SIAM Digital Library
 
 
 

SIAM J. on Discrete Mathematics

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

2001

Volume 14, Issue 4, pp. 433-564


Forcing Structures and Cliques in Uniquely Vertex Colorable Graphs

Amir Daneshgar

SIAM J. Discrete Math. 14, pp. 433-445 (13 pages) | Cited 1 time

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
Let G be a simple undirected uniquely vertex k-colorable graph, or a k-UCG for short. M. Truszczyński [Some results on uniquely colorable graphs, in Finite and Infinite Sets, North-Holland, Amsterdam, 1984, pp. 733--748] introduced $e^{^{*}}(G)=|V(G)|(k-1)-{k \choose 2}$ as the minimum number of edges for a k-UCG and S. J. Xu [J. Combin. Theory Ser. B, 50 (1990), pp. 319--320] conjectured that any minimal k-UCG contains a Kk as a subgraph. In this paper, first we introduce a technique called forcing. Then by applying this technique in conjunction with a feedback structure we construct a k-UCG with clique number k-t, for each $t \geq 1$ and each k, when k is large enough. This also improves some known results for the case t=1.
Second, we analyze the parameter $\Lambda(G)=|E(G)|-e^{^{*}}(G)$ for our constructions, and we obtain some bounds for the functions \lambda_{_{t}}(k)= \min \{\Lambda(G) \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}, $$ $$ \nu_{_{t}}(k)= \min \{|V(G)| \ : \ G \ {\rm is \ a} \ k\mbox{\rm -UCG and} \ cl(G)=k-t \}. Also, we introduce some possible applications of the technique in cryptography and data compression.

Blocking Semiovals of Type (1,m+1,n+1)

Lynn M. Batten and Jeremy M. Dover

SIAM J. Discrete Math. 14, pp. 446-457 (12 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We consider the existence of blocking semiovals in finite projective planes which have intersection sizes 1, m+1 or n+1 with the lines of the plane for $1 \leq m < n$. For those prime powers $q \leq 1024$, in almost all cases, we are able to show that, apart from a trivial example, no such blocking semioval exists in a projective plane of order q. We are also able to prove, for general q, that if q2+q+1 is a prime or three times a prime, then only the same trivial example can exist in a projective plane of order q.

An Optimal Online Algorithm for Bounded Space Variable-Sized Bin Packing

Steven S. Seiden

SIAM J. Discrete Math. 14, pp. 458-470 (13 pages) | Cited 5 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
An online algorithm for variable-sized bin packing, based on the Harmonic algorithm of Lee and Lee,[J. ACM, 32 (1985), pp. 562--572], is investigated. This algorithm was proposed by Csirik, [Acta Inform., 26 (1989), pp. 697--709], who proved that for all sets of bin sizes, 1.69103 upper bounds its performance ratio. The upper bound is improved in the sense that we give a method of calculating the performance ratio to any accuracy for any set of bin sizes. Further, it is shown that the algorithm is optimal among those which use bounded space. An interesting feature of the analysis is that, although it is shown that our algorithm achieves a performance ratio arbitrarily close to the optimum value, it is not known precisely what that value is. The case where bins of capacity 1 and $\alpha \in (0,1)$ are used is studied in greater detail. It is shown that among algorithms which are allowed to choose $\alpha$, the optimal performance ratio lies in [1.37530,1.37532].

Classification of Homomorphisms to Oriented Cycles and of k-Partite Satisfiability

Tomás Feder

SIAM J. Discrete Math. 14, pp. 471-480 (10 pages) | Cited 4 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We show that, for every choice of an oriented cycle H, the problem of whether an input digraph G has a homomorphism to H is either polynomially solvable or NP-complete. Along the way, we obtain simpler proofs for two known polynomial cases, namely, oriented paths and unbalanced oriented cycles, and exhibit two new simple polynomial cases of balanced oriented cycles. The more difficult cases of the classification are handled by means of a new problem, the bipartite boolean satisfiability problem. In general, the k-partite boolean satisfiability problems are shown to be either polynomially solvable or NP-complete, thus generalizing Schaefer's classification of boolean satisfiability problems.

Equireplicate Balanced Binary Codes for Oligo Arrays

Noga Alon, Charles J. Colbourn, Alan C. H. Ling, and Martin Tompa

SIAM J. Discrete Math. 14, pp. 481-497 (17 pages) | Cited 2 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
In the manufacture of oligo arrays for DNA hybridization experiments, manufacturing defects must be detected and their position determined. The design of manufacturing protocols for such oligo arrays leads to a combinatorial problem, requiring certain binary codes which have an additional balance property. Constructions using block designs and packings for these codes, within a range of interest in a practical manufacturing application, are developed. The focus is on equireplicate codes, constant weight codes in which every bit position is a one equally often.

