A note on the NP-hardness of the separation problem on some valid inequalities for the elementary shortest path problem

Abstract

In this paper, we investigate the separation problem on some valid inequalities for the s - t elementary shortest path problem in digraphs containing negative directed cycles. As we will see, these inequalities depend to a given parameter k ∈ ℕ. To show the NP-hardness of the separation problem of these valid inequalities, considering the parameter k ∈ ℕ, we establish a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s1, t1), (s2, t2) ... (sk+2, t k+2) in a digraph to the decision problem associated to the separation of these valid inequalities. Through some illustrative instances, we exhibit the evoked polynomial reduction in the cases k = 0 and k = 1.

polytope; digraphs; shortest path; valid inequality; separation

A note on the NP-hardness of the separation problem on some valid inequalities for the elementary shortest path problem

M.S. IbrahimI, ^*; N. Maculan^II; M. Minoux^III

^IUniversité A. Moumouni, Faculté des Sciences, BP 10.622, Niamey, Niger. E-mail: [e-mail]

^IIFederal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil. E-mail: [e-mail]

^IIIUniversite Pierre et Marie Curie, Paris, France. E-mail: [e-mail]

ABSTRACT

In this paper, we investigate the separation problem on some valid inequalities for the s - t elementary shortest path problem in digraphs containing negative directed cycles. As we will see, these inequalities depend to a given parameter k Є ℕ. To show the NP-hardness of the separation problem of these valid inequalities, considering the parameter k Є ℕ, we establish a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s₁, t1), (s2, t2) ... (sk+2, t_k+2) in a digraph to the decision problem associated to the separation of these valid inequalities. Through some illustrative instances, we exhibit the evoked polynomial reduction in the cases k = 0 and k = 1.

Keywords: polytope, digraphs, shortest path, valid inequality, separation.

1 INTRODUCTION

Let G = (V, E), be a connected directed graph and s Є V and t Є V two vertices of G. We suppose that G contains q elementary paths from s to t, and we denote p_1;p₂,..., pq,these s - t elementary paths. V (pi) and E (pi) are the set of vertices and the set of arcs corresponding to the s-t elementary path pi respectively.

Given a parameter k Є , let (S_k, A_k) be a pair of sets and in the digraph G = (V, E) such that:

• No arc in A_k has an endpoint in S_k;

•

.

In Ibrahim (2008) and Ibrahim et al. (2014), we call a pair (S_k, A_k) as defined above a k-subset pair with respect to vertices s and t. An element of S_k is called a k-vertex andanelement of A_k is called a k-arc.

For any pair (S_k, A_k) which is a k-subset pair, the inequality

is shown to be valid for P, the polytope induced by the s - t elementary paths in G. x_p and y_qr are binary variables associated to the vertex p and the arc (q, r) respectively, (see Ibrahim (2008), Ibrahim et al. (2014)). Such valid inequality is called a valid inequality of order k.

In this paper, we investigate the separation problem of the so-called valid inequalities of order k, first presented and exploited in cutting plane framework in view to solve the shortest path problem in digraphs possibly having negative cycles (see Ibrahim (2008), Ibrahim et al. (2014)). Considering a mixed integer linear model of the shortest elementary path problem, we use these valid inequalities in a cutting plane algorithm to build strong linear relaxations. For the mixed integer linear model of the shortest elementary path problem, one could refer to Maculan et al. (2003) and Ibrahim et al. (2009). We prove the NP-hardness of the separation problem of valid inequalities oforder k by establishing a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s₁, t1), (s2, t2),..., (sk+₂, tk+₂) in a digraph to the decision problem associated to the separation of these valid inequalities.

Figure 1 and Figure 2 illustrate the concept of k - arc and k - vertex. In Figure 1, the arc (11, 7) is a 0 - arc w.r.t. vertices s = 8 and t = 15 as it is not belonging to any elementary path between vertices 8 and 15. In Figure 2, (Ø, A₁) constitutes a 1-subset pair, with A₁ = {(3, 7), (8, 7), (11, 10)} and induces the valid inequality y3,7 + y8,7 + y_{11, 10}< 1. That is, all s - t elementary paths in Figure 2 passe by at most one of the 1 - arcs (3, 7), (8, 7) and (11, 10). One can also observe in Figure 2 that the vertices 14 and 15 are 0 - vertices. k - arcs and k - vertices induce what we call valid inequalities of order k and we will see that in general the problem consisting to detect such vertices and arcs is a difficult task.

