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Simple paths with exact and forbidden lengths
Alexandre Dolgui, Mikhail Kovalyov, Alain Quilliot
To cite this version:
Alexandre Dolgui, Mikhail Kovalyov, Alain Quilliot. Simple paths with exact and forbidden lengths.
Naval Research Logistics, Wiley-Blackwell, 2018, 65 (1), pp.78 - 85. �10.1002/nav.21783�. �hal-
01774847�
Simple Paths with Exact and Forbidden Lengths
Alexander Dolgui
a
, Mikhail Y. Kovalyov
b,∗
, Alain Quilliot
c
a
IMT Atlantique, LS2N, CNRS, La Chantrerie, 4, rue Alfred Kastler - B.P. 20722, F-44307
Nantes Cedex 3, France, E-mail: alexandre.dolgui@imt-atlantique.fr
b
United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk,
Belarus, E-mail: kovalyov my@newman.bas-net.by
c
LIMOS, UMR CNRS 6158, Bat. ISIMA, Universit´e Blaise Pascal, Campus des C´ezeaux,
BP 125, 63173 Aubiere, France, E-mail: alain.quilliot@isima.fr
Abstract
We study new decision and optimization problems of finding a simple path between two
given vertices in an arc weighted directed multigraph such that the path length is equal to a
given number or it does not fall into the given forbidden intervals (gaps). A fairly complete
computational complexity classification is provided and exact and approximation algorithms
are suggested.
Keywords: shortest path problem; longest path problem; exact path length; forbidden path
length; computational complexity; approximation
1 Introduction
Let G = (V, A) be an arbitrary arc weighted directed multigraph, which we further call graph
and where V = {1, . . . , n} is the set of vertices and A is the set of arcs, |A| = m. Arc a
(r)
ij
∈ A
is defined by its head vertex i ∈ V , tail vertex j ∈ V and its copy marker r, if there are
several arcs with the same head and the same tail. A length l(a), which is an arbitrary integer
number, is associated with each arc a ∈ A, and the length L(P ) of a path P from one specified
vertex of G to another is the total length of its arcs. A path with no vertex repetition is called
simple.
We study the following problems Exact Path(α), Path Gaps, Short Path Gaps and
Long Path Gaps of finding a simple path from a given vertex s to a given vertex t in the
graph G. Let integer numbers f
i
, f
i
, i = 1, . . . , k, be given such that L
−
Σ
≤ f
1
≤ f
1
< f
2
≤
f
2
< · · · < f
k
≤ f
k
≤ L
+
Σ
, where L
−
Σ
=
P
a∈A,l(a)≤0
l(a) is the total non-positive arc length
∗
Corresponding author
1
and L
+
Σ
:=
P
a∈A,l(a)>0
l(a) is the total positive arc length. Here we assume that any sum is
equal to zero if it is taken over an empty set. Denote [f, f] = {f, f + 1, . . . , f}.
Problem Exact Path(α): Given number α, find a simple path P from vertex s to
vertex t in the graph G such that its length L(P ) = α.
Problem Path Gaps: Find a simple path P from vertex s to vertex t in the graph G
such that the length L(P ) 6∈ {[f
1
, f
1
], . . . , [f
k
, f
k
]}. We call intervals [f
i
, f
i
], i = 1, . . . , k,
forbidden (path length) gaps. The length of a path should not fall into these gaps.
We also study three special cases of the problem Path Gaps: the case k = 2 of two
forbidden gaps [f
1
, f
1
] and [f
2
, f
2
], the case k = 1 of a single forbidden gap [f
1
, f
1
], and a
sub-case of the latter case, in which a single path length α is forbidden, that is, f
1
= f
1
= α.
We denote these cases as Path-2-Gaps, Path-1-Gap and Path No(α), respectively.
Problem Short (Long) Path Gaps: Differs from Path Gaps in that a shortest
(longest) simple path with the length not from the forbidden gaps has to be found.
Occasionally, we will assume that graph G is undirected, which will be explicitly indicated.
Problems Path Gaps and Short Path Gaps appear in route planning from one point of
a network to another such that the arrival time does not fall into the given time intervals
when the service required at the destination point is not available. This situation is similar
to that in the vehicle routing problems with time windows, see, for example, Br¨aysy and
Gendreau [6] for the formulation and a survey of the results for these problems. The problem
Long Path Gaps can be used for modeling a situation, in which profits are collected over
the route segments while traveling from one point of a network to another. If the total profit
is less than a given value B > 0, then it is not worth traveling. The problem is to find
a route such that the total profit is maximized and it does not fall into the forbidden gap
[0, B]. It is clear that the forbidden gap constraint is redundant if the goal is to find an exact
solution. However, the problem is NP-hard in general, and the gap constraint is essential for
an approximate solution. Vehicle routing problems with profits have been studied by Archetti
et al. [3]. In a situation of goods collection over segments of a path and their transportation
in containers, there can be a requirement of container capacity utilization, leading to the
forbidden gaps. For example, assume that the capacity of any container is 30 goods, and each
container is required to be filled with at least 25 goods. A fixed number of goods associated
with a path segment has to be collected if this segment is visited. In this situation, the total
number of collected goods should fall into the intervals [25,30], [50,60], [75,90], [100,120],
2
[125, ∞], and the forbidden intervals are [0,24], [31,49], [61,74], [91,99] and [121,124]. Feasible
loads of the containers can be decided after a feasible path has been determined.
