Hamiltonian decomposition problem to analyze the 1-skeletonof the traveling salesperson polytope. Let x = ( V, E x ) and y = ( V, E y ) be twoHamiltonian cycles on the vertex set V . We denote by x ∪ y a multigraph ( V, E x ∪ E y ) that contains all edges of both cycles x and y . Lemma 1 (Sufficient condition for nonadjacency, Rao [29]).
Given twoHamiltonian cycles x and y , if the multigraph x ∪ y contains a Hamiltoniandecomposition into edge-disjoint cycles z and w different from x and y , then thecorresponding vertices x v and y v of the polytope TSP( n ) (or ATSP( n ) ) are notadjacent. n iterative ILP approach for constructing a Hamiltonian decomposition 31 2 3 456 x y x ∪ y z w Fig. 1.
The multigraph x ∪ y has two different Hamiltonian decompositions From a geometric point of view, the sufficient condition means that the seg-ment connecting two vertices x v and y v intersects with the segment connectingtwo other vertices z v and w v of the polytope TSP( n ) (or ATSP( n ) correspond-ingly). Thus, the vertices x v and y v cannot be adjacent in 1-skeleton. An exampleof a satisfied sufficient condition is shown in Fig. 1.Thus, the sufficient condition for vertex nonadjacency of the traveling sales-person polytope can be formulated as a combinatorial problem. Instance.
Let x and y be two Hamiltonian cycles. Question.
Does the multigraph x ∪ y contain a pair of edge-disjoint Hamil-tonian cycles z and w different from x and y ?Note that testing of whether a graph has a Hamiltonian decomposition isNP-complete, even for 4-regular undirected graphs and 2-regular directed graphs[27]. Therefore, polynomially solvable special cases of the vertex nonadjacencyproblem have been studied in the literature. In particular, the polynomial suf-ficient conditions for the pyramidal tours [8], pyramidal tours with step-backs[24], and pedigrees [2,3] are known. However, all of them are weaker than theconsidered sufficient condition by Rao.Previously, the Hamiltonian decomposition and vertex nonadjacency problemwas studied in [22,25], where two heuristic algorithms were proposed. The set offeasible solutions in both algorithms consists of pairs z and w of vertex-disjointcycle covers of the multigraph x ∪ y . Recall that a vertex-disjoint cycle cover ofa graph G is a set of cycles with no vertices in common which are subgraphs of G and contain all vertices of G . The differences are as follows: – the simulated annealing algorithm from [22] rebuilds the cycle covers z and w through the reduction to perfect matching [30] until it obtains a pair ofHamiltonian cycles different from x and w ; A. Kostenko, A. Nikolaev – the general variable neighborhood search algorithm from [25] adds to theprevious algorithm several neighborhood structures and cycle merging oper-ations combined in the basic variable neighborhood descent approach [14].Heuristic algorithms have proven to be very efficient on instances with an exist-ing solution, especially on undirected graphs. However, on instances without asolution, the heuristics face significant difficulties.In this paper, we propose two exact algorithms for solving the problem of con-structing the Hamiltonian decomposition of a 4-regular multigraph and verifyingvertex nonadjacency of the traveling salesperson polytope. The first algorithmiteratively generates integer linear programming models for the problem. Thesecond algorithm combines the first one and the modified local search heuristicfrom [25]. Let x = ( V, E x ), y = ( V, E y ), x ∪ y = ( V, E = E x ∪ E y ). With each edge e ∈ E we associate the variable x e = ( , if e ∈ z, , if e ∈ w. We adapt the classical ILP formulation of the traveling salesperson problem[13] into the following ILP model for the considered Hamiltonian decompositionproblem: X e ∈ E x e = | V | , (1) X e ∈ E v x e = 2 , ∀ v ∈ V, (2) X e ∈ E x \ E y x e ≤ | V | − | E x ∩ E y | − , (3) X e ∈ E y \ E x x e ≤ | V | − | E x ∩ E y | − , (4) X e ∈ E S x e ≤ | S | − , ∀ S ⊂ V, (5) X e ∈ E S x e ≥ | E S | − | S | + 1 , ∀ S ⊂ V, (6) x e ∈ { , } , ∀ e ∈ E. (7)Let’s take a brief overview of the model. The multigraph x ∪ y contains 2 | V | edges. The constraint (1) guarantees that both components z ( x e = 1) and w ( x e = 0) receive exactly | V | edges.We denote by E v the set of all edges incident to the vertex v in x ∪ y . Thenthe vertex degree constraint (2) ensures that the degree of each vertex in z and w is equal to 2. n iterative ILP approach for constructing a Hamiltonian decomposition 5 If we consider the problem on directed graphs, then the vertex degree con-straints should be slightly modified. Let for some vertex v ∈ V : e and e be twoincoming edges, and u and u be two outgoing