The paper is organized as follows. In section 2, we address some cases for which the separation problem of valid inequalities of order k can be solved in polynomial time. Then, we show the NP-hardness of the problem of separation of these valid inequalities in digraphs for a given general k. In section 3, considering the cases k = 0 and k = 1, we present some instances illustrating the evoked polynomial reduction between the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s₁, t₁), (s₂, t2),(sk+2, tk+2) and the decision problem associated to the separation of these valid inequalities in a digraph.

2 SEPARATION PROBLEM FOR VALID INEQUALITIES OF ORDER k

Given (x, y) an optimal (fractional) solution of a shortest path linear model, the separation problem w.r.t. valid inequalities of order k consists in finding in G,a k-subset pair (S_k, A_k) such that:

The problem of separating valid inequalities of order k corresponds in looking for a k-subset pair (S_k, A_k) in G such that the valid inequality of order k is violated.

2.1 Some polynomial cases

We have polynomial algorithms for some special cases:

• if , then α Є S_k, k = 0;

• if and there is no elementary path from s to α, then αЄ S_k, k = 0;

• if and there is no elementary path from α to t,then α_i Є S_k, k = 0.

Where and denote the sets of arcs coming into and going out of the vertex α, respectively. That is, considering a digraph and .

In the case of undirected graphs, for k = 0, the first polynomial algorithms solving the problem of the existence of k + 2 vertex-disjoint elementary paths between k + 2 pairs of vertices (s₁, t1),(s2, t2),..., (sk+2, tk+2) are due to Ohtsuzi (1981), Seymour (1980), Shiloach (1980) and Thomassen (1985). Robertson & Seymour (1995) treat the general case of k.

In digraphs, there exist particular cases for which the problem of the existence of k + 2vertex-disjoint elementary paths between k + 2 pairs of vertices (s₁, t₁), (s₂, t₂),...,(sk₊₂, tk₊₂) is solvable in polynomial time. Perl & Shiloach (1978) present a polynomial algorithm that solves such problem, with k = 0, in three connected directed planar and directed acyclic graphs. The latter result concerning directed acyclic graphs is extended for a given k by Fortune et al. (1980). One can remark that in directed acyclic graphs, the separation problem of these inequalities is not interesting, as in such digraph the shortest path problem can be solved easily. On other hand, Schrijver (1994) present a polynomial method for planar digraphs for a given k.

2.2 The general case for the problem

In the general case, for a given k Є , the separation problem consists in finding k + 1-uplet (θ₁,θ₂,..., θ_k+₁) in G such that θ_i Є S_k or θ_i Є A_k and i = 1,...,k + 1. If θ_i Є S_k, we set θ_i = α_i, otherwise θ_i Є A_k and we set θ_i = (α_i ,β_i),where α_i and β_i are the endpoints of the arc θ_i .

Let П_k be the following decision problem associated to the separation problem of valid inequalities of order k:

"Given k + 1 vertices and/or arcs θ₁,θ₂,..., θ_k+₁ in G, is θ_i a k-vertex or a k-arc?" With i = 1,...,k + 1.

Consider the problem П'_k defined as follow:

"Given 2k + 4 distinct vertices s₁, t₁, s₂, t₂, ..., sk₊₂, tk₊₂,arethereno k + 2 elementary paths, P_s1,t1,P_s2,t2 ... Psk+1,tk+1, Psk+2,tk+2 in G such that

Where V (P_si,ti) are vertex sets of the elementary path P_Si,ti between s_i and t_i. П'_k is well known to be NP-complete in general digraph even if k = 0 (see Fortune, Hopcroft & Wyllie (1980), Garey & Johnson (1979)).

