Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s$s$ and t$t$ in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t$s-t$ path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s$s$ and t$t$ for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k$k$ pairwise vertex/edge disjoint properly edge-colored s−t$s-t$ paths/trails in a c$c$-edge-colored graph G^c$G^c$ is NP-complete even for k=2$k=2$ and c=Ω(n²)$c=\Omega \left(n^2\right)$, where n$n$ denotes the number of vertices in G^c$G^c$. Moreover, we prove that these problems remain NP-complete for c$c$-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n)$c=\Omega \left(n\right)$. We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.     

State-complexity hierarchies of uniform languages of alphabet-size length

We study the state complexity of certain simple languages. If A$A$ is an alphabet of k$k$ letters, then a k$k$-language   is a nonempty set of words of length k$k$, that is, a uniform language of length k$k$. We show that the minimal state complexity of a k$k$-language is k+2$k+2$, and the maximal, (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. We prove constructively that, for every i$i$ between the minimal and maximal bounds, there is a language of state complexity i$i$. We introduce a class of automata accepting sets of words that are permutations of A$A$; these languages define a complete hierarchy of complexities between k²−k+3$k^2-k+3$ and 2^k+1$2^k+1$. The languages of another class of automata, based on k$k$-ary trees, define a complete hierarchy of complexities between 2^k+1$2^k+1$ and (k^k−1−1)/(k−1)+2^k+1$(k^{k-1}-1)/(k-1)+2^k+1$. This provides new examples of uniform languages of maximal complexity.     

A parameterized view on matroid optimization problems

Matroid theory gives us powerful techniques for understanding combinatorial optimization problems and for designing polynomial-time algorithms. However, several natural matroid problems, such as 3-matroid intersection, are NP-hard. Here we investigate these problems from the parameterized complexity point of view: instead of the trivial n^O(k)$n^{O\left(k\right)}$ time brute force algorithm for finding a k$k$-element solution, we try to give algorithms with uniformly polynomial (i.e., f(k)⋅n^O(1)$f\left(k\right)\cdot n^{O\left(1\right)}$) running time. The main result is that if the ground set of a represented linear matroid is partitioned into blocks of size ?$?$, then we can determine in randomized time f(k,?)⋅n^O(1)$f(k,?)\cdot n^{O\left(1\right)}$ whether there is an independent set that is the union of k$k$ blocks. As a consequence, algorithms with similar running time are obtained for other problems such as finding a k$k$-element set in the intersection of ?$?$ matroids, or finding k$k$ terminals in a network such that each of them can be connected simultaneously to the source by ?$?$ disjoint paths.     

Minimum leaf out-branching and related problems

Given a digraph D$D$, the Minimum Leaf Out-Branching problem (MinLOB) is the problem of finding in D$D$ an out-branching with the minimum possible number of leaves, i.e., vertices of out-degree 0. We prove that MinLOB is polynomial-time solvable for acyclic digraphs. In general, MinLOB is NP-hard and we consider three parameterizations of MinLOB. We prove that two of them are NP-complete for every value of the parameter, but the third one is fixed-parameter tractable (FPT). The FPT parameterization is as follows: given a digraph D$D$ of order n$n$ and a positive integral parameter k$k$, check whether D$D$ contains an out-branching with at most n−k$n-k$ leaves (and find such an out-branching if it exists). We find a problem kernel of order O(k²)$O\left(k^2\right)$ and construct an algorithm of running time O(2^O(klogk)+n⁶)$O(2^{O(klogk)}+n^6)$, which is an ‘additive’ FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal out-tree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.     

Spanners for bounded tree-length graphs

This paper concerns construction of additive stretched spanners with few edges for n$n$-vertex graphs having a tree-decomposition into bags of diameter at most δ$\delta$, i.e., the tree-length δ$\delta$ graphs. For such graphs we construct additive 2δ$2\delta$-spanners with O(δn+nlogn)$O(\delta n+nlogn)$ edges, and additive 4δ$4\delta$-spanners with O(δn)$O\left(\delta n\right)$ edges. This provides new upper bounds for chordal graphs for which δ=1$\delta =1$. We also show a lower bound, and prove that there are graphs of tree-length δ$\delta$ for which every multiplicative δ$\delta$-spanner (and thus every additive (δ−1)$(\delta -1)$-spanner) requires Ω(n^1+1/Θ(δ))$\Omega \left(n^{1+1/\Theta \left(\delta \right)}\right)$ edges.     

