All posts under category "Computer Science / Discrete Mathematics"
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 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
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 .
Message passing algorithms are popular in many combinatorial optimization problems. For example, experimental results show that {em survey propagation} (a certain message passing algorithm) is effective in finding proper $k$-colorings of random graphs in the near-threshold regime. In 1962 Gallager i
Partition functions, also known as homomorphism functions, form a rich family of graph invariants that contain combinatorial invariants such as the number of k-colourings or the number of independent sets of a graph and also the partition functions of certain 'spin glass' models of statistical physi
We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.
We consider positional numeration system with negative base $-beta$, as introduced by Ito and Sadahiro. In particular, we focus on arithmetical properties of such systems when $beta$ is a quadratic Pisot number. We study a class of roots $beta>1$ of polynomials $x^2-mx-n$, $mgeq ngeq 1$, and show th
This paper presents an application of the Infinite Unit Axiom, introduced by Yaroslav Sergeyev, (see [11] - [14]) to the development of one-dimensional cellular automata. This application allows the establishment of a new and more precise metric on the space of definition for one-dimensional cellula
This paper presents a novel technique for state space reduction of probabilistic specifications, based on a newly developed notion of confluence for probabilistic automata. We prove that this reduction preserves branching probabilistic bisimulation and can be applied on-the-fly. To support the techn
We consider the design and analysis of the efficiently-encodable rate-compatible ($E^2RC$) irregular LDPC codes proposed in previous work. In this work we introduce semi-structured $E^2RC$-like codes and protograph $E^2RC$ codes. EXIT chart based methods are developed for the design of semi-structur
Properties of systems with majority voting rules have been exhaustingly studied. In this work we focus on the randomized case - where the system is initialized by randomized initial set of seeds. Our main aim is to give an asymptotic estimate for sampling probability, such that the initial set of se
A polytopal digraph $G(P)$ is an orientation of the skeleton of a convex polytope $P$. The possible non-degenerate pivot operations of the simplex method in solving a linear program over $P$ can be represented as a special polytopal digraph known as an LP digraph. Presently there is no general chara
An edge-coloring of a multigraph G with colors 1,2,...,t is called an interval t-coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In this paper we prove that if G is a connected cubic multigraph (a connected cubic gra
An interval t-coloring of a graph G is a proper edge-coloring of G with colors 1,2,...,t such that at least one edge of G is colored by i, i=1,2,...,t, and the edges incident to each vertex vin V(G) are colored by d_{G}(v) consecutive colors, where d_{G}(v) is the degree of the vertex v in G. In thi
Finding the maximum clique is a known NP-Complete problem and it is also hard to approximate. This work proposes two efficient algorithms to obtain it. Nevertheless, the first one is able to fins the maximum for some special cases, while the second one has its execution time bounded by the number of
In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let Ain{RR^{ntimes m}} and BinRR^{n times p} be two matrices and eps>0. We approximate the product A^top B using two down-sampled sketches, tilde{A}inRR^{ttimes m} and tilde{B}inRR^{ttimes
We present a tight lower bound for the spanning tree congestion of Hamming graphs.
Let G be a graph cellularly embedded in a surface S. Given two closed walks c and d in G, we take advantage of the RAM model to describe linear time algorithms to decide if c and d are homotopic in S, either freely or with fixed basepoint. We restrict S to be orientable for the free homotopy test, b
We study the relation between the performance of the randomized rumor spreading (push model) in a d-regular graph G and the performance of the same algorithm in the percolated graph G_p. We show that if the push model successfully broadcast the rumor within T rounds in the graph G then only (1 + eps
In this paper we study the complexity of the firefighter problem and related problems on trees when more than one firefighter is available at each time step, and answer several open questions of Finbow and MacGillivray 2009. More precisely, when $b geq 2$ firefighters are allowed at each time step,
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.
![A note on the paper 'Minimizing total tardiness on parallel machines with preemptions' by Kravchenko and Werner [2010]](/images/papers/1112.4396/cover.png)
In this note, we point out two major errors in the paper 'Minimizing total tardiness on parallel machines with preemptions' by Kravchenko and Werner [2010]. More precisely, they proved that both problems P|pmtn|sum(Tj) and P|rj, pj = p, pmtn|sum(Tj) are NP-Hard. We give a counter-example to their pr
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 $K_{frac{n-1}{2},frac{n+1}{2}}$.
The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.
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
In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-tau, where tau is the golden ratio, can be described by a finite set of fat (-1-tau)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean that every fat Hoffman graph with smallest eigenval
This lecture discusses the mathematical relationship between network structure and network utilization of transportation network. Network structure means the graph itself. Network utilization represent the aggregation of trajectories of agents in using the network graph. I show the similarity and re
We study the following inverse graph-theoretic problem: how many vertices should a graph have given that it has a specified value of some parameter. We obtain asymptotic for the minimal number of vertices of the graph with the given number $n$ of maximal independent sets for a class of natural numbe
In this paper we deal with the problem of finding the smallest and the largest elements of a totally ordered set of size $n$ using pairwise comparisons if $k$ of the comparisons might be erroneous where $k$ is a fixed constant. We prove that at least $(k+1.5)n+Theta(k)$ comparisons are needed in the
This paper investigates questions related to the modularity in discrete models of biological interaction networks. We develop a theoretical framework based on the analysis of their asymptotic dynamics. More precisely, we exhibit formal conditions under which agents of interaction networks can be gro
Let N^{+}(k)= 2^{k/2} k^{3/2} f(k) and N^{-}(k)= 2^{k/2} k^{1/2} g(k) where 1=o(f(k)) and g(k)=o(1). We show that the probability of a random 2-coloring of {1,2,...,N^{+}(k)} containing a monochromatic k-term arithmetic progression approaches 1, and the probability of a random 2-coloring of {1,2,...
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.
This paper proposes an optimum version of the recently advanced scheme for generalized unary coding. In this method, the block of 1s that identifies the number is allowed to be broken up, which extends the count. The result is established by a theorem. The number count is now n(n-k-1)+1 rather than
In the past few decades there has been a good deal of papers which are concerned with optimization problems in different areas of mathematics (along 0-1 words, finite or infinite) and which yield - sometimes quite unexpectedly - balanced words as optimal. In this note we list some key results along
We consider a system of m linear equations in n variables Ax=b where A is a given m x n matrix and b is a given m-vector known to be equal to Ax' for some unknown solution x' that is integer and k-sparse: x' in {0,1}^n and exactly k entries of x' are 1. We give necessary and sufficient conditions fo
We study the combinatorial two-player game Tron. We answer the extremal question on general graphs and also consider smaller graph classes. Bodlaender and Kloks conjectured in [2] PSPACE- completeness. We proof this conjecture.
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
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
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
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
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 $p$
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
A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. In this paper, the total coloring conjecture is completely confirmed for pseud
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
The study of regular linear conjunctive normal form (LCNF) formulas is of interest because exact satisfiability (XSAT) is known to be NP-complete for this class of formulas. In a recent paper it was shown that the subclass of regular exact LCNF formulas (XLCNF) is of sub-exponential complexity, i.e.
The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $Deltageq12$ and $d=2$, the (2,1)-total labelling number $lambda_2^T(G)$ is at most $Delta+2$.
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