edges. Then the constraints (2)will take the form: x e + x e = 1 ,x u + x u = 1 , thus, exactly one incoming edge and exactly one outgoing edge for each vertexin the solution.The constraints (3)-(4) forbid Hamiltonian cycles x and y as a solution. Ifwe consider the problem of constructing a Hamiltonian decomposition of a 4-regular multigraph without reference to the vertex adjacency in the travelingsalesperson polytope, then these constraints can be omitted.Finally, the inequalities (5)-(6) are known as the subtour eliminations con-straints , which forbid solutions consisting of several disconnected tours. Here S is a subset of V , E S is the set of all edges from E with both vertices belongingto S : E S = { ( u, v ) ∈ E : u, v ∈ S } . The main problem with the subtour elimination constraints is that thereare exponentially many of them: two for each subset of S ⊂ V , i.e. Ω (2 | V | ).Therefore, the idea of the first algorithm is as follows. We start with the relaxedmodel (1)-(4),(7) of the basic 2-matching problem with O ( V ) constraints. ByILP-solver we obtain an integer point that corresponds to the pair of cycle covers z and w . Then we find all subtours in z and w , add the corresponding subtourelimination constraints (5) and (6) into the model, and iteratively repeat thisprocedure. The algorithm stops either by finding the Hamiltonian decompositioninto cycles z and w or by obtaining an infeasible model that does not containany integer points. This approach is inspired by the algorithm for the travelingsalesperson problem from [28] and is summarized in Algorithm 1. Algorithm 1
Iterative integer linear programming algorithm procedure
IterativeILP ( x ∪ y )define current model as (1)-(4),(7) ⊲ relaxed 2-matching problem while the model is feasible do z, w ← an integer point of the current model by an ILP-solver if z and w is a Hamiltonian decomposition thenreturn Hamiltonian decomposition z and w end if find all subtours in z and w and add the corresponding subtour eliminationconstraints (5) and (6) into the model end whilereturn Hamiltonian decomposition does not exist end procedure
A. Kostenko, A. Nikolaev
To improve the performance, we enhance Algorithm 1 with the local searchheuristic. The neighborhood structure is a modified version of the first neigh-borhood in the GVNS algorithm [25] with a new chain edge fixing procedure.
Every solution of the ILP model (1)-(4),(7) with partial subtour eliminationconstraints corresponds to the pair z and w of vertex-disjoint cycle covers of themultigraph x ∪ y (Fig. 2).We compose a set of feasible solutions for the local search algorithm from allpossible pairs of vertex-disjoint cycle covers of the multigraph x ∪ y . As the objective function to minimize, we choose the total number of connectedcomponents in the vertex-disjoint cycle covers z and w . If it equals 2, then z and w are Hamiltonian cycles. The main difference between the neighborhood structure presented in this chap-ter and those described in [25] is the chain edge fixing procedure. We divide theedges of z and w into two classes: – unfixed edges that can be moved between z and w to get a neighboringsolution; – edges that are fixed in z or w and cannot be moved.The idea is that one fixed edge starts a recursive chain of fixing other edges.For example, we consider a directed 2-regular multigraph x ∪ y with all indegreesand outdegrees are equal to 2. Let us choose some edge ( i, j ) and fix it in thecomponent z , then the second edge ( i, k ) outgoing from i and the second edge( h, j ) incoming into j obviously cannot get into z . We will fix these edges in w (Fig. 3). In turn, the edges ( i, k ) and ( h, j ), fixed in w , start recursive chains offixing edges in z , etc.At the initial step, we fix in z and w a copy of each multiple edge of the x ∪ y , since both copies obviously cannot end up in the same Hamiltonian cycle.We construct a neighboring solution by choosing an edge of z , moving it to w ,and running the chain edge fixing procedure to restore the correct cycle covers.If the number of connected components in z and w has decreased, we proceedto a new solution and restart the local search. This procedure is summarized inAlgorithm 2.Note that although the chain edge fixing procedure at each step can call upto two of its recursive copies, the total complexity is linear ( O ( V )), since eachedge can only be fixed once, and | E | = 2 | V | . Thus, the size of the neighborhoodis equal to the number of edges in z , i.e. O ( V ), and the total complexity ofexploring the neighborhood is quadratic O ( V ). n iterative ILP approach for constructing a Hamiltonian decomposition 71 2 3 456 x ∪ y z w = ( x ∪ y ) \ z Fig. 2.