For a given k, we show the NP-completeness of П_k, by exhibiting the following polynomial reduction from П'_k toП_k:

For any instance of П'_k, considering a k + 1-uplet (θ₁,θ₂,..., θ_k+1) such that θ_i Є S_k or θ_i Є A_k and i = 1,...,k + 1, its corresponding instance of П_k is obtained:

• by adding a vertex αi and the arcs (ti ,αi) and (αi, s_i+1),if θi is the vertex αi;

• by adding the arcs (αi ,βi), (ti,αi) and (βi, s_i+1),if θi is the arc (αi ,βi);

• and by setting s = s₁ and t = tk₊₂.

Lemma 2.1.The answer to the instance of П'_k is YES iff the answer to the instance of П_k is also YES.

Proof.i) ⇒: Let Ps,t be a path that visits the nodes α1,α2,... αk+1 (in this order) in the graph of the instance of Пk. This path can be decomposed into the sub-paths Psi, ti, Pti,si+1 and Psi+1,ti+1, i = 1,...,k + 1 where s = s₁, t = tk+2 and paths Pti,si+1 , i = 1,...,k + 1 are the sequences ti ,αi, si+1. Ps,t cannot be elementary because , since the answer to the instance of П'k is YES.

ii) ⇐: Let P_s1,t1, P_s2,t2,..., Psk+2,tk+2 be paths in the graph of the instance of П'k. Consider a path Ps,t in the graph of the instance of Пk. Ps,t can be decomposed into the sub-paths Psi,ti, Pti,si+1 and Psi_{+1 ,ti+1} , i = 1 ,... , k + 1 wheres s = s₁, t = tk₊₂ and paths Pti,si ₊₁, i = 1 ,..., k+ 1 are the sequences ti ,αi, si+1. Since the answer to the instance of Пk is YES, Ps,t is not elementary. This implies that .

Theorem.The decision problem П_k is NP-complete.

Proof. As the decision problem П'_k is known to be NP-complete in general digraph even if k = 0 (see Fortune, Hopcroft & Wyllie (1980), Garey & Johnson (1979)), by Lemma 2.1, it's obvious that the problem П_k is also NP-complete.

After such polynomial transformation, for a given k Є , we conclude that the problem of separation of valid inequalities of order k is NP-hard as its associated decision problem П_k is NP-complete.

3 SEPARATION PROBLEM IN THE CASES k = 0 AND k = 1

3.1 Separation problem in the case k = 0

In the case k = 0, the decision problem associated to the separation problem of valid inequalities of order 0, П₀, is formulated as follow:

"Given a vertex α or an arc (α, β) in G, is α a 0-vertex or is (α, β) a 0-arc w.r.t. vertices s and t ?"

To answer the complexity issue, let us consider the problem П'₀ defined as:

"Given four distinct vertices s₁, t₁, s₂, t₂, are there no two elementary paths, P_s1,t1 and P_s2,t2 in G such that is well known to be NP-complete in general digraphs, see Fortune, Hopcroft & Wyllie (1980), Garey & Johnson (1979).

The NP-completeness of problem П₀ is readily obtained by considering the following polynomial reduction from П_'0 to П₀:

For any instance of П'₀ the corresponding instance of П₀ is obtained by adding a vertex α,two arcs (t₁,α) and (α, s₂), and by setting s = s₁ and t = t₂, or by adding the arcs (α, β), (t₁,α), (β, s₂) and we set s = s₁ and t = t₂.

Lemma 3.1.The answer to the instance of П'₀ is YES iff the answer to the instance of П₀ is YES.

Proof.i) ⇒: Let Ps,t be a path that visits node a in the graph of the instance of П₀.This path can be decomposed into the sub-paths Ps1,t1, Pt1,s2, Ps2,t₂, where s = s₁, t = t₂ and path Pt1,s2 is the sequence t1,α,s₂. Ps,t cannot be elementary because , since the answer to the instance of П'₀ is YES.

ii) ⇐: Let P_s1,t1 and P_s2,t2 be paths in the graph of the instance of П'₀ Consider a path Ps,t in the graph of the instance of П₀. Ps,t can be decomposed into the sub-paths P_s1,t1, P_t1,s2, P_s2,t2, where s = s₁, t = t₂ and path Pt1,s2 is the sequence t₁,α, s₂. Since the answer to the instance of П₀ is YES, Ps,t is not elementary. This implies that

Example. Consider the next instance of П'₀ such that s₁= 6, s₂= 5, t₁= 4, and t₂= 3:

The elementary paths between s₁= 6 and t₁= 4 represented by vertices [6, 1, 3, 4] and [6, 1, 2, 3, 4] are not vertex-disjoints with [5, 6, 1, 2, 3] and [5, 6, 1, 3] the elementary paths between s₂= 5and t₂= 3, thus {6, 1, 3, 4} ∩ {5, 6, 1, 2, 3} ≠ Ø, {6, 1, 2, 3, 4} ∩{5, 6, 1, 2, 3} ≠ Ø, {6, 1, 3, 4} ∩ {5, 6, 1, 3} ≠ Ø and {6, 1, 2, 3, 4} ∩ {5, 6, 1, 3} ≠ Ø.

As explained above, to obtain the following instance of П₀ from П´₀, we add the vertex α and the arcs (4,α) and (α, 5) or by adding the arcs (α, β), (4, α) and (β, 5)

The fact that the answer of the problem П´₀ is YES, i.e, elementary paths represented by vertices [6, 1, 3, 4], [6, 1, 2, 3, 4] and [5, 6, 1, 3], [5, 6, 1, 2, 3] are not vertex-disjoints, it follows that the answer of the problem П₀ is also YES. Then, α is a 0 - vertex. One can observe that α does not belong to any elementary path between vertices s1 = 6 and t₂= 3 (see Fig.4).

3.2 Separation problem in the case k = 1

In the case k = 1, the associate decision problem П₁ is as follow : "Given two vertices α, β in G, are α and β being 1-vertices w.r.t s and t?". Consider the problem П'₁:

"Given six distinct vertices s₁, t₁, s₂, t₂, s₃, t₃, are there no three elementary paths, P_s1,t1, P_s2,t2 and P_s3,t3 in G such that V (PSi,ti) ∩ V(Psj,tj) = Ø, 1 < i < j < 3?"

As П'₀ is a special case of П'₁. П'₁ is NP-complete in general digraph, (see Fortune, Hopcroft & Wyllie (1980), Garey & Johnson (1979)). We show the NP-completeness of П₁, by exhibiting the following polynomial reduction from П'₁ to П₁: For any instance of П'₁, the corresponding instance of П₁ is obtained by adding the vertices α, β and the four arcs (t1,α), (α, s2), (t2,β) and (β, s₃), and by setting s = s₁ and t = t₃.

W.r.t arcs (α1,α2) and (β1,β2) to create the corresponding instance of П₁ from any instance of П'₁ we add the arcs (α₁,α₂), (β₁,β₂), (t₁,α₁), (α₂, s₂), (t2,β1) and (β₂, s₃).

Lemma 3.2.The answer to the instance of П'₁is YES iff the answer to the instance of П1 is YES.

Proof.i) ⇒: Let Ps,t be a path that visits the nodes α and β (in this order) in the graph of the instance of П₁. This path can be decomposed into the sub-paths P_s1,t1, P_t1,s2, P_s2,t2, P_t2,s3, P_s3,t3, where s = s₁, t = t₃ and path Pt1,s2 is the sequence t1,α, s2 and path Pt2,s3 is the sequence t2, β, s3. Ps,t cannot be elementary because V (Ps1,t1) ∩ V (Ps2,t2) ∩ V(Ps3,t3) ≠ Ø, since the answer to the instance of П'₁ is YES.

ii) ⇐: Let Ps1,t1 , Ps2,t2 and P_{s3 ,t3} be paths in the graph of the instance of Consider a path Ps,t in the graph of the instance of П₁. Ps,t can be decomposed into the sub-paths Ps1,t1, Pt1,s2, Ps2,t2, Pt2,s3 and Ps3,t3 ,where s = s₁, t = t₃ and path Pt1,s2 is the sequence t₁,α, s₂ and path Pt2,s3 is the sequence t2,β, s₃. Since the answer to the instance of П₁ is YES, Ps,t is not elementary. This implies that V(Ps1,t1) ∩ V(Ps2,t2) ∩ V(P_s3,t3) ≠ Ø.

