Approximability of Connected Factors

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard - finding a minimal connected 2-factor is just the traveling salesman problem (TSP). Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected $d$-factor. We give a 3-approximation for all $d$ and improve this to an (r+1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP. Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.Comment: To appear in the proceedings of WAOA 201

Hitting Diamonds and Growing Cacti

We consider the following NP-hard problem: in a weighted graph, find a minimum cost set of vertices whose removal leaves a graph in which no two cycles share an edge. We obtain a constant-factor approximation algorithm, based on the primal-dual method. Moreover, we show that the integrality gap of the natural LP relaxation of the problem is \Theta(\log n), where n denotes the number of vertices in the graph.Comment: v2: several minor changes

Parametrized Complexity of Weak Odd Domination Problems

Given a graph $G=(V,E)$, a subset $B\subseteq V$ of vertices is a weak odd dominated (WOD) set if there exists $D \subseteq V {\setminus} B$ such that every vertex in $B$ has an odd number of neighbours in $D$. $\kappa(G)$ denotes the size of the largest WOD set, and $\kappa'(G)$ the size of the smallest non-WOD set. The maximum of $\kappa(G)$ and $|V|-\kappa'(G)$, denoted $\kappa_Q(G)$, plays a crucial role in quantum cryptography. In particular deciding, given a graph $G$ and $k>0$, whether $\kappa_Q(G)\le k$ is of practical interest in the design of graph-based quantum secret sharing schemes. The decision problems associated with the quantities $\kappa$, $\kappa'$ and $\kappa_Q$ are known to be NP-Complete. In this paper, we consider the approximation of these quantities and the parameterized complexity of the corresponding problems. We mainly prove the fixed-parameter intractability (W$[1]$-hardness) of these problems. Regarding the approximation, we show that $\kappa_Q$, $\kappa$ and $\kappa'$ admit a constant factor approximation algorithm, and that $\kappa$ and $\kappa'$ have no polynomial approximation scheme unless P=NP.Comment: 16 pages, 5 figure

Online Admission Control and Embedding of Service Chains

The virtualization and softwarization of modern computer networks enables the definition and fast deployment of novel network services called service chains: sequences of virtualized network functions (e.g., firewalls, caches, traffic optimizers) through which traffic is routed between source and destination. This paper attends to the problem of admitting and embedding a maximum number of service chains, i.e., a maximum number of source-destination pairs which are routed via a sequence of to-be-allocated, capacitated network functions. We consider an Online variant of this maximum Service Chain Embedding Problem, short OSCEP, where requests arrive over time, in a worst-case manner. Our main contribution is a deterministic O(log L)-competitive online algorithm, under the assumption that capacities are at least logarithmic in L. We show that this is asymptotically optimal within the class of deterministic and randomized online algorithms. We also explore lower bounds for offline approximation algorithms, and prove that the offline problem is APX-hard for unit capacities and small L > 2, and even Poly-APX-hard in general, when there is no bound on L. These approximation lower bounds may be of independent interest, as they also extend to other problems such as Virtual Circuit Routing. Finally, we present an exact algorithm based on 0-1 programming, implying that the general offline SCEP is in NP and by the above hardness results it is NP-complete for constant L.Comment: early version of SIROCCO 2015 pape

Subventricular zone stem cells are heterogeneous with respect to their embryonic origins and neurogenic fates in the adult olfactory bulb

Wedetermined the embryonic origins of adult forebrain subventricular zone (SVZ) stem cells by Cre-lox fate mapping in transgenic mice. We found that all parts of the telencephalic neuroepithelium, including the medial ganglionic eminence and lateral ganglionic eminence (LGE) and the cerebral cortex, contribute multipotent, self-renewing stem cells to the adult SVZ. Descendants of the embryonic LGE and cortex settle in ventral and dorsal aspects of the dorsolateral SVZ, respectively. Both populations contribute new (5-bromo-2(')-deoxyuridine- labeled) tyrosine hydroxylase- and calretinin-positive interneurons to the adult olfactory bulb. However, calbindin-positive interneurons in the olfactory glomeruli were generated exclusively by LGE- derived stem cells. Thus, different SVZ stem cells have different embryonic origins, colonize different parts of the SVZ, and generate different neuronal progeny, suggesting that some aspects of embryonic patterning are preserved in the adult SVZ. This could have important implications for the design of endogenous stem cell-based therapies in the future