The multigraph x ∪ y and its two complementary cycle covers i jkh zww Fig. 3.
Fixing edge ( i, j ) in z Algorithm 2
Local search for directed graphs procedure
Chain Edge Fixing Directed (( i, j ) in z )Fix the edge ( i, j ) in z and mark it as checked to avoid double checking if the edge ( i, k ) is not fixed then Chain Edge Fixing Directed (( i, k ) in w ) end ifif the edge ( h, j ) is not fixed then Chain Edge Fixing Directed (( h, j ) in w ) end ifend procedureprocedure Local Search Directed ( z, w )Fix the multiple edges in z and w repeat Shuffle the edges of z in random order for each unchecked and unfixed edge ( i, j ) in z do Chain Edge Fixing Directed (( i, j ) in w ) ⊲ Move ( i, j ) from z to w if the number of connected components in z and w has decreased then Proceed to a new solution and restart the local search end if
Restore the original z and w and unfix all non-multiple edges end foruntil z and w is a local minimum return z and w is a local minimum end procedure A. Kostenko, A. Nikolaev
The neighborhood structure for undirected graphs is similar, we choose an edgeof z , move it to w and run the chain edge fixing procedure.The key difference is that after the exchange of edges and the chain edgefixing procedure, there will remain some broken vertices in z and w with a degreenot equal to 2. We restore the degree of each broken vertex by moving randomunfixed incident edges between components z and w (Fig. 4). This procedure issummarized in Algorithm 3.Since at each step we pick a random edge to restore a broken vertex, thelocal search for undirected graphs is a randomized algorithm. Therefore, we runseveral attempts (parameter attemptLimit ) while constructing each neighboringsolution, i.e. exploring several random branches in the search tree. Thus, the sizeof the neighborhood is equal to O ( V · attemptLimit ), and the total complexityof exploring the neighborhood is O ( V · attemptLimit ). We add the local search heuristic into the iterative ILP algorithm between it-erations to improve the performance on instances with an existing Hamiltoniandecomposition.If the ILP-solver returns a pair of cycle covers z and w that are not a Hamil-tonian decomposition, then we call the local search to minimize the number ofconnected components. If the heuristic also fails, then we modify the model byadding the corresponding subtour elimination constraints and restart the ILP-solver until a Hamiltonian decomposition is found, or the resulting model isinfeasible. This procedure is summarized in Algorithm 4.Note that we generate the subtour elimination constraints for all subtoursin z and w twice: first for an integer solution to the model, and then after thelocal search gets stuck in a local minimum. Thus, we implement the memorystructure and prohibit the algorithm from returning to feasible solutions thathave already been explored. The algorithms were tested on random directed and undirected Hamiltoniancycles. For comparison, we chose two algorithms presented in this paper and twoknown heuristic algorithms: – Iterative ILP algorithm from Section 3 (Algorithm 1); – Iterative ILP + LS algorithm from Section 4 (Algorithm 4); – SA: the simulated annealing algorithm which rebuilds the cycle covers throughthe reduction to perfect matching from [22]; – GVNS: the general variable neighborhood search algorithm from [25]. n iterative ILP approach for constructing a Hamiltonian decomposition 91 234 5 67858 ⇒ Fig. 4.
