More facets from fences for linear ordering and acyclic subgraph polytopes

We present new facets for the linear ordering polytope. These new facets generalize facets induced by subgraphs called fences, introduced by Grijtschel et al. (1985) and augmented fences, introduced by McLennan (1990). One novelty of the facets introduced here is that each subgraph induces not one but a family of facets, which are not generally rank inequalities. Indeed, we provide the smallest known example of a facet-representing inequality for the linear ordering polytope that is not a rank inequality. Gilboa (1990) introduced the diagonal inequalities for the linear ordering polytope, and Fishburn (1991) posed the question of identifying precisely which diagonal inequalities represent facets. We completely resolve Fishburn’s question. Some of our results can be transported to the acyclic subgraph polytope. These new facets for the acyclic subgraph polytope are the first ones that are not represented by rank inequalities. 1. The linear ordering polytype A linear ordering of an n-element set N is a bijection 7t : { 1,2, . . . , n} + N. Let c,, be the value of having u before v in the linear ordering. The value of the linear ordering n is c c,,: 71 -l(u) < 71-l (u, U)EN x N *Corresponding author. Research supported in part by Grant No. DDM-8909660 from the National Science Foundation. ** Research conducted in part at CORE, Universitk Catholique de Louvain, Louvain-la-Neuve, Belgium, and supported in part by a CORE fellowship and a Yale Senior Faculty Fellowship. 0166-218X/94/$07.00 0 199kElsevier Science B.V. All rights reserved SSDI 0166-218X(92)00151-F 186 J. Lung, J. Lee/ Discrete Applied Mathematics 50 (1994) 185-200 The problem of determining if there is a linear ordering with value less than an input constant is NP-complete. Let G = (N, A) be a strict directed graph (i.e. having no loops or multiple arcs) with n_ode s$ N and arc set A c N x N. For fl c N and A c A n (fl x fl), the graph G = (N, A”) is a subgraph of G. The graph G is acyclic if it contains no directed cycles. A directed graph (N, T) is a tournament on N if for every pair of distinct nodes u, u E N, exactly one of (u, v) and (u, u) is in T. Given a linear ordering of the nodes N of a directed graph, the arc set {(u, u): nnl(u) < Xl(u)} forms an acyclic tournament on N; conversely, an acyclic tournament (N, T) induces a unique linear ordering of N. Following [7], we let (ii, i2 ,...,i,)denotethearcset{(ij,i,):ldj<kdn}ofthe acyclic tournament induced by the linear ordering rc where rc( j ) = ij for j = 1,2, . . . , n. For other graph-theoretic terminology, see [l]. Let G, = (N, A,) be the complete strict directed graph on the n-element node set { 1,2, . . . , n}. Let & E; 2A” be the collection of the arc sets of acyclic tournaments on N. The linear ordering polytope YE0 is the convex hull of the incidence vectors of elements of Yn. The linear ordering polytope is important in the understanding of binary choice probabilities in mathematical psychology and voting proportions in the theory of social choice (see [3]). The linear ordering problem is also of interest in economics, specifically, in the triangulation of input-output matrices (see [14]). Other areas of application include scheduling (see [ll]) and anthropology (see [S]). Investigations into the characterization of 8” Lo by linear inequalities dates from 1953 (see [8]) and has attracted considerable attention in mathematical psychology in the past decade (see C31). The dimension of a polytope 9 c R A, denoted dim(S), is one less than the maximum number of affinely independent points in 9’. Grdtschel et al. [7] established that dim(g’“,o) = $n(n 1) and showed that x(i,j)+x(j,i)=l Vi,jEN,withi#j, (1) is a maximal irredundant equation system for gto. For a~ RA, a # 0, a0 E R, the inequality aTx 6 a0 is valid for 9 if 9 E {XE [WA: aTx < uo}. A valid inequality uTx d a, represents a facet of B if dim(Y n {xE IFP: uTx = ao}) = dim(S) 1. Grotschel et al. [7] also showed that the nonnegativity inequalities and the upperbound inequalities O< x(&j)< 1, V(i,j)EA,, represent facets of 9 lo. (2) We are interested in studying the structure of Y Lo, and this paper presents two collections of facet-representing inequalities of PLO. The first collection of these inequalities, which we call t-reinforced k-fence inequalities, are induced by subgraphs of G, known as k-fences. That these inequalities represent facets generalizes a result of Grotschel et al. [7]. Indeed, we will demonstrate that the t-reinforced k-fence inequalities, together with the well-known triangle inequalities and the trivial inequalities and equations, imply the full set of diagonal inequalities introduced by Gilboa [4], thus resolving a question of Fishburn [3]. The second collection of facet-representing inequalities, which we call augmented t-reinforced k-fence inequalities, are induced by J. Leung. J. Lee/Discrete Applied Mathematics 50 (1994) 185-200 187 subgraphs called augmented k-fences. That these inequalities represent facets generalizes a result of McLennan [15]. In many other combinatorial problems defined on graphs, facets of the associated polytope are often induced by subgraphs. A rank inequality is an inequality of the form c eeS x(e) < r(S), where r(S) is the minimum number for which the inequality is valid. One novelty of the facets introduced in this paper is that each subgraph induces not one but a family of facets, which are not generally rank inequalities. We provide the smallest known example of a facet-representing inequality for the linear ordering polytope that is not a rank inequality. We transport some of our results to the acyclic subgraph polytope. In doing so, we establish the first facets of the acyclic subgraph polytope that are not represented by rank inequalities. 2. New facets for the linear ordering polytope For every integer k 2 3, a directed graph is a simple k-fence if it is isomorphic to D = (U u W, F, u F,), where U=(ul,uZ ,..., uk} and W=(wI,w2 ,..., wk} withUnW=& Fi = {(n,, WI), (uz, w,), . . . >&r w,)}, Fz= fi {(wi,u)Ia~U\{ui)}. i=l (3) (4)