Uri N. Peled

Facet of regular 0–1 polytopes

The role of 0–1 programming problems having monotone or regular feasible sets was pointed out in [6]. The solution sets of covering and of knapsack problems are examples of monotone and of regular sets respectively. Some connections are established between prime implicants of a monotone or a regular Boolean functionβ on the one hand, and facets of the convex hullH of the zeros ofβ on the other. In particular (Corollary 2) a necessary and sufficient condition is given for a constraint of a covering problem to be a facet of the corresponding integer polyhedron. For any prime implicantP ofβ, a nonempty familyF(P) of facets ofH is constructed. Proposition 17 gives easy-to-determine sharp upper bounds for the coefficients of these facets whenβ is regular. A special class of prime implicants is described for regular functions and it is shown that for anyP in this class,F(P) consists of one facet ofH, and this facet has 0–1 coefficients. Every nontrivial facet ofH with 0–1 coefficients is obtained from this class.

Polynomial-time algorithms for regular set-covering and threshold synthesis

Abstract A set-covering problem is called regular if a cover always remains a cover when any column in it is replaced by an earlier column. From the input of the problem - the coefficient matrix of the set-covering inequalities - it is possible to check in polynomial time whether the problem is regular or can be made regular by permuting the columns. If it is, then all the minimal covers are generated in polynomial time, and one of them is an optimal solution. The algorithm also yields an explicit bound for the number of minimal covers. These results can be used to check in polynomial time whether a given set-covering problem is equivalent to some knapsack problem without additional variables, or equivalently to recognize positive threshold functions in polynomial time. However, the problem of recognizing when an arbitrary Boolean function is threshold is NP-complete. It is also shown that the list of maximal non-covers is essentially the most compact input possible, even if it is known in advance that the problem is regular.

Threshold Numbers and Threshold Completions

The threshold number t(f) of a positive Boolean function f ( x 1 , …, x n ) is the least number of linear inequalities whose solution set in 0–1 variables is the set of zeroes of f. These inequalities can be taken with nonnegative coefficients. If P is the collection of the prime implicants of f and S ⊆ P , then f s denotes the Boolean sum of the prime implicants in S. S is called a threshold subcollection in f if there exists a threshold function g satisfying f s ⩽ g ⩽ f. The threshold number t(f) is equal to the least number of threshold subcollections in f that cover P. Thus t(f) ⩽ | P | ⩽([ n n /2]). A graph G f with the vertex set P is defined, and it is shown that t(f) is not less than its chromatic number, generalizing the results of Chvatal and Hammer. This is used to show that the maximum value of t(f) is at least ([ n n /2])/ n. When each prime implicant has the form x i x j , f is called graphic and corresponds naturally to a graph G with vertex set { x 1 ,…, x n } and edge set P. In that case x i x j , and x k x l , are adjacent in G f if and only if x i x k , x j x l ∉ P or x i x l , x j x k ⊆ P. . Also for f graphic, a subset S ⊆ P is a threshold subcollection in f if and only if G does not contain a closed walk alternating between S -edges and non-edges of G. This is proved both from the separation theorem for polytypes and by an O ( n 3 ) algorithm. When G f is bipartite and S is one of the colours in a bicolouring of G f , a further simplification of the condition is achieved.

(n,e)-Graphs with maximum sum of squares of degrees*

It was conjectured in [Wang, to appear in The Australasian Journal of Combinatorics] that, for each integer k ≥ 2, there exists M(k) such that if G= (V1, V2, E) is a bipartite graph with |V1| = |V2| = n ≥ M(k) and d(x) + d(y) ≥ n + k for each pair of nonadjacent vertices x and y of G with x e V1 and y e V2, then, for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck in G such that ei e E(Ci for all i e {1, …, k} and V(C1 ⊎ ··· ∪ Ck) = V(G). This conjecture is also verified for k = 2, 3 in [Wang, to appear; Wang, manuscript]. In this article, we prove this conjecture to be true if n ≥ 3k, i.e., M(k) ≤ 3k. We will also show that, if n ≥ 3k, then for any k independent edges e1, …, ek of G, there exist k vertex-disjoint cycles C1, …, Ck of length at most 6 in G such that ei e E(Ci) for all i e {1, …, k}.