Restoring the broken vertex 8 (gray colored) by moving the random incidentedge (8 ,
7) from w (dashed edges) to z (solid edges). Algorithm 3
Local search for undirected graphs procedure
Chain Edge Fixing Undirected (( i, j ) in z )Fix the edge ( i, j ) in z if vertex i in z has two incident fixed edges then Chain Edge Fixing Undirected (( i, k ) and ( i, h ) in w ) end ifif vertex j in z has two incident fixed edges then Chain Edge Fixing Undirected (( j, k ) and ( j, h ) in w ) end ifend procedureprocedure Local Search Undirected ( z, w, attemptLimit )Fix the multiple edges in z and w repeat Shuffle the edges of z in random order for each unchecked and unfixed edge ( i, j ) in z do Chain Edge Fixing Undirected (( i, j ) in w ) for i ← to attemptLimit dowhile z contains a broken vertex i with degree not equal to 2 doif vertex degree of i is equal to 1 then Pick a random unfixed edge ( i, k ) of w ⊲ one missing edge
Chain Edge Fixing Undirected (( i, k ) in z ) end ifif vertex degree of i is equal to 3 then Pick a random unfixed edge ( i, k ) of z ; ⊲ one extra edge Chain Edge Fixing Undirected (( i, k ) in w ) end ifend whileif the number of connected components has decreased then Proceed to a new solution and restart the local search end if
Restore z and w and unfix all non-multiple edges except ( i, j ) end for Unfix the edge ( i, j ) and mark it as checked end foruntil z and w is a local minimum return z and w is a local minimum end procedure Algorithm 4
Iterative ILP algorithm with local search procedure
IterativeILP+LS ( x ∪ y, attemptLimit )define the current model as (1)-(4),(7) ⊲ relaxed 2-matching problem while the model is feasible do z, w ← an integer point of the current model by ILP-solver if z and w is a Hamiltonian decomposition thenreturn Hamiltonian decomposition z and w end if find all subtours in z and w and add the corresponding subtour eliminationconstraints (5) and (6) into the model if the graph is directed then z, w ← Local Search Directed ( z, w ); else z, w ← Local Search Undirected ( z, w, attemptLimit ); end ifif z and w is a Hamiltonian decomposition thenreturn Hamiltonian decomposition z and w end if find all subtours in z and w and add the corresponding subtour eliminationconstraints (5) and (6) into the model end whilereturn Hamiltonian decomposition does not exist end procedure
The ILP algorithms are implemented in C++, for the SA and GVNS al-gorithms the existing implementation in Node.js [25] is taken. Computationalexperiments were performed on an Intel (R) Core (TM) i5-4460 machine with a3.20GHz CPU and 16GB RAM. As the ILP-solver we used SCIP 7.0.2 [16].The results of computational experiments are presented in Tables 1 and 2. Foreach graph size, 100 pairs of random permutations with a uniform probabilitydistribution were generated by the Fisher-Yates shuffle algorithm [21]. For twoILP algorithms, a limit of 2 hours was set for each set of 100 instances. Therefore,the tables indicate how many instances out of 100 the algorithms managed tosolve in 2 hours For both heuristic algorithms, a limit of 60 seconds per testwas set, as well as a limit on the number of iterations: 2 500 for SA and 250for GVNS. The reason is that the heuristic algorithms have a one-sided error. Ifthe algorithm finds a solution, then the solution exists. However, the heuristicalgorithms cannot guarantee that the solution to the problem does not exist, onlythat the solution has not been found in a given time or number of iterations. Foreach set of 100 instances, the tables show the average running time in secondsand the average number of iterations separately for problems with and withouta solution.It is known that random undirected regular graphs have a Hamiltonian de-composition with a very high probability [20]. Indeed, for all 1 000 instances onundirected multigraphs (Table 1), there was a Hamiltonian decomposition intocycles different from the original ones, and the vertices of the traveling sales-person polytope were not adjacent. From a geometric point of view, this means n iterative ILP approach for constructing a Hamiltonian decomposition 11
Table 1.
Computational results for 100 random undirected Hamiltonian cycles
Iterative ILP Iterative ILP + LSSolution No solution Solution No solution | V | N time (s) Iter N time (s) Iter N time (s) Iter N time (s) Iter192 100 2 .
029 23 . − − −
100 0 .
052 1 . − − −
256 100 4 .
800 30 . − − −
100 0 .
094 1 . − − −
384 100 12 .
168 34 . − − −
100 0 .
150 1 . − − −
512 100 24 .
914 44 . − − −
100 0 .
217 1 . − − −
768 100 67 .
382 54 . − − −
100 0 .
