Showing papers in "SIAM Journal on Discrete Mathematics in 1990"

A hierarchy of relaxation between the continuous and convex hull representations

Abstract: In this paper a reformulation technique is presented that takes a given linear zero-one programming problem, converts it into a zero-one polynomial programming problem, and then relinearizes it into an extended linear program. It is shown that the strength of the resulting reformulation depends on the degree of the terms used to produce the polynomial program at the intermediate step of this method. In fact, as this degree varies from one up to the number of variables in the problem, a hierarchy of sharper representations is obtained with the final relaxation representing the convex hull of feasible solutions. The reformulation technique readily extends to produce a similar hierarchy of linear relaxations for zero-one polynomial programming problems. A characterization of the convex hull in the original variable space is also available through a projection process. The structure of this convex hull characterization (or its other relaxations) can be exploited to generate strong or facetial valid inequaliti...

Singularity analysis of generating functions

Abstract: This work presents a class of methods by which one can translate, on a term-by-term basis, an asymptotic expansion of a function around a dominant singularity into a corresponding asymptotic expans...

The random walk construction of uniform spanning trees and uniform labelled trees

Abstract: A random walk on a finite graph can be used to construct a uniform random spanning tree. It is shown how random walk techniques can be applied to the study of several properties of the uniform random spanning tree: the proportion of leaves, the distribution of degrees, and the diameter.

Monotone circuits for connectivity require super-logarithmic depth

Abstract: It is proved here that every monotone circuit which tests $st$-connectivity of an undirected graph on n nodes has depth $\Omega (\log^2 \,n)$. This implies a superpolynomial $(n^{\Omega (\log n)} )$ lower bound on the size of any monotone formula for $st$-connectivity.The proof draws intuition from a new characterization of circuit depth in terms of communication complexity. Within the same framework, a very simple and intuitive proof is given of a depth analogue of a theorem of Khrapchenko concerning formula size lower bounds.

Harmonic Analysis of Polynomial Threshold Functions

Abstract: The analysis of linear threshold Boolean functions has recently attracted the attention of those interested in circuit complexity as well as ofthose interested in neural networks. Here a generalization oflinear threshold functions is defined, namely, polynomial threshold functions, and its relation to the class of linear threshold functions is investigated. A Boolean function is polynomial threshold if it can be represented as a sign function ofa polynomial that consists ofa polynomial (in the number ofvariables) number ofterms. The main result ofthis paper is showing that the class ofpolynomial threshold functions (which is called PT1 is strictly contained in the class ofBoolean functions that can be computed by a depth 2, unbounded fan-in polynomial size circuit of linear threshold gates (which is called LT2). Harmonic analysis ofBoolean functions is used to derive a necessary and sufficient condition for a function to be an S-threshold function for a given set S of monomials. This condition is used to show that the number of different S-threshold functions, for a given S, is at most 2 t'/ 1)lsl. Based on the necessary and sufficient condition, a lower bound is derived on the number of terms in a threshold function. The lower bound is expressed in terms of the spectral representation of a Boolean function. It is found that Boolean functions having an exponentially small spectrum are not polynomial threshold. A family of functions is exhibited that has an exponentially small spectrum; they are called "semibent" functions. A function is constructed that is both semibent and symmetric to prove thatPT is properly contained in LT2.

Integer polyhedra arising from certain network design problems with connectivity constraints

Abstract: In this paper a general integer linear programming model is presented for the important practical problem of designing minimum-cost survivable networks, and this model is related to concepts in graph theory and polyhedral combinatorics. In particular, several interesting special cases of this general model are considered, including the minimum spanning tree problem, the Steiner tree problem, and the minimum cost k-edge connected and k-node connected network design problems. The integer polyhedra associated with these problems are studied, those inequalities from natural ILP-formulations that define facets are identified, the separation problem for these facets is addressed, and how good lower bounds can be obtained from the models studied here is indicated.

Irregular assignments of trees and forests

Abstract: Let G be a graph on n vertices. An irregular assignment of G is a weighting $ w:E ( G ) \to \{ 1, \cdots ,m \} $ of the edge-set of G such that all weighted degrees $w( v ) = \sum_{v \in e} w ( e ) $ are distinct. The minimal number m for which this is possible is called the irregularity strength$s( G )$ of G. Lehel and others have shown that $s ( G ) < \infty $ implies $s ( G )\leqq n- 1$ for connected graphs on $n \geqq 4$ vertices, and $s( G )\leqq 2n - 3$ for arbitrary graphs. By using decompositions of the additive group $\mathbf{Z}_r $ (integers mod r), these results are strengthened. Main Theorem: $s ( G )\leqq n + 1$ for any graph with $s( G ) < \infty $.