Swapping a failing edge of a shortest paths tree by minimizing the average stretch factor

We consider a two-edge connected, undirected graph G=(V,E)$G=(V,E)$, with n$n$ nodes and m$m$ non-negatively real weighted edges, and a single source shortest paths tree (SPT) T$T$ of G$G$ rooted at an arbitrary node r$r$. If an edge in T$T$ is temporarily removed, it makes sense to reconnect the nodes disconnected from the root by adding a single non-tree edge, called a swap edge  , instead of rebuilding a new optimal SPT from scratch. In the past, several optimality criteria have been considered to select a best possible swap edge. In this paper we focus on the most prominent one, that is the minimization of the average distance between the root and the disconnected nodes. To this respect, we present an O(mlog²n)$O\left(m{log}^{2}n\right)$ time and O(m)$O\left(m\right)$ space algorithm to find a best swap edge for every edge of T$T$, thus improving for m=o(n²/log²n)$m=o(n^2/{log}^{2}n)$ the previously known O(n²)$O\left(n^2\right)$ time and space complexity algorithm.     

Divergence bounded computable real numbers

A real x$x$ is called h$h$-bounded computable  , for some function h:N→N$h:\mathbb{N}\to \mathbb{N}$, if there is a computable sequence (_xs)$(x_s)$ of rational numbers which converges to x$x$ such that, for any n∈N$n\in \mathbb{N}$, at most h(n)$h(n)$ non-overlapping pairs of its members are separated by a distance larger than 2^-n$2^{-n}$. In this paper we discuss properties of h$h$-bounded computable reals for various functions h$h$. We will show a simple sufficient condition for a class of functions h$h$ such that the corresponding h$h$-bounded computable reals form an algebraic field. A hierarchy theorem for h$h$-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h$h$-bounded computability for special functions h$h$.     

Bounding the radii of balls meeting every connected component of semi-algebraic sets

We prove an explicit bound on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set S⊂R^k$S\subset {\mathbb{R}}^k$ defined by a weak sign condition involving s$s$ polynomials in Z[_X1,…,_Xk]$\mathbb{Z}[X_1,\dots ,X_k]$ having degrees at most d$d$, and whose coefficients have bitsizes at most τ$\tau$. Our bound is an explicit function of s,d,k$s,d,k$ and τ$\tau$, and does not contain any undetermined constants. We also prove a similar bound on the radius of a ball guaranteed to intersect every connected component of S$S$ (including the unbounded components). While asymptotic bounds of the form 2^τdO(k)$2^{\tau d^{O\left(k\right)}}$ on these quantities were known before, some applications require bounds which are explicit and which hold for all values of s,d,k$s,d,k$ and τ$\tau$. The bounds proved in this paper are of this nature.     

An approximation algorithm to the k-Steiner Forest problem

Given a graph G$G$, an integer k$k$, and a demand set D={(_s1,_t1),…,(_sl,_tl)}$D=\{(s_1,t_1),\dots ,(s_l,t_l)\}$, the k$k$-Steiner Forest problem finds a forest in graph G$G$ to connect at least k$k$ demands in D$D$ such that the cost of the forest is minimized. This problem was proposed by Hajiaghayi and Jain in SODA’06. Thereafter, using a Lagrangian relaxation technique, Segev et al. gave the first approximation algorithm to this problem in ESA’06, with performance ratio O(n^2/3logl)$O(n^{2/3}logl)$. We give a simpler and faster approximation algorithm to this problem with performance ratio O(n^2/3logk)$O(n^{2/3}logk)$ via greedy approach, improving the previously best known ratio in the literature.     

An efficient construction of one-to-many node-disjoint paths in folded hypercubes

A folded hypercube is basically a hypercube with additional links augmented, where the additional links connect all pairs of nodes with longest distance in the hypercube. In an n$n$-dimensional folded hypercube, it has been shown that n+1$n+1$ node-disjoint paths from one source node to other n+1$n+1$ (mutually) distinct destination nodes, respectively, can be constructed in O(n⁴)$O\left(n^4\right)$ time so that their maximal length is not greater than ⌈n/2⌉+1$\lceil n/2\rceil +1$, where n+1$n+1$ is the connectivity and ⌈n/2⌉$\lceil n/2\rceil$ is the diameter. Besides, their maximal length is minimized in the worst case. In this paper, we further show that by minimizing the computations of minimal routing functions, these node-disjoint paths can be constructed in O(n³)$O\left(n^3\right)$ time, which is more efficient, and is hard to be reduced because it must take O(n³)$O\left(n^3\right)$ time to compute a minimal routing function by solving a corresponding maximum weighted bipartite matching problem with the best known algorithm.     

