Polynomial inequalities representing polyhedra

Our main result is that every ro-dimensional polytope can be described by at most (2n — 1) polynomial inequalities and, moreover, these polynomials can explicitly be constructed. For an n-dimensional pointed polyhedral cone we prove the bound 2n — 2 and for arbitrary polyhedra we get a constructible representation by 2n polynomial inequalities. 1. I N T R O D U C T I O N By a striking result of Bröcker and Scheiderer (see [Sch89], [Brö91], [BCR98] and [Mah89]), every basic closed semi-algebraic set of the form S = {xeR : f i ( x ) > 0 , . . . , f , O E ) > 0 } , where f$ £ R[x], 1 < i < I, are polynomials, can be represented by at most n(n + l ) / 2 polynomials, i.e., there exist polynomials pi , • • • ,pn(n+i)/2 £ ^ W such that 5 = { x G r : P l ( S ) > 0 , . . . , P„(„+i)/2(aO > 0} . Moreover, in the case of basic open semi-algebraic sets, i.e., > is replaced by strict inequality, one can even bound the maximal number of polynomials needed by the dimension n instead of n(n + l ) / 2 . Rephrasing the results in terms of semi-algebraic geometry, the stability index of every basic closed or open semi-algebraic set is n(n + l ) / 2 or n, respectively. Both bounds are best possible. No explicit constructions, however, of such systems of polynomials are known nor whether the upper bound n(n + l ) / 2 can be improved for semi-algebraic sets having additional structure such as convexity. Even in the very special case of n-dimensional polyhedra almost nothing was known. In [Brö91, Example 2.10] or in [ABR96, Example 4.7] a description of a regular convex m-gon in the plane by two polynomials is given. This result was generalised to arbitrary convex polygons and three polynomial inequalities by vom Hofe [vH92]. Bernig [Ber98] proved that , for n = 2, every convex polygon can even be represented by two polynomial inequalities. In [GH03] a construction of 0(n) polynomial inequalities representing an n-dimensional simple polytope is given. Based on ideas from [Bos03], here we give, in particular, an explicit construction of (2n — 1) polynomials describing an arbitrary n-dimensional polytope. Hence the general upper bound of n(n + l ) / 2 polynomials can be improved (at • Supported by the DFG Research Center "Mathematics for key technologies" (FZT 86) in Berlin. 1 2 HARTWIG BOSSE, MARTIN GRÖTSCHEL, AND MARTIN HENK least) for n-dimensional polytopes, and we conjecture that the dimension itself is the right value for this special case. In order to state our results we fix some notation. A polyhedron P C 1 " is the intersection of finitely many closed halfspaces, i.e., we can write it as P = {xGR :ai-x(pi , . . . ,p,) := {x G R : pi(x) > 0 , . . . ,p,(a:) > 0} the associated basic closed semi-algebraic set generated by the polynomials. T h e o r e m 1.1. Let C c l TM be an n-dimensional pointed polyhedral cone. Then we can construct (2n — 2) polynomials pi G R[x], 1 < i < In — 2, such that C = V(pU...,p2n-2). The case of polytopes can be derived as a consequence of the construction behind Theorem 1.1 and here we get T h e o r e m 1.2. Let P C ffi be an n-dimensional polytope. Then we can construct (2n — 1) polynomials pi G M[x], 1 < i < 2n — I, such that P = P ( p i , . . . , p 2 „ i ) . At the end of Section 3 (see Definition 3.3) we will give an explicit description of the polynomials we employ. The construction behind the proof of Theorem 1.2 or Theorem 1.1 can also be applied to the interior of a polytope or a cone which are open semi-algebraic sets. Furthermore, in [GH03, Proposition 2.5] it is shown how a representation of a polytope by polynomial inequalities can be used to get a representation of a polyhedron by polynomials. Applying this proposition to Theorem 1.2 leads to Corollary 1.3. Let P C W be an n-dimensional polyhedron. Then we can construct 2n polynomials pi G R[x], 1 < i < 2n, such that P = "P(pi , . . . ,p2n)The paper is organised as follows. In Section 2 we give, for a pointed cone C, a construction of two polynomials pc*,e>Po such that C is "nicely approximated" by V(pc,e,Po)Then, for a face F = C n {x e W : a, • x = 0, i G IF} of C, we apply this construction to the cone Cp = {x G ffi" : a, • x < 0, i G IF}, where Ip denotes the index set of active constraints of F. In that way we get an approximation of Cp by a semi-algebraic set of the type V(pcF,e, PF)In Section 3 we study the relations between the set V(pcFnG,£,pFnG) and 'P(PCF,S,PF), V(PCG,£->PG) for two different faces F and G of the same dimension. Thereby, it turns out that we may multiply all polynomials pcF,e belonging to faces of the same dimension as well as the polynomials pi? in order to get a representation of a pointed polyhedral cone by polynomials. In Section 4 we give a brief outlook why we are interested in such a polynomial representation of polytopes POLYNOMIAL INEQUALITIES REPRESENTING POLYHEDRA 3 and what might be achievable by such a representation with respect to hard combinatorial optimisation problems. 2. A P P R O X I M A T I N G CONES In the following we use some standard terminology and facts from the theory of polyhedra for which we refer to the books [MS71] and [Zie95]. For the approximation of a cone by a closed semi-algebraic set consisting of two polynomials we need a lemma about the approximation of a polytope by a strictly convex polynomial which was already shown in [GH03, Lemma 2.6]. Since it is essential for the explicit construction of the polynomials we state it here. To this end, let B be the n-dimensional unit ball centred at the origin. The diameter of a polytope is denoted by d iam(P) , i.e., d iam(P) = max{||a; — y\\ : x,y G P}, where || • || denotes the Euclidean norm. L e m m a 2.1 . Let P = {x G W : en • x < 6j, 1 < % < m} be an n-dimensional polytope. For 1 < % < m let 2d • x — h(üi) + h(—üi) Vl{x) : = h(ai) + h(-at) ' where h(a) := max{a • x : x G P} is the support function of P. Let e > 0, choose an integer k such that k > ln(m)/(21n(l + (ra+1)dLm(p)))> ^ ^ m -. pP,e(x) := J2 [<x)f k and K£ := {x£R n : pPy£{x) < 1}. i=i m Then we have P C K£ C P + e B . Proof. [GH03, Lemma 2.6]. D