Example. Consider the below instance of П'₁:

Let s₁= 8, t1 = 6, s2 = 7, t2 = 5, s3 = 4, and t₃= 10, we observe that there is no three vertex-disjoint elementary paths between (s₁, t₁), (s₂, t₂) and (s₃, t₃) in the above instance of digraph. By transformation, we can obtain in polynomial time an instance of П₁ as follow:

The fact that the answer of the problem П'₁ is YES, i.e, there is no vertex-disjoint elementary paths between vertices (8, 6), (7, 5) and (4, 10) in the considered instance of П'₁, it follows that the answer of the problem П₁ is also YES. Thus, it doesn't exist any elementary path between vertices s₁ = 8 and t₃ = 10 containing both vertices α and β or arcs (α₁, α₂) and (β₁, β₂).

4 CONCLUSION

In this paper, we prove theNP-hardness of the separation problem of the so-called valid inequalities of order k. We establish a polynomial reduction from the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s₁, t₁), (s₂, t2) ... (sk+2, tk+2) in a digraph to the decision problem associated to the separation of valid inequalities of order k. We recall that the problem of the existence of k + 2 vertex-disjoint paths between k + 2 pairs of vertices (s₁, t₁), (s₂, t2)... (sk+2, tk+2) in a digraph is known to be NP-complete.

ACKNOWLEDGMENTS

We gratefully acknowledge the referees for their careful reading and insightful and constructive comments.

REFERENCES

[1] FORTUNE S, HOPCROFT J& WYLLIE J. 1980. The directed subgraph homeomorphism problem. Theoretical Computer Science, 10: 111-121.

[2] GAREY M & JOHNSON D. 1979. Computer and Intractibility: a guide to the theory of NP-completeness. Freeman, San Francisco.

[3] IBRAHIM MS. 2008. Etude de formulations et inégalités valides pour le problème du plus court chemin dans un graphe avec des circuits absorbants. PhD dissertation,LIP6, Université Pierre et Marie Curie, Paris, France.

[4] IBRAHIM MS, MACULAN N & MINOUX M. 2009. Strong flow-based formulation for the shortest path problem in digraphs with negative cycles. International Transaction in Operations Research (ITOR), 16: 361-369.

[5] IBRAHIM MS, MACULAN N & MINOUX M. 2014. Valid inequalities and lifting procedures for the shortest path problem in digraphs with negative cycles. Accepted to be published in Optimization Letters.

[6] MACULAN N, PLATEAU G & LISSER A. 2003. Integer linear models with a polynomial number of variables and constraints for some classical combinatorial optimization problems. Pesquisa Operacional, 23: 161-168.

[7] OHTSUKI T. 1981. The two disjoint path problem and wire routing design. Proceeding Symposium On Graph Theory and Applications, Lecture Notes in Computer Science, 108: 207-216, Berlin.

[8] PERL Y & SHILOACH Y. 1978. Finding two disjoint paths between two pairs of vertices in a graph. J. ACM, 25: 1-9.

[9] ROBERTSON N & SEYMOUR PD. 1995. Graphs minors XIII, The disjoint paths problem. J. Combinatorial Theory,Ser.B, 63: 65-110.

[10] SCHRIJVER A. 1994. Finding k-disjoint paths in directed planar graph. SIAM, J. Comput., 23: 780-788.

[11] SEYMOUR PD. 1980. Disjoint paths in graphs. Discrete Mathematics, 29: 293-309.

[12] SHILOACH Y. 1980. A polynomial solution to the undirected two paths problems. J. ACM, 27: 445-456.

[13] THOMASSEN C. 1985. The two linkage problem for acyclic digraphs. Discrete Mathematics, 55: 73-87.

Received September 27, 2012

Accepted March 23, 2014

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Corresponding author

Publication Dates

• Publication in this collection
  08 May 2014
• Date of issue
  Apr 2014

History

• Received
  27 Sept 2012
• Accepted
  23 Mar 2014