Conjunctive query evaluation by search-tree revisited

The most natural and perhaps most frequently used method for testing membership of an individual tuple in a conjunctive query is based on searching trees of partial solutions, or search-trees. We investigate the question of evaluating conjunctive queries with a time-bound guarantee that is measured as a function of the size of the optimal search-tree. We provide an algorithm that, given a database D$D$, a conjunctive query Q$Q$, and a tuple a$a$, tests whether Q(a)$Q\left(a\right)$ holds in D$D$ in time bounded by a polynomial in (sn)^logk(sn)^loglogn${\left(sn\right)}^{logk}{\left(sn\right)}^{loglogn}$ and n^r$n^r$, where n$n$ is the size of the domain of the database, k$k$ is the number of bound variables of the conjunctive query, s$s$ is the size of the optimal search-tree, and r$r$ is the maximum arity of the relations. In many cases of interest, this bound is significantly smaller than the n^O(k)$n^{O\left(k\right)}$ bound provided by the naive search-tree method. Moreover, our algorithm has the advantage of guaranteeing the bound for any given conjunctive query. In particular, it guarantees the bound for queries that admit an equivalent form that is much easier to evaluate, even when finding such a form is an NP-hard task. Concrete examples include the conjunctive queries that can be non-trivially folded into a conjunctive query of bounded size or bounded treewidth. All our results translate to the context of constraint-satisfaction problems via the well-publicized correspondence between both frameworks.     

Approximately optimal trees for group key management with batch updates

We investigate the group key management problem for broadcasting applications. Previous work showed that, in handling key updates, batch rekeying can be more cost effective than individual rekeying. One model for batch rekeying is to assume that every user has probability p$p$ of being replaced by a new user during a batch period with the total number of users unchanged. Under this model, it was recently shown that an optimal key tree can be constructed in linear time when p$p$ is a constant and in O(n⁴)$O\left(n^4\right)$ time when p→0$p\to 0$. In this paper, we investigate more efficient algorithms for the case p→0$p\to 0$, i.e., when membership changes are sparse. We design an O(n)$O\left(n\right)$ heuristic algorithm for the sparse case and show that it produces a nearly 2-approximation to the optimal key tree. Simulation results show that its performance is even better in practice. We also design a refined heuristic algorithm and show that it achieves an approximation ratio of 1+?$1+?$ for any fixed ?>0$?>0$ and n$n$, as p→0$p\to 0$. Finally, we give another approximation algorithm for any p∈(0,0.693)$p\in (0,0.693)$ which is shown to be quite good by our simulations.     

Simulating one-reversal multicounter machines by partially blind multihead finite automata

This work is concerned with simulating nondeterministic one-reversal multicounter automata (NCMs) by nondeterministic partially blind multihead finite automata (NFAs). We show that any one-reversal NCM with k$k$ counters can be simulated by a partially blind NFA with k$k$ blind heads. This provides a nearly complete categorization of the computational power of partially blind automata, showing that the power of a (k+1)$(k+1)$-NFA lies between that of a k$k$-NCM and a (k+1)$(k+1)$-NCM.     

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.     

Algorithms for near solutions to polynomial equations

Let F(x,y)$F(x,y)$ be a polynomial over a field K$K$ and m$m$ a nonnegative integer. We call a polynomial g$g$ over K$K$ an m$m$-near solution of F(x,y)$F(x,y)$ if there exists a c∈K$c\in K$ such that F(x,g)=cx^m$F(x,g)=c{x}^m$, and the number c$c$ is called an m$m$-value of F(x,y)$F(x,y)$ corresponding to g$g$. In particular, c$c$ can be 0. Hence, by viewing F(x,y)=0$F(x,y)=0$ as a polynomial equation over K[x]$K\left[x\right]$ with variable y$y$, every solution of the equation F(x,y)=0$F(x,y)=0$ in K[x]$K\left[x\right]$ is also an m$m$-near solution. We provide an algorithm that gives all m$m$-near solutions of a given polynomial F(x,y)$F(x,y)$ over K$K$, and this algorithm is polynomial time reducible to solving one variable equations over K$K$. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions.