All posts under tag "Discrete Mathematics"
The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Si
A pair of unit clauses is called conflicting if it is of the form $(x)$, $(bar{x})$. A CNF formula is unit-conflict free (UCF) if it contains no pair of conflicting unit clauses. Lieberherr and Specker (J. ACM 28, 1981) showed that for each UCF CNF formula with $m$ clauses we can simultaneously sati
We prove that there is an absolute constant $C>0$ so that for every natural $n$ there exists a triangle-free emph{regular} graph with no independent set of size at least $Csqrt{nlog n}$.
Several different measures for digraph width have appeared in the last few years. However, none of them shares all the 'nice' properties of treewidth: First, being emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all $MS1$-definable problems on digraphs of bounded width. An
Message passing type algorithms such as the so-called Belief Propagation algorithm have recently gained a lot of attention in the statistics, signal processing and machine learning communities as attractive algorithms for solving a variety of optimization and inference problems. As a decentralized,
This paper considers heuristics for well known resource-constrained project scheduling problem (RCPSP). First a feasible schedule is constructed using randomized best insertion algorithm. The construction is followed by a local search where a new solution is generated as follows: first we randomly d
A solution to a problem of ErdH{o}s, Rubin and Taylor is obtained by showing that if a graph $G$ is $(a:b)$-choosable, and $c/d > a/b$, then $G$ is not necessarily $(c:d)$-choosable. The simplest case of another problem, stated by the same authors, is settled, proving that every 2-choosable graph is
We show the following. begin{theorem} Let $M$ be an finite-state ergodic time-reversible Markov chain with transition matrix $P$ and conductance $phi$. Let $lambda in (0,1)$ be an eigenvalue of $P$. Then, $$phi^2 + lambda^2 leq 1$$ end{theorem} This strengthens the well-known~cite{HLW,Dod84, AM85, A
We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesp
The configurations of single and double bonds in polycyclic hydrocarbons are abstracted as Kekul'e states of graphs. Sending a so-called soliton over an open channel between ports (external nodes) of the graph changes the Kekul'e state and therewith the set of open channels in the graph. This switch
We provide linear-time algorithms for geometric graphs with sublinearly many crossings. That is, we provide algorithms running in O(n) time on connected geometric graphs having n vertices and k crossings, where k is smaller than n by an iterated logarithmic factor. Specific problems we study include
The class of 2-interval graphs has been introduced for modelling scheduling and allocation problems, and more recently for specific bioinformatic problems. Some of those applications imply restrictions on the 2-interval graphs, and justify the introduction of a hierarchy of subclasses of 2-interval
On-line chain partition is a two-player game between Spoiler and Algorithm. Spoiler presents a partially ordered set, point by point. Algorithm assigns incoming points (immediately and irrevocably) to the chains which constitute a chain partition of the order. The value of the game for orders of wid
Let $S$ be a set of $n^2$ symbols. Let $A$ be an $ntimes n$ square grid with each cell labeled by a distinct symbol in $S$. Let $B$ be another $ntimes n$ square grid, also with each cell labeled by a distinct symbol in $S$. Then each symbol in $S$ labels two cells, one in $A$ and one in $B$. Define
In this paper we show how to extend the known algorithm of nodal analysis in such a way that, in the case of circuits without nullors and controlled sources (but allowing for both, independent current and voltage sources), the system of nodal equations describing the circuit is partitioned into one
The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix C_{nm} of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The second mapping is from a perfect matchi
Many real discrete optimization problems (DOPs) are $NP$-hard and contain a huge number of variables and/or constraints that make the models intractable for currently available solvers. Large DOPs can be solved due to their special tructure using decomposition approaches. An important example of dec
The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx: Ax >= b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >=
The concept of group divisible codes, a generalization of group divisible designs with constant block size, is introduced in this paper. This new class of codes is shown to be useful in recursive constructions for constant-weight and constant-composition codes. Large classes of group divisible codes
In this paper, we show a connection between a certain online low-congestion routing problem and an online prediction of graph labeling. More specifically, we prove that if there exists a routing scheme that guarantees a congestion of $alpha$ on any edge, there exists an online prediction algorithm w
Community structure is an important area of research. It has received a considerable attention from the scientific community. Despite its importance, one of the key problems in locating information about community detection is the diverse spread of related articles across various disciplines. To the
In a graph $G$ of maximum degree $Delta$ let $gamma$ denote the largest fraction of edges that can be $Delta$ edge-coloured. Albertson and Haas showed that $gamma geq 13/15$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a grap
The sun is the graph obtained from a cycle of length even and at least six by adding edges to make the even-indexed vertices pairwise adjacent. Suns play an important role in the study of strongly chordal graphs. A graph is chordal if it does not contain an induced cycle of length at least four. A g
We consider the Seidel complementation on ($P_5, bar{P_5}, Bull)$-free graphs
Let $mathcal {T}^{Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $Delta$. Suppose that every tree in $mathcal {T}^{Delta}_n$ is equally likely. We show that the number of vertices of degree $j$ in $mathcal {T}^{Delta}_n$ is asymptoticall
We investigate the complexity of approximately counting stable matchings in the $k$-attribute model, where the preference lists are determined by dot products of 'preference vectors' with 'attribute vectors', or by Euclidean distances between 'preference points' and 'attribute points'. Irving and Le
This paper presents an approach for the development of a number theoretic discrete Hilbert transform. The forward transformation has been applied by taking the odd reciprocals that occur in the DHT matrix with respect to a power of 2. Specifically, the expression for a 16-point transform is provided
The paper is organized as a self-contained literate Haskell program that implements elements of an executable finite set theory with focus on combinatorial generation and arithmetic encodings. The code, tested under GHC 6.6.1, is available at http://logic.csci.unt.edu/tarau/research/2008/fSET.zip .