488 1 . − − − .
215 95 . − − −
100 0 .
721 1 . − − − .
598 33 − − −
100 1 .
518 1 . − − − .
87 235 . − − −
100 3 .
281 1 . − − − .
42 143 . − − −
100 6 .
746 1 . − − − .
19 168 . − − −
100 14 .
447 1 . − − − SA (perfect matching) GVNSSolved Not solved Solved Not solved | V | N time (s) Iter N time (s) Iter N time (s) Iter N time (s) Iter192 100 0 .
884 105 . − − −
100 0 .
023 1 . − − −
256 100 1 .
904 124 . − − −
100 0 .
035 1 . − − −
384 100 7 .
734 228 . − − −
100 0 .
073 1 . − − −
512 99 12 .
880 236 .
39 1 60 .
000 1016 100 0 .
133 1 . − − −
768 70 21 .
223 194 .
74 30 60 .
000 498 .
96 100 0 .
291 1 . − − − .
548 124 .
23 54 60 .
000 313 .
14 100 0 .
511 1 . − − − .
140 70 .
24 75 60 .
000 157 .
05 100 1 .
085 1 . − − − .
540 54 .
33 88 60 .
000 91 .
71 100 1 .
824 1 . − − − .
225 19 .
50 94 60 .
000 41 .
11 100 4 .
235 1 . − − − − − −
100 60 .
000 22 .
22 100 7 .
593 1 . − − − that the degrees of vertices in 1-skeleton are much less than the total number ofvertices, so two random vertices are not adjacent with a high probability.Summary for random undirected multigraphs: both iterative ILP + LS andGVNS solved all 1 000 instances, SA solved 558 instances, and ILP solved only531 instances in a given time (Table 1). It can be concluded that the pure iterativeILP algorithm was not very successful for undirected graphs and showed similarresults to the SA algorithm. On instances up to 768 vertices, where all tests weresolved, the ILP was on average 2.3 times slower than the SA. The problem isthat undirected multigraphs contain a large number of subtours that have tobe forbidden. On average, the ILP algorithm took about 86 iterations to finda solution. On the other hand, the addition of the local search heuristic to theILP algorithm reduced the running time by an average of 200 times, and thenumber of iterations by 65 times. The ILP + LS algorithm showed results similarto GVNS, solving all test instances and being on average only 1.8 times slower.This time loss is due to two factors. Firstly, the GVNS has a more complex Table 2.
Computational results for 100 random directed Hamiltonian cycles
Iterative ILP Iterative ILP + LSSolution No solution Solution No solution | V | N time (s) Iter N time (s) Iter N time (s) Iter N time (s) Iter192 21 0 .
028 4 .
23 79 0 .
029 4 .
22 21 0 .
022 2 .
00 79 0 .
037 3 . .
111 7 .
04 75 0 .
060 5 .
74 25 0 .
075 3 .
12 75 0 .
090 4 . .
054 4 .
65 80 0 .
082 5 .
75 20 0 .
080 2 .
60 80 0 .
156 4 . .
105 5 .
45 78 0 .
114 5 .
58 22 0 .
125 2 .
36 78 0 .
248 4 . .
148 6 .
05 81 0 .
125 5 .
43 19 0 .
226 2 .
26 81 0 .
439 4 . .
182 5 .
70 83 0 .
222 6 .
21 17 0 .
277 1 .
88 83 0 .
925 4 . .
325 6 .
43 84 0 .
404 6 .
83 16 1 .
099 2 .
50 84 2 .
312 5 . .
568 7 .
33 85 0 .
503 6 .
70 15 3 .
137 3 .
13 85 3 .
829 5 . .
130 7 .
95 79 1 .
009 7 .
65 21 4 .
661 2 .
42 79 10 .
722 5 . .
681 8 .
16 82 1 .
522 7 .
95 18 18 .
560 4 .
44 82 21 .
283 5 . | V | N time (s) Iter N time (s) Iter N time (s) Iter N time (s) Iter192 21 0 .
594 183 .
95 71 8 .
196 2500 21 0 .
015 3 .
76 79 1 .
618 250256 20 2 .
557 466 .
75 80 13 .
776 2500 25 0 .
084 10 .
72 75 2 .
168 250384 17 5 .
824 500 .
70 83 28 .
136 2500 20 0 .
