Graphs of interval count two with a given partition

We describe a polynomial time algorithm to decide for a given connected graph G and a given partition of its vertex set into two sets A and B  , whether it is possible to assign a closed interval I(u)$I(u)$ to each vertex u of G such that two distinct vertices u and v of G   are adjacent if and only if I(u)$I(u)$ and I(v)$I(v)$ intersect, all intervals assigned to vertices in A   have some length _LA$L_A$, and all intervals assigned to vertices in B   have some length _LB$L_B$ where _LA<_LB$L_A<L_B$. Our result is motivated by the interval count problem whose complexity status is open.     

On the partition dimension of a class of circulant graphs

For a vertex v   of a connected graph G(V,E)$G(V,E)$ and a subset S of V, the distance between a vertex v and S   is defined by d(v,S)=min{d(v,x):x∈S}$d(v,S)=\min \{d(v,x):x\in S\}$. For an ordered k  -partition π={_S1,_S2…_Sk}$\pi =\{S_1,S_2\dots S_k\}$ of V, the partition representation of v with respect to π is the k  -vector r(v|π)=(d(v,_S1),d(v,_S2)…d(v,_Sk))$r(v|\pi )=(d(v,S_1),d(v,S_2)\dots d(v,S_k))$. The k-partition π is a resolving partition if the k  -vectors r(v|π)$r(v|\pi )$, v∈V(G)$v\in V(G)$ are distinct. The minimum k for which there is a resolving k-partition of V is the partition dimension of G. Salman et al. [1] in their paper which appeared in Acta Mathematica Sinica, English Series   proved that partition dimension of a class of circulant graph G(n,±{1,2})$G(n,\pm \{1,2\})$, for all even n?6$n?6$ is four. In this paper we prove that it is three.     

The pessimistic diagnosability of alternating group graphs under the PMC model

The process of identifying faulty processors is called the diagnosis of the system. Several diagnostic models have been proposed, the most popular is the PMC (Preparata, Metze and Chen) diagnostic model. The pessimistic diagnosis strategy is a classic strategy based on the PMC model in which isolates all faulty nodes within a set containing at most one fault-free node. A system is t/t$t/t$-diagnosable if, provided the number of faulty processors is bounded by t, all faulty processors can be isolated within a set of size at most t with at most one fault-free node mistaken as a faulty one. The pessimistic diagnosability of a system G  , denoted by _tp(G)$t_p(G)$, is the maximal number of faulty processors so that the system G   is t/t$t/t$-diagnosable. Jwo et al. [11] introduced the alternating group graph as an interconnection network topology for computing systems. The proposed graph has many advantages over hypercubes and star graphs. For example, for all alternating group graphs, every pair of vertices in the graph are connected by a Hamiltonian path and the graph can embed cycles with arbitrary length with dilation 1. In this article, we completely determine the pessimistic diagnosability of an n  -dimensional alternating group graph, denoted by _AGn${\mathit{AG}}_n$. Furthermore, _tp(_AGn)=4n−11$t_p({\mathit{AG}}_n)=4n-11$ for n≥4$n\ge 4$.     

Computing phylogenetic roots with bounded degrees and errors is NP-complete

In this paper we study the computational complexity of the following optimization problem: given a graph G=(V,E)$G=(V,E)$, we wish to find a tree T such that (1) the degree of each internal node of T   is at least 3 and at most Δ$\Delta$, (2) the leaves of T are exactly the elements of V, and (3) the number of errors, that is, the symmetric difference between E   and {{u,v}:u,v$\{\{u,v\}:u,v$ are leaves of T   and _dT(u,v)≤k}$d_T(u,v)\le k\}$, is as small as possible, where _dT(u,v)$d_T(u,v)$ denotes the distance between u$u$ and v$v$ in tree T  . We show that this problem is NP-hard for all fixed constants Δ,k≥3$\Delta ,k\ge 3$.     

Asynchronous deterministic rendezvous in graphs

Two mobile agents (robots) having distinct labels and located in nodes of an unknown anonymous connected graph have to meet. We consider the asynchronous version of this well-studied rendezvous problem and we seek fast deterministic algorithms for it. Since in the asynchronous setting, meeting at a node, which is normally required in rendezvous, is in general impossible, we relax the demand by allowing meeting of the agents inside an edge as well. The measure of performance of a rendezvous algorithm is its cost: for a given initial location of agents in a graph, this is the number of edge traversals of both agents until rendezvous is achieved. If agents are initially situated at a distance D   in an infinite line, we show a rendezvous algorithm with cost O(D|_Lmin|²)$\mathrm{O}(D|L_{\min }{|}^2)$ when D   is known and O((D+|_Lmax|)³)$\mathrm{O}((D+\left|L_{\max }\right|{)}^3)$ if D   is unknown, where |_Lmin|$\left|L_{\min }\right|$ and |_Lmax|$\left|L_{\max }\right|$ are the lengths of the shorter and longer label of the agents, respectively. These results still hold for the case of the ring of unknown size, but then we also give an optimal algorithm of cost O(n|_Lmin|)$\mathrm{O}(n|L_{\min }|)$, if the size n   of the ring is known, and of cost O(n|_Lmax|)$\mathrm{O}(n|L_{\max }|)$, if it is unknown. For arbitrary graphs, we show that rendezvous is feasible if an upper bound on the size of the graph is known and we give an optimal algorithm of cost O(D|_Lmin|)$\mathrm{O}(D|L_{\min }|)$ if the topology of the graph and the initial positions are known to agents.     