The Obnoxious Center Problem on a Tree

Rainer E. Burkard, Helidon Dollani, Yixun Lin, and Günter Rote

SIAM J. Discrete Math. 14, pp. 498-509 (12 pages) | Cited 3 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The obnoxious center problem in a graph G asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. We derive algorithms with linear running time for the cases when G is a path or a star, thus improving previous results of Tamir [SIAMJ. Discrete Math, 1 (1988), pp. 377--396]. For subdivided stars we present an algorithm of running time O(n log n). For general trees, we improve an algorithm of Tamir [SIAM J. Discrete Math, 1 (1988), pp. 377--396] by a factor of log n. Moreover, a linear algorithm for the unweighted center problem on an arbitrary tree with neutral and obnoxious vertices is described.

Structural Diagnosis of Wiring Networks: Finding Connected Components of Unknown Subgraphs

Weiping Shi and Douglas B. West

SIAM J. Discrete Math. 14, pp. 510-523 (14 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
Given a graph $G=(V,\cal E)$, we want to find the vertex sets of the components of an unknown subgraph F=(V,E) of G such that $E \subseteq \cal E$. We learn about F by sending an oracle a query set $S \subseteq V$, and the oracle tells us the vertices connected to S in F. The objective is to use the minimum number of queries to partition the vertex set V into components of F. In electronic circuit design, the problem is also known as structural diagnosis of wiring networks.

Reserving Resilient Capacity in a Network

G. Brightwell, G. Oriolo, and F. B. Shepherd

SIAM J. Discrete Math. 14, pp. 524-539 (16 pages) | Cited 7 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
We examine various problems concerning the reservation of capacity in a given network, where each arc has a per-unit cost, so as to be "resilient" against one or more arc failures. For a given pair (s,t) of nodes and demand T, we require that, on the failure of any k arcs of the network, there is sufficient reserved capacity in the remainder of the network to support an (s,t) flow of value T. This problem can be solved in polynomial time for any fixed k, but we show that it is NP-hard if we are required to reserve an integer capacity on each arc.
We concentrate on the case where the reservation has to consist of a collection of arc-disjoint paths: here we give a very simple algorithm to find a minimum cost fractional solution, based on finding successive shortest paths in the network. Unlike traditional network flow problems, the integral version is NP-hard: we do, however, give a polynomial time $\frac{15}{14}$-approximation algorithm in the case k=1 and show that this bound is best possible unless P = NP.

When is Individual Testing Optimal for Nonadaptive Group Testing?

S. H. Huang and F. K. Hwang

SIAM J. Discrete Math. 14, pp. 540-548 (9 pages) | Cited 2 times

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
The combinatorial group testing problem is, assuming the existence of up to d defectives among n items, to identify the defectives by as few tests as possible. In this paper, we study the problem for what values of n, given d, individual testing is optimal in nonadaptive group testing. Let N(d) denote the largest n for fixed d for which individual testing is optimal. We will show that N(d)=(d+1)2 under a prevalent constraint in practical nonadaptive algorithms and prove that N(d)=(d+1)2 for d=1, 2, 3, 4 without any constraint.

Asymptotic Minimum Covering Radius of Block Codes

Po-Ning Chen and Yunghsiang S. Han

SIAM J. Discrete Math. 14, pp. 549-564 (16 pages)

Online Publication Date: August 01, 2006

Full Text: | Download PDF

Show Abstract
In this paper, we restudy the covering radius of block codes from an information theoretic point of view by ignoring the combinatorial formulation of the problem. In the new setting, the formula of the statistically defined minimum covering radius, for which the probability mass of uncovered space by M spheres can be made arbitrarily small, is reduced to a minimization of a statistically defined spectrum formula among codeword-selecting distributions. The advantage of the new view is that no assumptions need to be made on the code alphabet (such as finite, countable, etc.) and the distance measure (such as additive, symmetric, bounded, etc.) in the problem transformation, and hence the spectrum formula can be applied in most general situations. We next address a sufficient condition under which uniform codeword-selecting distribution minimizes the spectrum formula. With the condition, the asymptotic minimum covering radius for block codes under J-ary quantized channels and constant weight codes under Hamming distance measure are determined to display the usage of the spectrum formula.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Mean Field Games: Numerical Methods for the Planning Problem
  2. A Convex Condition for Robust Stability Analysis via Polyhedral Lyapunov Functions
  3. Stability and Resolution Analysis for a Topological Derivative Based Imaging Functional
  4. Experimental Design for Biological Systems
  5. Linear Programming and Constrained Average Optimality for General Continuous-Time Markov Decision Pr...
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close