Worst case and probabilistic analysis of the 2-Opt algorithm for the TSP

2-Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2-Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on 2-dimensional Euclidean instances was known so far. We clarify this issue by presenting, for every p∈N , a family of L p instances on which 2-Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0,1]2, where it was shown that the expected number of steps is bounded by O~(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed independently according to general distributions on [0,1] d , for an arbitrary d≥2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the maximal density ϕ of the probability distributions. We show an upper bound on the expected length of any 2-Opt improvement path of O~(n4+1/3⋅ϕ8/3) . When starting with an initial tour computed by an insertion heuristic, the upper bound on the expected number of steps improves even to O~(n4+1/3−1/d⋅ϕ8/3) . If the distances are measured according to the Manhattan metric, then the expected number of steps is bounded by O~(n4−1/d⋅ϕ) . In addition, we prove an upper bound of O(ϕ√d) on the expected approximation factor with respect to all L p metrics. Let us remark that our probabilistic analysis covers as special cases the uniform input model with ϕ=1 and a smoothed analysis with Gaussian perturbations of standard deviation σ with ϕ∼1/σ d

The Complexity of Finding Reset Words in Finite Automata

We study several problems related to finding reset words in deterministic finite automata. In particular, we establish that the problem of deciding whether a shortest reset word has length k is complete for the complexity class DP. This result answers a question posed by Volkov. For the search problems of finding a shortest reset word and the length of a shortest reset word, we establish membership in the complexity classes FP^NP and FP^NP[log], respectively. Moreover, we show that both these problems are hard for FP^NP[log]. Finally, we observe that computing a reset word of a given length is FNP-complete.Comment: 16 pages, revised versio

Interval Scheduling: A Survey

In interval scheduling, not only the processing times of the jobs but also their starting times are given. This article surveys the area of interval scheduling and presents proofs of results that have been known within the community for some time. We first review the complexity and approximability of different variants of interval scheduling problems. Next, we motivate the relevance of interval scheduling problems by providing an overview of applications that have appeared in literature. Finally, we focus on algorithmic results for two important variants of interval scheduling problems. In one variant we deal with nonidentical machines: instead of each machine being continuously available, there is a given interval for each machine in which it is available. In another variant, the machines are continuously available but they are ordered, and each job has a given "maximal" machine on which it can be processed. We investigate the complexity of these problems and describe algorithms for their solution

Freezing in random graph ferromagnets

Using T=0 Monte Carlo and simulated annealing simulation, we study the energy relaxation of ferromagnetic Ising and Potts models on random graphs. In addition to the expected exponential decay to a zero energy ground state, a range of connectivities for which there is power law relaxation and freezing to a metastable state is found. For some connectivities this freezing persists even using simulated annealing to find the ground state. The freezing is caused by dynamic frustration in the graphs, and is a feature of the local search-nature of the Monte Carlo dynamics used. The implications of the freezing on agent-based complex systems models are briefly considered.Comment: Published version: 1 reference deleted, 1 word added. 4 pages, 5 figure

An Epistemic Perspective on Consistency of Concurrent Computations

Consistency properties of concurrent computations, e.g., sequential consistency, linearizability, or eventual consistency, are essential for devising correct concurrent algorithms. In this paper, we present a logical formalization of such consistency properties that is based on a standard logic of knowledge. Our formalization provides a declarative perspective on what is imposed by consistency requirements and provides some interesting unifying insight on differently looking properties