The Computational Complexity of Untrapped Choice Procedures

In this paper, we define two versions of Untrapped set (weak and strong Untrapped sets) over a finite set of alternatives. These versions, considered as choice procedures, extend the notion of Untrapped set in a more general case (i.e. when alternatives are not necessarily comparable). We show that they all coincide with Top cycle choice procedure for tournaments. In case of weak tournaments, the strong Untrapped set is equivalent to Getcha choice procedure and the Weak Untrapped set is exactly the Untrapped set studied in the litterature. We also present a polynomial-time algorithm for computing each set.


Introduction
A common way to model a decision maker's preferences is to consider a binary relation R over a set A of alternatives (teams, projects, candidates, goods, etc. …).
In many different contexts (Sports league, Social Choice Theory, Economics, Operational Research, etc …), the binary relation R is used to make a choice between alternatives of A. Very often this relation is assumed to be complete and asymmetric (we say that R is a tournament) or sometimes complete (R is said to be a weak tournament). The general case concerning incomplete binary relations has received less attention (see [1] [2] [3]). Incomplete preferences have been increasingly recognized as important [4] [5]. The origin of these preferences is twofold: a lack of information about the alternatives or a lack of information of the decision maker about her own tastes on the alternatives [6] [7]. choice set is infeasible, the applicability of the corresponding solution concept is seriously undermined [17]. Most of the familiar procedures mentioned above are demonstrated to be tractable [17] i.e. belonging to class P of problems which can be solved by an algorithm whose running time is polynomial in the size of the problems instance. These procedures are then considered useful because if the computation of a choice set is intractable, the associated choice procedure is virtually rendered useless for large problem instances.
In this article, we consider the Untrapped choice procedure (UT) defined by Duggan [18] for (weak) tournaments. The resulting set is composed of alternatives x that are not directly beaten or that beat indirectly some other alternatives (especially alternatives that directely beat x). Duggan [18] proves that this choice procedure coincides with the Top cycle choice procedure in the case of tournaments and is nested between the Getcha and the Gocha choice procedures for weak tournaments. UT strongly depends on the asymmetric part of the binary relation considered.
We particularly focus, in this paper, on pseudo tournaments and we deduce another notion of the Untrapped (Strong Untrapped: SUt) choice procedure directely dependent on the pseudo tournament R studied. We also discuss the computational complexity of identifying the choice set for each of the choice procedures studied.
The rest of this article is structured as follows. Concepts that are used throughout this paper are given in Preliminaries (Section 2). Section 3 introduces the two extensions of the Untrapped choice procedures which are compared with two extensions of the Top cycle choice procedure. Computational complexity of Untrapped choice procedures is then explored in Section 4. Section 5 ends with an overview of the results.

Preliminaries
A represents a finite set of alternatives and R a binary relation defined on A (i.e. R is a subset of A A × not yRx . It can be noticed that I is reflexive and symmetric, P is asymmetric (P is also called the asymmetric part of R) and J is symmetric. xPy (resp. xIy ) can be interpreted as x beats or is better than (resp. x is indifferent to) y.
x Px . A is acyclic (resp. P-acyclic) if it contains no circuit (resp. no P-circuit).
The transitive closure * R of R is defined as follows: In other words * xR y if and only if there exists at least a path of length k from x to y (we also say that y is reachable from x). The transitive closure * P of P can also be defined in the same way (we then say that y is P-reachable from x).
The predecessor with respect to R (resp. with respect to P) of an alternative x A ∈ is the set We also define the set y Cl x ∈ ) if y is P-reachable from x. A choice procedure is a function C that maps each pseudo tournament R to a nonempty subset ( ) C R of A called the choice set. If R is a tournament (resp. a weak tournament) the choice procedure is called a tournament solution (resp. a generalized or weak tournament solution) (see [15]).
We say that a choice procedure C is contained in a choice procedure C′ if Tournaments are always supposed to be asymmetric. We suppose without lost of generality that tournament may be reflexive. 2 Pseudo tournaments should not be confound with partial tournaments for which binary relations are asymmetric and not necessarily reflexive.
It is also obvious that the asymmetric part of the transitive closure * R is without circuit and because * R An attractive property of TC is that any alternative that beats another alternative in the Top Cycle is indirectly beaten by the latter.
The notion of (minimal) dominant set has been extended to the case of weak tournaments in two directions.
D is a minimal dominant (resp. minimal undominated) set if D is dominant (resp. undominated) and if no subset of D is dominant (resp. undominated). For pseudo tournaments, we adopt the same definition for dominant and undominated sets. It is then easy to see that the dominant set is no more unique, so we have the following definition.
Definition 4. Let R be a pseudo-tournament on A 4 . The Gocha choice procedure is defined as the union of all minimal undominated sets for R in A.
The Getcha choice procedure is defined as the union of all minimal dominant sets for R in A. Lemma The result of Deb [20] for pseudo tournaments is then generalized as follow.
Proposition 1. For a pseudo tournament R defined on A, we have: Proof