An almost linear time algorithm for generalized matrix searching

Abstract: An $O( m\alpha ( n ) + n )$ time algorithm is given for finding row-maxima and minima in totally monotone partial $n \times n$ matrices. As a result, faster algorithms are obtained for some optimization problems concerning distance and visibility between vertices of two convex polygons. Also shown is how the algorithm can be modified to give an $O( n \alpha ( n ) )$ algorithm for a class of dynamic programming problems satisfying convex quadrangle inequalities. This results in faster algorithms for a number of problems arising in molecular biology, speech recognition, and geology.

On the 1.1 edge-coloring of multigraphs

Abstract: A new upper bound is proved for the chromatic index $q^* ( G )$ of multigraphs $G = ( V,E )$. Let $d ( G )$ be the maximum degree of G, and let \[ p ( G ) = \text{MAX} \{ \lceil 2 | E(X) |/ ( | X | - 1) \rceil :X \subset V, | X | e 1\,\text{and odd} \} \] where $E( X )$ is the set of edges in the subgraph of G induced by X. The upper bound is expressed in terms of the two trivial lower bounds $d ( G )$ and $p ( G )$ as follows: \[ q^* ( G )\leqq \text{MAX} \{ p ( G ), \lfloor 1.1 d ( G ) + 0.8 \rfloor \}. \] The proof yields an algorithm which edge-colors any given multigraph G with at most $\lfloor 1.1q^* ( G ) + 0.8 \rfloor $ colors. The running time is $O ( | E | ( d ( G ) + | V | ) )$ and the storage space is $O ( | E | )$.

On Approximate Solutions for Combinatorial Optimization Problems

Abstract: The usefulness of a special kind of approximability-preserving transformations (called continuous reductions) among combinatorial optimization problems is demonstrated. One common measure for the approximability of an optimization problem is its best performance ratio. This parameter attains the same value for two problems (up to a bounded factor) whenever they are mutually related by continuous reductions. Therefore, lower and upper bounds or gap-theorems valid for a particular problem are transferred along reduction chains. In this paper, continuous reductions are used for the analysis of several basic combinatorial problems including graph coloring, consistent deterministic finite automaton, covering by cliques, covering by complete bipartite subgraphs, independent set, set packing, and others. The results obtained and the methods involved are a contribution towards a systematic classification of NP-complete problems with regard to their approximability.

Bit serial multiplication in finite fields

Abstract: Bit serial multiplication schemes for hardware implementation of arithmetic in a finite field of characteristic two are considered. In addition, certain aspects of polynomial bases in finite fields are investigated.

An isoperimetric inequality on the discrete torus

Abstract: The discrete torus is the graph on $\mathbb{Z}_k^n = ( \mathbb{Z}/k\mathbb{Z} )^n $ in which $x = (x_i )_1^n $ is joined to $y = (y_i )_1^n $ if for some i there is $x_i = y_i \pm 1$ and $x_j = y_j $ for all $j e i$. For a set $A \subset \mathbb{Z}_k^n $ and a natural number t, let $A_{( t )} $ be the set of vertices of $\mathbb{Z}_k^n $ within distance t of A. The main aim of this paper is to give a best possible lower bound for $| A_{(t)} |$ in terms of $| A |$, for even values of k.

Computing the bandwidth of interval graphs

Abstract: In this note, an $O ( | V |k )$ algorithm is described for determining whether an interval graph on $| V |$ vertices has a bandwidth less than or equal to a given integer k. While the algorithm is not the first to resolve this problem, it does admit a shorter proof of its correctness than a previous algorithm of the same complexity due to Kratsch (Information and Computation, 74 (1987), pp. 140–158).

Finding critical independent sets and critical vertex subsets are polynomial problems

Abstract: An independent set $J_c $ of a graph G is called critical if \[ | J_c | - | N ( J_c ) | = \max \{ | J | - | N ( J ) |:J\,\text{is an independent set of }G \}, \] and a vertex subset $U_c $ is called critical if \[ | U_c | - | N ( U_c ) | = \max \{ | U | - | N ( U ) |:U\,\text{is a vertex subset of }G \} . \] In this paper, it will be shown that finding a critical independent set and a critical vertex subset of a graph are solvable in polynomial time.

Recognition of tree metrics

Abstract: A tree metric on a set of n points (objects) is realized by a weighted undirected tree whose tips are labelled by the objects. It is well known that tree metrics are characterized by the so-called 4-point condition. In this note it is shown that a metric d is a tree metric if and only if the Farris transform of d qualifies as an ultrametric. Based on this, an $O(n^2 \log n)$ procedure is established for recognizing tree metrics on an n-set.

On the structure of minimum-weight k-connected spanning networks

Abstract: The problem of finding a minimum-weight k-connected spanning subgraph of a complete graph, assuming that the edge weights satisfy the triangle inequality, is studied. It is shown that the class of minimum-weight k-edge connected spanning subgraphs can be restricted to those subgraphs which, in addition to the connectivity requirements, satisfy the following two conditions: (I) Every vertex has degree k or $k + 1$; (II) Removing any $1, 2, \cdots ,$ or k edges does not leave the resulting connected components all k-edge connected. For the k-vertex connected case, the parallel result is obtained with “k-edge” replaced by “k-vertex,” with the added technical restriction that $| V |\geqq 2k$ for condition (I) to hold. This generalizes recent work of Monma, Munson, and Pulleyblank for the case $k = 2$.

Transversals of Vertex Partitions in Graphs

Abstract: This paper studies graph properties of the following forms: For every partition of the vertex set that satisfies an upper (or lower) bound on the number of elements in each partition class, there is a transversal of the partition that is an independent (or dominating) set. A possible application to fault-tolerant data storage is discussed, and bounds for the parameters that are functions of minimum and maximum degree are established. The complexity of associated decision problems is also addressed.

On the distribution of lengths of evolutionary trees

Abstract: This paper presents the results of the authors’ investigation of a combinatorial problem arising from the study of evolutionary trees. In graph theoretic terms it can be expressed as a problem of colouring vertices of a binary tree. For a given colouring of the pendant vertices of a binary tree there is a simple algorithm for assigning colours to internal vertices minimising the number of edges of the tree whose end vertices have differing colours. This minimal number is called the length of the tree. The question posed is: For given numbers of pendant vertices of assigned colours, how many trees of a particular length can be constructed on those vertices? This question is answered in two special cases. Answers to this problem are needed to establish the distribution of lengths of evolutionary trees, by which the significance of the maximum parsimony principle for selecting evolutionary trees can be judged.

Random paths and cuts, electrical networks, and reversible Markov chains

Abstract: Let G be a graph representing an electrical network having source a and sink b, where edge e has conductivity $C( e )$ and resistance $R ( e )( R ( e ) = 1/C ( e ) )$. Let $C_{\text{eff}} $ and $R_{\text{eff}} $ denote the effective conductivity and effective resistance of G, respectively, $(R_{\text{eff}} = 1/C_{\text{eff}} )$. Let $\mathcal{P}$ denote the set of all paths joining a and b, and $\mathcal{K}$ denote the set of all cuts separating a and b. Let P be a random path from $\mathcal{P}$ having probability function p, and K be a random cut from $\mathcal{K}$ having probability function q. Denote by $\pi ( e )( \kappa ( e ) )$ the expected number of paths (cuts) containing edge e. For w a weighting defined on E and H a subset of E (or subgraph ), let $| H |_w = \sum_{e \in H} w ( e ) $. In this paper it is shown that $C_{\text{eff}} $ equals the maximum expected value of $1/ | P |_{R\pi } $ over all probability functions p. Further, the maximum is achieved when p is chosen so that $\pi $ is the cur...

On the number of distinct forests

Abstract: This paper contains a simple explicit formula, a recurrence formula and an asymptotic expansion for the number of distinct forests with n labeled vertices.

Binary probabilities induced by rankings

Abstract: A system $[ p_{ij} : i, j \in \{ 1, 2, \cdots ,n \}, i e j,p_{ij} + p_{ji} = 1 ]$ of binary probabilities is said to be induced by rankings if there is a probability distribution P on the set of $n!$ linear orders of $\{ 1,2, \cdots ,n \}$ such that, for all distinct i and $j,p_{ij} $ is the sum of the P values over all linear orders in which i precedes j. It has been known for some time that the triangle inequality $p_{ij} + p_{jk} \geqq p_{ik} $ is necessary for $\{ p_{i j} \} $ to be induced by rankings and that it is also sufficient if $n\leqq 5$. The insufficiency of the triangle inequality when $n\geqq 6$ has been known since about 1970, and other necessary conditions for $n\geqq 6$ have been known since 1978.The present paper generates additional necessary conditions that pertain to $n\geqq 6$ and shows that they are independent of previous necessary conditions. It then observes that the set of conditions on the $p_{ij} $ that are sufficient for $\{ p_{ij} \} $ to be induced by rankings regardles...

Medians and Majorities in Semimodular Lattices

Abstract: Given a $ u $-tuple $(x_1 , \cdots ,x_ u )$ in a metric space $(X,d)$, a median is an element m of X minimizing the sum $\sum_i d(m, x_i )$. One of the basic facts about medians in a finite space is that their obtainment by the majority rule is characteristic of a distributive-like structure on X, called a median semilattice. It is shown here that, in the case where X is a finite lattice, the semimodularity property is characterized by a weaker relation between the medians and the majority rule. Some consequences of this result are investigated.

The boolean basis problem and how to cover some polygons by rectangles

Abstract: For $S \subseteq \{ 0,1\} ^n $ the Boolean basis (or set basis) problem is to find a minimum size set $B \subseteq \{ 0,1\} ^n $ such that each $s \in S$ is a Boolean sum of vectors from B. This paper examines tractable special cases of this NP-complete problem. In particular a construction preserving tractability is given. The rectangle cover problem—expressing a rectilinear polygon as the union of a minimum number of rectangles—is a main application.

A new basis for trades

Abstract: A very fast algorithm to generate a semitriangular basis for trades consisting of minimal trades (sparsest basis) is given. By augmenting the elements of this basis, we construct an infinite family of 2-designs.

Sorting, minimal feedback sets, and Hamilton paths in tournaments

Abstract: A general method is presented for translating sorting by comparisons algorithms to algorithms that compute a Hamilton path in a tournament. The translation is based on the relation between minimal feedback sets and Hamilton paths in tournaments. It is proven that there is a one to one correspondence between the set of minimal feedback sets and the set of Hamilton paths. In the comparison model, all the tradeoffs for sorting between the number of processors and the number of rounds hold as well for computing Hamilton paths. For the CRCW model, with $O( n )$ processors, we show the following: (i) Two paths in a tournament can be merged in $O(\log \log n)$ time (Valiant’s algorithm [SIAM J. Comput., 4 (1975), pp. 348–355], (ii) a Hamilton path can be computed in $O(\log n)$ time (Cole’s algorithm). This improves a previous algorithm for computing a Hamilton path whose running time was $O(\log^2 n)$ using $O(n^2 )$ processors.

The domatic number problem interval graphs

Abstract: A set of vertices D is a dominating set of a graph $G = ( V,E )$ if every vertex in $V - D$ is adjacent to a vertex in D. The domatic number $d ( G )$ of a graph $G = ( V,E )$ is the maximum number k such that V can be partitioned into k disjoint dominating sets $D_1 , \cdots ,D_k $. The main purpose of this paper is to give linear algorithms for the domatic number problem in interval graphs. This paper also proves that $d ( G ) = \delta ( G ) + 1$ for any interval graph G, where $\delta ( G )$ is the minimum degree of a vertex in G.

A minimax arc theorem for reducible flow graphs

Abstract: A conjecture of Frank and Gyarfas is established by proving that the cardinality of a minimum feedback arc set in a reducible flow graph is equal to the cardinality of a maximum collection of arc disjoint cycles.

Maximal Chains and Antichains in Boolean Lattices

Abstract: The following equivalent results in the Boolean lattice $2^n $ are proven. (a) Every fibre of $2^n $ contains a maximal chain. (b) Every cutset of $2^n $ contains a maximal antichain. (c) Every red-blue colouring of the vertices of $2^n $ produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in $2^n $ there is a disjoint maximal antichain.Statement (a) is then improved to: (a') Every fibre of $2^n $ contains at least $n!/2^{n - 1} $ maximal chains. One conjecture of Lonc and Rival is supported, and another conjecture disproved, by showing: (i) Every fibre of $2^n $ has order $\Omega ( 1.25^n )$elements. (ii) There is a minimal fibre of $2^n ( n\geqq 4 )$ of size $2^{n - 1} + 2$.

Polynomial algorithms for finding cycles and paths in bipartite tournaments

Abstract: Efficient algorithms for finding Hamiltonian cycles, Hamiltonian paths, and cycles through two given vertices in bipartite tournaments are given.

Brooks coloring in parallel

Abstract: A theorem of Brooks guarantees that a maximum-degree-D graph can be properly colored with D colors if the graph does not contain a large complete subgraph. It is proved that finding such a coloring is in NC.