To the best of our knowledge, no literature exists on problems with the forbidden objective
values gaps. The only exception is our recent paper [13] on a 0-1 knapsack problem with the
forbidden objective function values. On the other hand, several exact value (or exact cost)
combinatorial problems have been studied in the literature, which concern the existence of
a solution with a given objective function value. The exact value assignment problem has
been studied by Papadimitriou and Yannakakis [31] who proved its NP-completeness in the
ordinary sense, and Karzanov [22] who developed a polynomial time algorithm for the case of
0-1 costs. Pseudo-polynomial time algorithms for the exact value spanning tree problem, the
exact value perfect matching problem on planar graph, the exact value cycle sum problem
on planar directed graph and the exact value cut problem on planar or toroidal graphs have
been presented by Leclerc [25] and Barahona and Pulleyblank [4]. A number of computational
complexity and algorithmic results for the exact weight (maximum) independent set problem
on various classes of graphs have been obtained by Milanic and Monnot [30]. Computational
complexity of the exact weight subgraph problems, in which the number of vertices of the
subgraph is a constant, has been studied by Vassilevska and Williams [35] and Abboud and
Lewi [1]. Lop´ez et al. [27] studied the problem Exact Path( α), which they showed to
be NP-complete and suggested modifications of the goal search (A
∗
) and bidirectional search
algorithms for the solution. The problem Exact Path(α) can also be formulated as a special
case of a constrained path problem, which is to maximize path length, subject to the constraint
that the path length does not exceed a certain value, or to minimize the path length, subject
to the constraint that the path length is at least a certain value. There exists a bulk of the
literature on the constrained and bi-objective path problems, see, for example, Joksch [20],
Dial [12], Hansen [18], Aneja et al. [2], Desrochers [11], Warburton [36], Hassin [19], Lorenz
and Raz [28], Ergun et al. [14], Righini and Salani [32], Boland et al. [5], Garcia [16],
Tsaggouris and Zaroliagis [34] and Bruegem et al. [8]. We will present new observations
about the relations between the problem Exact Path(α) and other path problems with the
forbidden gaps.
The rest of the paper is organized as follows. In the next section, we describe connections
of the new problems with the earlier studied problems that provide a fairly complete com-
putational complexity classification of the new problems and some new algorithmic results.
3
Section 3 studies the case in which the graph is acyclic and the arc lengths are non-negative.
While the problem Exact Path(α) and any of the problems Path Gaps, Short Path
Gaps and Long Path Gaps with at least two forbidden gaps are NP-hard for this special
case, an efficient approximation scheme is suggested, which delivers a solution with value close
to the optimum but, possibly, violating a gap constraint with a given relative error. Poly-
nomial time algorithms are presented for a more restrictive special case of these problems,
in which forbidden gaps are polynomially bounded but the arc lengths are not. The paper
concludes with a table of the obtained results and suggestions for future research.
2 Connections with the earlier studied problems
Observe that if graph G contains directed cycles, then the problems Exact Path(α) and
Path-1-Gap are NP-complete in the strong sense even if the arc lengths are all equal to one
because the NP-complete problem Hamiltonian Path (Garey and Johnson [17]) reduces to
Exact Path(α) by setting α = n − 1 and it reduces to Path-1-Gap by setting [f
1
, f
1
] =
[1, n − 2].
We further abbreviate directed acyclic (multi)graph to DAG. Assume that G is a DAG.
In this case, Exact Path(α) is NP-complete in the ordinary sense as it was mentioned by
Lop´ez et al. [27]. Furthermore, it is pseudo-polynomially solvable by the following folkloric
dynamic programming algorithm, denoted as DP-All-Lengths. This algorithm scans ver-
tices in a topological order (cf. Cormen et al. [10]) and constructs paths from vertex s to the
successor vertices j. A state (j, f) is associated with a path from vertex s to vertex j, where
f is the length of this path. If a complete path P
0
from vertex s to vertex t with length
L(P
0
) goes via a vertex j and a sub-path of P
0
is in the state (j, f), then any (incomplete)
path in this state can be extended to a complete path P from s to t with the same length
L(P ) = L(P
0
).
Algorithm DP-All-Lengths recursively generates states (j, f ). Let S(j) denote the set
of states (j, f) generated for vertex j, which differ by the path lengths f. The initialization
is S(s) = {(s, 0)}. Vertex s is labeled. In the general recursion step, a vertex j is identified
whose predecessor vertices i are all labeled. Since G is a DAG, such a vertex always exists.
Set S(j) of states of the identified vertex is calculated as follows.
S(j) :=
n
j, f + l(a
(r)
ij
)
| a
(r)
ij
∈ A, (i, f) ∈ S(i)
o
.
4