A note on scaling the Linpack benchmark

Dimensional analysis yields a new scaling formula for the Linpack benchmark. The computational power r(_p0,_q0)$r(p_0,q_0)$ on a set of processors decomposed into a (_p0,_q0)$(p_0,q_0)$ grid determines the computational power r(p,q)$r(p,q)$ on a set of processors decomposed into a (p,q)$(p,q)$ grid by the formula r(p,q)=(p/_p0)^α(q/_q0)^βr(_p0,_q0)$r(p,q)=(p/p_0{)}^{\alpha }(q/q_0{)}^{\beta }r(p_0,q_0)$. The two scaling parameters α$\alpha$ and β$\beta$ measure interprocessor communication overhead required by the algorithm. A machine that scales perfectly corresponds to α=β=1$\alpha =\beta =1$; a machine that scales not at all corresponds to α=β=0$\alpha =\beta =0$. We have determined the two scaling parameters by imposing a fixed-time constraint on the problem size such that the execution time remains constant as the number of processors changes. Results for a collection of machines confirm that the formula suggested by dimensional analysis is correct. Machines with the same values for these parameters are self-similar. They scale the same way even though the details of their specific hardware and software may be quite different.     

A sufficient condition involving implicit degree and neighborhood intersection for long cycles

Let id(v)$\mathit{id}(v)$ denote the implicit degree of a vertex v in a graph G. In this paper, we prove that: Let G   be a 2-connected graph. If max?{id(u),id(v)}≥c/2$\max ?\{\mathit{id}(u),\mathit{id}(v)\}\ge c/2$ for each pair of nonadjacent vertices u and v   in an induced claw, and |N(x)∩N(y)|≥2$|N(x)\cap N(y)|\ge 2$ for each pair of nonadjacent vertices x and y in an induced modified claw, then G contains either a hamiltonian cycle or a cycle of length at least c.     

On mimicking networks representing minimum terminal cuts

Given a capacitated undirected graph G=(V,E)$G=(V,E)$ with a set of terminals K⊂V$K\subset V$, a mimicking network   is a smaller graph H=(_VH,_EH)$H=(V_H,E_H)$ which contains the set of terminals K   and for every bipartition [U,K−U]$[U,K-U]$ of the terminals, the cost of the minimum cut separating U   from K−U$K-U$ in G is exactly equal to the cost of the minimum cut separating U   from K−U$K-U$ in H.     

Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f   a labeling of its vertices with positive integers; denote by S(v)$S(v)$ the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering   if S(u)<S(v)$S(u)<S(v)$ for each arc (u,v)$(u,v)$ of the digraph. The problem asks to find the minimum number k for which D   has a topological additive numbering with labels belonging to {1,…,k}$\{1,\dots ,k\}$, denoted by _ηt(D)${\eta }_t(D)$.     

Compatibility of unrooted phylogenetic trees is FPT

A collection of _T1,_T2,…,_Tk$T_1,T_2,\dots ,T_k$ of unrooted, leaf labelled (phylogenetic) trees, all with different leaf sets, is said to be compatible   if there exists a tree T$T$ such that each tree _Ti$T_i$ can be obtained from T$T$ by deleting leaves and contracting edges. Determining compatibility is NP-hard, and the fastest algorithm to date has worst case complexity of around Ω(n^k)$\mathrm{\Omega }(n^k)$ time, n$n$ being the number of leaves. Here, we present an O(nf(k))$\mathrm{O}(\mathit{nf}(k))$ algorithm, proving that compatibility of unrooted phylogenetic trees is fixed parameter tractable   (FPT) with respect to the number k$k$ of trees.     

Finding the nucleolus of any n-person cooperative game by a single linear program

In this paper we show a new method for calculating the nucleolus by solving a unique minimization linear program with O(4ⁿ)$O(4^n)$ constraints whose coefficients belong to {−1,0,1}$\{-1,0,1\}$. We discuss the need of having all these constraints and empirically prove that they can be reduced to O(_kmax2ⁿ)$O(k_{\mathit{max}}{2}^n)$, where k_max is a positive integer comparable with the number of players. A computational experience shows the applicability of our method over (pseudo)random transferable utility cooperative games with up to 18 players.     

A representation theorem for (q-)holonomic sequences

Chomsky and Schützenberger showed in 1963 that the sequence _dL(n)$d_L(n)$, which counts the number of words of a given length n in a regular language L, satisfies a linear recurrence relation with constant coefficients for n  , i.e., it is C-finite. It follows that every sequence s(n)$s(n)$ which satisfies a linear recurrence relation with constant coefficients can be represented as _dL1(n)−_dL2(n)$d_{L_1}(n)-d_{L_2}(n)$ for two regular languages. We view this as a representation theorem for C-finite sequences. Holonomic or P-recursive sequences are sequences which satisfy a linear recurrence relation with polynomial coefficients. q-Holonomic sequences are the q-analog of holonomic sequences. In this paper we prove representation theorems of holonomic and q-holonomic sequences based on position specific weights on words, and for holonomic sequences, without using weights, based on sparse regular languages.     