Untrapped Choice Procedures
We study in this section two choice procedures (strong and weak Untrapped choice procedures) for pseudo tournaments. These choice procedures generalize the concept of Untrapped choice procedure defined by Duggan [19] for weak tournaments.
Definition 5. Let R be a pseudo tournament on A. We say that x weakly (resp. strongly) traps y with respsect to R and we write xTy (resp xTy  ) if xPy and if ( ) Relation T (resp T  ) is not necessary transitive but is P-acyclic (resp. acyclic) x P x : this is not possible since 1 2 x Tx ]. So, we can define the set of its maximal elements. This leads to two choice procedures called weak Untrapped (resp. strong Untrapped) choice procedure, denoted by WUt (resp. SUt ) and defined as follow: It is easy to see that an element x of A is in So any alternative x which is not directly beaten or which beats indirectly some other alternatives (specially alternatives that beat directly x) is in the weak Untrapped set.
It is also easy to see that an element x of A is in SUt if and only if ( ) We can say that an alternative x strongly traps another alternative y ( xTy  ) if xPy and if ( ) When relation R is a tournament (resp. weak tournament) Duggan [19], . It is obvious that for weak tournaments, The following proposition gives inclusion relations between the different choice procedures mentionned above.
Proposition 2. For a pseudo tournament R defined on A, we have the following relations (Table 1).
Legend: The symbol ⊇ (resp.=, ∅ ) indicates that the choice set in column is always contained in (resp. is equaled to, intersects) the choice set in row.
Proof. See Appendix.
The previous proposition can be summarized by the following Hasse diagram.
We can then notice that WUt is nested between Gocha and Getcha ( Gocha WUt Getcha ⊆ ⊆ ) and that Getcha SUt ⊆ . Missing arrows between two choice sets indicates that the two always intersect and none is included in the other.  2) Similar to the previous one.

Computational Complexity
In this section we analyze the computational complexity of the weak (resp.  Let us mention that deciding whether an alternative is contained in a choice set is computationally equivalent to finding the set [18]. Algorithm  It has been shown that the transitive closure of each x A ∈ is computable in polynomial time. The same holds for the computation of predecessors of x (see [21] page 137), we can then conclude that deciding whether an alternative is contained in the weak (resp. strong) Untrapped set is in P (class of problems that can be solved in polynomial time).

Conclusions
Duggan [18] has defined the concept of Untrapped choice procedure for weak Untrapped (WUt) choice procedure also depends on the asymmetric part of the pseudo tournament while the strong Untrapped choice procedure (SUt) is directly defined by the given pseudo tournament.
We have shown that each of the new choice procedures coincides with the familiar Top cycle choice procedure for tournaments. In case of weak tournaments, the strong Untrapped set is equivalent to Getcha choice procedure and the Weak Untrapped set is exactly the Untrapped set studied by Duggan [18]. We know (see [18]) that for a weak tournament R, we have When R is a pseudo tournament, we've seen (from proposition 2) that the three choice procedures (WUt, Getcha and Gocha) are all contained in SUt.
In terms of computational complexity, we present an algorithm to compute both the strong and the weak Untrapped choice procedure. This algorithm allows us to show that deciding whether an alternative is contained in the strong (or in the weak) Untrapped set is in P (class of problems that can be solved in polynomial time).

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.