110 8 .
00 80 5 .
958 250512 16 6 .
525 315 .
68 84 47 .
791 2500 22 0 .
136 5 .
45 78 10 .
903 250768 13 13 .
384 292 .
15 87 60 .
000 1420 19 0 .
643 10 .
52 81 25 .
673 2501024 9 11 .
576 127 .
88 91 60 .
000 749 .
81 17 1 .
746 16 .
94 83 42 .
272 2501536 − − −
100 60 .
000 361 .
40 16 1 .
072 5 .
81 84 60 .
000 196 . .
717 38 99 60 .
000 249 .
98 15 3 .
201 12 .
00 85 60 .
000 151 . .
719 42 .
33 97 60 .
000 185 .
57 21 5 .
554 12 .
28 79 60 .
000 115 . .
713 37 99 60 .
000 141 .
15 18 9 .
395 17 .
38 82 60 .
000 98 . heuristic with several neighborhood structures, which made it possible to findall solutions in just 1 iteration. Secondly, one iteration of the ILP-solver is muchmore expensive than constructing cycle covers through the reduction to a perfectmatching.It should be noted that although all 1 000 random instances on undirectedgraphs had a solution, in the general case, the traveling salesperson polytopecontains adjacent vertices for which, accordingly, the Hamiltonian decompositiondoes not exist. Moreover, the 1-skeleton of the traveling salesperson polytope hascliques with an exponential number of vertices [7]. Thus, the ILP + LS algorithmmay turn out to be more promising, since it will be able to prove that there isno Hamiltonian decomposition for the given problem.The situation for random directed multigraphs is fundamentally different(Table 2), only 194 out of 1 000 test instances have a solution. Three algorithms:iterative ILP, iterative ILP + LS, and GVNS correctly solved all instances inthe given time, while SA found only 101 Hamiltonian decompositions of 194.It can be seen that directed multigraphs contain fewer subtours compared to n iterative ILP approach for constructing a Hamiltonian decomposition 13 undirected ones. Thus, the iterative ILP algorithm requires on average only 6.3iterations to find a solution, and 6.2 iterations to prove that a solution does notexist. The addition of the local search heuristic to the algorithm makes it possibleto reduce the number of iterations by an average of 2.4 times for problems witha solution and 1.3 times for problems without a solution. In some cases, as forgraphs on 192 and 256 vertices, this speeds up the algorithm. However, in mostcases, the heuristic does not give an improvement in runtime. On average, ILP +LS is 3 times slower on problems with a solution and 5 times slower on problemswithout a solution, and the gap only increases with the growth of the graph size.We can conclude that a few extra iterations of the ILP-solver turn out to becheaper in runtime than using an additional heuristic.Regarding the heuristic algorithms, the performance of GVNS on instanceswith the existing solution is on average 3.8 times slower than ILP and is com-parable to ILP + LS. While SA completely dropped out of the competition,finding only 101 solutions out of 194, and being an order of magnitude slower.As for instances without a solution, both heuristic algorithms are not able todetermine this scenario and exit only when the limit on the running time or thenumber of iterations is reached. In this case, GVNS turns out to be on average100 times slower than ILP. However, it is difficult to compare performance heresince the time and iteration limits in both heuristic algorithms are set as param-eters. Note that, in the GVNS, the limit was 250 iterations, while the algorithmfound a solution, if it exists, on average in 10 iterations. This means that thelimit can potentially be lowered to speed up the algorithm. However, this willincrease the danger of losing the existing solution. We introduced two iterative ILP algorithms to find a Hamiltonian decompositionof the 4-regular multigraph. On random undirected multigraphs, the versionenhanced by the local search heuristic turned out to be much more efficientthan the basic ILP algorithm, showing results comparable to the known generalvariable neighborhood search heuristic. While for random directed multigraphsthe iterative ILP algorithm significantly surpassed in speed the previously knownalgorithms. A key feature that distinguishes the considered ILP algorithms frompreviously known heuristics is the fact that they can prove that the Hamiltoniandecomposition does not exist in the graph.The presented algorithms were developed for the problem of verifying vertexnonadjacency in 1-skeleton of a traveling salesperson polytope. However, theycan also be applied directly to the problem of finding the Hamiltonian decom-position of a regular multigraph and many of its applications.
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