Choosing starting values for certain Newton–Raphson iterations

We aim at finding the best possible seed values when computing a^1/p$a^{1/p}$ using the Newton–Raphson iteration in a given interval. A natural choice of the seed value would be the one that best approximates the expected result. It turns out that in most cases, the best seed value can be quite far from this natural choice. When we evaluate a monotone function f(a)$f(a)$ in the interval [_amin,_amax]$[a_{\min },a_{\max }]$, by building the sequence _xn$x_n$ defined by the Newton–Raphson iteration, the natural choice consists in choosing _x0$x_0$ equal to the arithmetic mean of the endpoint values. This minimizes the maximum possible distance between _x0$x_0$ and f(a)$f(a)$. And yet, if we perform n$n$ iterations, what matters is to minimize the maximum possible distance between _xn$x_n$ and f(a)$f(a)$. In several examples, the value of the best starting point varies rather significantly with the number of iterations.     

Optimal fault-tolerant embedding of paths in twisted cubes

The twisted cube is an important variation of the hypercube. It possesses many desirable properties for interconnection networks. In this paper, we study fault-tolerant embedding of paths in twisted cubes. Let _TQn(V,E)${\mathit{TQ}}_n(V,E)$ denote the n-dimensional twisted cube. We prove that a path of length l   can be embedded between any two distinct nodes with dilation 1 for any faulty set F⊂V(_TQn)∪E(_TQn)$F\subset V({\mathit{TQ}}_n)\cup E({\mathit{TQ}}_n)$ with |F|?n-3$\left|F\right|?n-3$ and any integer l   with 2^n-1-1?l?|V(_TQn-F)|-1$2^{n-1}-1?l?|V({\mathit{TQ}}_n-F)|-1$ (n?3$n?3$). This result is optimal in the sense that the embedding has the smallest dilation 1. The result is also complete in the sense that the two bounds on path length l   and faulty set size |F|$\left|F\right|$ for a successful embedding are tight. That is, the result does not hold if l?2^n-1-2$l?2^{n-1}-2$ or |F|?n-2$\left|F\right|?n-2$. We also extend the result on (n-3)$(n-3)$-Hamiltonian connectivity of _TQn${\mathit{TQ}}_n$ in the literature.     

Scheduling machine-dependent jobs to minimize lateness on machines with identical speed under availability constraints

In this paper we consider the problem of scheduling n$n$ preemptive jobs on m$m$ machines with identical speed under machine availability and eligibility constraints when minimizing maximum lateness (_Lmax$(L_{\max }$). The lateness of a job is defined to be its completion time minus its due date, and _Lmax$L_{\max }$ is the maximum value of lateness among all jobs. We assume that each machine is not continuously available at all time throughout the planning horizon and each job is only allowed to be processed on specific machines. Network flow technique is used to formulate this scheduling problem into a series of maximum flow problems. We propose a polynomial time two-phase binary search algorithm to verify the feasibility of the problem and to solve the scheduling problem optimally if a feasible schedule exists. Finally, we show that the time complexity of the algorithm is O((n+(2n+2x))³log(UB-LB))$O((n+(2n+2x){)}^3\log (\text{<italic>UB</italic>}-\text{<italic>LB</italic>}))$. Most literature in parallel machine scheduling assume that all machines are continuously available for processing and all jobs can be processed at any available machine throughout the planning horizon. But both assumptions might not be true in some practical environment, such as machine preventive maintenance and machines that have different capabilities to process jobs. This type of scheduling problem is seldom studied in the literature. The purpose of this paper is to examine the scheduling problem with machines with identical speed under machine availability and eligibility constraints. The objective is to minimize maximum lateness. We formulate this scheduling problem into a series of maximum flow problems with different values of _Lmax$L_{\max }$. A polynomial time two-phase binary search algorithm is proposed to verify the feasibility of the problem and to determine the optimal _Lmax$L_{\max }$.     

A new model for path planning with interval data

In this paper, we establish a new model for path planning with interval data which arises in a variety of applications. It is formulated as minimum risk-sum path problem  : given a source-destination pair in a network G=(V,E)$G=(V,E)$, traveling on each link e in G   may take time _xe$x_e$ in a prespecified interval [_le,_ue]$[l_e,u_e]$ and take risk (_ue-_xe)/(_ue-_le)$(u_e-x_e)/(u_e-l_e)$, the goal is to find a path in G from the source to the destination, together with an allocation of travel times along each link on the path, so that the total travel time of links on the path is no more than a given time bound and the risk-sum over the links on the path is minimized. Our study shows that this new model has two features that make it different from the existing models. First, the minimum risk-sum path problem is polynomial-time solvable, and second, it provides many solutions that vary with time bounds and risk sums and leaves the choice for decision makers. Therefore, the new model is more flexible and easier to use for the path planning with interval data.