In this article, I present a conjecture on the number of independent sets on graph covers. I also show that the conjecture implies that the partition function of a binary pairwise attractive model is greater than that of the Bethe approximation.
We show that if $G$ is a simple triangle-free graph with $ngeq 3$ vertices, without a perfect matching, and having a minimum degree at least $frac{n-1}{2}$, then $G$ is isomorphic either to $C_5$ or to $K_{frac{n-1}{2},frac{n+1}{2}}$.
We study the bandwidth and the pathwidth of multi-dimensional grids. It can be shown for grids, that these two parameters are equal to a more basic graph parameter, the vertex boundary width. Using this fact, we determine the bandwidth and the pathwidth of three-dimensional grids, which were known o
We show that in any subset of the vertices of $n$-dimensional cube that contains at least $2^{n-1}+1$ vertices ($ngeq 4$), there are four vertices that induce a claw, or there are eight vertices that induce the cycle of length eight.
The paper deals with a non-linear system largely used in biology, which, in certain conditions and for particular coefficient values, becomes linear, with a linear diagram over a large range of time. It can be used as a veritable regulator in systems' control
Huffman coding is often presented as the optimal solution to Twenty Questions. However, a caveat is that Twenty Questions games always end with a reply of 'Yes,' whereas Huffman codewords need not obey this constraint. We bring resolution to this issue, and prove that the average number of questions
A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with at least three letters, we show that multiplication by an int
The differential encoding/decoding setup introduced by Kiran et al, Oggier-Hassibi and Jing-Jafarkhani for wireless relay networks that use codebooks consisting of unitary matrices is extended to allow codebooks consisting of scaled unitary matrices. For such codebooks to be usable in the Jing-Hassi
We show how to construct an explicit Hamilton cycle in the directed Cayley graph Cay({sigma_n, sigma_{n-1}} : mathbb{S}_n), where sigma_k = (1 2 >... k). The existence of such cycles was shown by Jackson (Discrete Mathematics, 149 (1996) 123-129) but the proof only shows that a certain directed grap
The lower bound W(K_{2n})>=3n-2 is proved for the greatest possible number of colors in an interval edge coloring of the complete graph K_{2n}.
The (-beta)-integers are natural generalisations of the beta-integers, and thus of the integers, for negative real bases. They can be described by infinite words which are fixed points of anti-morphisms. We show that they are not necessarily uniformly discrete and relatively dense in the real number
In 2001, Brualdi and Hollingsworth proved that an edge-colored balanced complete bipartite graph Kn,n with a color set C = {1,2,3,..., 2n-1} has a heterochromatic spanning tree if the number of edges colored with colors in R is more than |R|^2 /4 for any non-empty subset R subseteq C, where a hetero
A graph $G$ with a list of colors $L(v)$ and weight $w(v)$ for each vertex $v$ is $(L,w)$-colorable if one can choose a subset of $w(v)$ colors from $L(v)$ for each vertex $v$, such that adjacent vertices receive disjoint color sets. In this paper, we give necessary and sufficient conditions for a w
In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The {em chromatic edge strength} of a graph is the minimum number of colors required in a min
G is a Koenig-Egervary graph provided alpha(G)+ mu(G)=|V(G)|, where mu(G) is the size of a maximum matching and alpha(G) is the cardinality of a maximum stable set. S is a local maximum stable set of G if S is a maximum stable set of the closed neighborhood of S. Nemhauser and Trotter Jr. proved tha
A run is a maximal occurrence of a repetition $v$ with a period $p$ such that $2p le |v|$. The maximal number of runs in a string of length $n$ was studied by several authors and it is known to be between $0.944 n$ and $1.029 n$. We investigate highly periodic runs, in which the shortest period $p$
Using specializations of unfold and fold on a generic tree data type we derive unranking and ranking functions providing natural number encodings for various Hereditarily Finite datatypes. In this context, we interpret unranking operations as instances of a generic anamorphism and ranking operatio
More precise estimation of the greedy algorithm complexity for a special case of the set cover problem is given in this paper.
At some places (see the references) Martin Erickson describes a certain game: 'Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an n x n grid. The first player (if any) to occupy four cells at the vertices of a square with horizontal and vertical
Two Diophantine equation generator function for integer residuals produced by integer division over closed intervals are presented. One each for the closed intervals [1,Floor(n^0.5)] and [Ceiling(n^0.5),n], respectively.
The notion of the Fibonacci cobweb poset from [1] has been naturally extended to any admissible sequence $F$ in [2] where it was also recognized that the celebrated prefab notion of Bender and Goldman [3] - (see also [4,5]) - admits such an extension so as to encompass the new type combinatorial obj
We study suitable parameters and relations in a numerical semigroup S. When S is the Weierstrass semigroup at a rational point P of a projective curve C, we evaluate the Feng-Rao order bound of the associated family of Goppa codes. Further we conjecture that the order bound is always greater than a
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