An O( nm )-time algorithm for computing the dual of a regular Boolean function

Abstract We consider the problem of dualizing a positive Boolean function ƒ: Bn → B given in irredundant disjunctive normal form (DNF), that is, obtaining the irredundant DNF form of its dual ƒ d (x) = ƒ ( x ) . The function f is said to be regular if there is a linear order ≳ on {1,…,n} such that i≳j, xi = 0, and xj = 1 imply ƒ(x) ⩽ ƒ(x + u i − u j ) , where uk denote unit vectors. A previous algorithm of the authors, the Hop-Skip-and-Jump algorithm, dualizes a regular function in polynomial time. We use this algorithm to give an explicit expression for the irredundant DNF of ƒd in terms of the one for ƒ. We show that if the irredundant DNF for ƒ has m ⩾ 2 terms, then the one for ƒd has at most (n − 2)m + 1, and can be computed in O(nm) time. This can be applied to solve regular set-covering problems in O(nm) time.

The Polytope of Degree Sequences

Abstract A nonnegative integer sequence ( d 1 , d 2 ,…, d n ) is called a degree sequence if there exists a simple graph on the vertex set V = {1,2,…, n } such that deg ( i )= d i for all i . The degree sequence of a threshold graph is a threshold sequence . Let D n = Convex Hull {( x 1 , x 2 ,…, x n )|( x 1 ,…, x n ) is a degree sequence}. It is proved that: (1) A degree sequence f is an extreme point of D n if and only if f is a threshold sequence. (2) Two threshold sequences f and g are adjacent extreme points of D n if and only if f can be obtained from g by either adding 1 to two components of g or subtracting 1 from two components of g . (3) D n is determined bythe following system of inequalities: ∑ i∈ x i − ∑ i∈T x i ⩽|S|(n−1−|T|) for all sets S , T with ⊘ ≠ S ∪ T ⊆ {1,2,…,n}, S ∩T = ⊘ . Moreover, this system is totally dual integral. Furthermore, for n ⩾ 4, ( S , T ) determines a facet of D n if and only if either ⋎S ∪ T| = 1 or else S ≠ ⊘,T ≠ ⊘, |S ∪T| ≠n - 1,n - 2 . (4) f is a threshold sequence if and only if the only degree sequences majorizing f in the sense of Hardy, Littlewood, and Polya are the rearrangements of f . Consequently, every degree sequence is a convex combination of isomorphic threshold sequences (i.e., threshold sequences that are rearrangements of each other).

Restrictions and preassignments in preemptive open shop scheduling

Abstract Preemptive open shop scheduling can be viewed as an edge coloring problem in a bipartite multigraph. In some applications, restrictions of colors (in particular preassignments) are made for some edges. We give characterizations of graphs where some special preassignments can be embedded in a minimum coloring (number of colors = maximum degree). The case of restricted colorings of trees is shown to be solvable in polynomial time.

On the Maximization of a Pseudo-Boolean Function

A b ranch-and-bound method is proposed for the maximizat ion of real va lued functions wi th var iables assuming only the values 0 and 1. The impor tance of the problem cons i s t s a s has been shown by Hammer and R u d e a n u i n the fact t h a t numerous problems in operations research, g raph theory, combinator ia l mathemat ics , etc., can be b rough t to this form. The method has been successfully tes ted on an IBM 360/50 computer .

Constraint Pairing In Integer Programming

AbstractSingle linear constraints can be used in a straightforward way for deriving bounds on the variables of discrete optimization problems. More powerful conclusions can be obtained from the examination of all the surrogate constraints associated to pairs of constraints. The paper shows that for problems involving n integer variables, the examination of at most n + 2 surrogates (constructively) associated to any pair of constraints in the problem produces all the conclusions derivable from any linear combination of the given constraint pair. The bounds obtained are shown to be the same as those obtained by maximizing or minimizing the individual variables under the pair of constraints. If m denotes the number of constraints in the problem, then 0(m2n2) elementary operations will be performed for testing all constraint-pairs of the problem in order to improve the bounds on the variables and/or to derive binary relations among them.

Hamiltonian threshold graphs

Abstract The vertices of a threshold graph G are partitioned into a clique K and an independent set I so that the neighborhoods of the vertices of I are totally ordered by inclusion. The question of whether G is hamiltonian is reduced to the case that K and I have the same size, say r , in which case the edges of K do not affect the answer and may be dropped from G , yielding a bipartite graph B . Let d 1 ≤ d 2 ≤…≤ d r and e 1 ≤ e 2 ≤…≤ e r be the degrees in B of the vertices of I and K , respectively. For each q = 0, 1,…, r −1, denote by S q the following system of inequalities: d j ⩾ j + 1, j = 1,2,…, q , e j ⩾ j + 1, j = 1, 2,…, r −1−1. Then the following conditions are equivalent: 1. (1) B is hamiltonian, 2. (2) S q holds for some q = 0, 1,…, r −1, 3. (3) S q holds for each q = 0, 1,…, r −1.