A quadratic lower bound for Rocchio’s similarity-based relevance feedback algorithm with a fixed query updating factor

Rocchio’s similarity-based relevance feedback algorithm, one of the most important query reformation methods in information retrieval, is essentially an adaptive supervised learning algorithm from examples. In practice, Rocchio’s algorithm often uses a fixed query updating factor. When this is the case, we strengthen the linear Ω(n) lower bound obtained by Chen and Zhu (Inf. Retr. 5:61–86, 2002) and prove that Rocchio’s algorithm makes Ω(k(n−k)) mistakes in searching for a collection of documents represented by a monotone disjunction of k relevant features over the n-dimensional binary vector space {0,1} ⁿ , when the inner product similarity measure is used. A quadratic lower bound is obtained when k is linearly proportional to n. We also prove an O(k(n−k)³) upper bound for Rocchio’s algorithm with the inner product similarity measure in searching for such a collection of documents with a constant query updating factor and a zero classification threshold.     

Improving an upper bound on the size of k-regular induced subgraphs

This paper considers the NP-hard graph problem of determining a maximum cardinality subset of vertices inducing a k-regular subgraph. For any graph G, this maximum will be denoted by α _k (G). From a well known Motzkin-Straus result, a relationship is deduced between α _k (G) and the independence number α(G). Next, it is proved that the upper bounds υ _k (G) introduced in Cardoso et al. (J. Comb. Optim., 14, 455–463, 2007) can easily be computed from υ ₀(G), for any positive integer k. This relationship also allows one to present an alternative proof of the Hoffman bound extension introduced in the above paper. The paper continues with the introduction of a new upper bound on α _k (G) improving υ _k (G). Due to the difficulty of computing this improved bound, two methods are provided for approximating it. Finally, some computational experiments which were performed to compare all bounds studied are reported.     

On lazy bureaucrat scheduling with common deadlines

Lazy bureaucrat scheduling is a new class of scheduling problems introduced by Arkin et al. (Inf. Comput. 184:129–146, 2003). In this paper we focus on the case where all the jobs share a common deadline. Such a problem is denoted as CD-LBSP, which has been considered by Esfahbod et al. (Algorithms and data structures. Lecture notes in computer science, vol. 2748, pp. 59–66, 2003). We first show that the worst-case ratio of the algorithm SJF (Shortest Job First) is two under the objective function [min-time-spent], and thus answer an open question posed in (Esfahbod, et al. in Algorithms and data structures. Lecture notes in computer science, vol. 2748, pp. 59–66, 2003). We further present two approximation schemes A _k and B _k both having worst-case ratio of , for any given integer k>0, under the objective functions [min-makespan] and [min-time-spent], respectively. Finally, we prove that the problem CD-LBSP remains NP-hard under several objective functions, even if all jobs share the same release time. A preliminary version of the paper appeared in Proceedings of the 7th Latin American Symposium on Theoretical Informatics, pp 515–523, 2006. Research of G. Zhang supported in part by NSFC (60573020).     

A new recombination lower bound and the minimum perfect phylogenetic forest problem

Understanding recombination is a central problem in population genetics. In this paper, we address an established computational problem in this area: compute lower bounds on the minimum number of historical recombinations for generating a set of sequences (Hudson and Kaplan in Genetics 111, 147–164, 1985; Myers and Griffiths in Genetics 163, 375–394, 2003; Gusfield et al. in Discrete Appl. Math. 155, 806–830, 2007; Bafna and Bansal in IEEE/ACM Trans. Comput. Biol. Bioinf. 1, 78–90, 2004 and in J. Comput. Biol. 13, 501–521, 2006; Song et al. in Bioinformatics 421, i413–i244, 2005). In particular, we propose a new recombination lower bound: the forest bound. We show that the forest bound can be formulated as the minimum perfect phylogenetic forest problem, a natural extension to the classic binary perfect phylogeny problem, which may be of interests on its own. We then show that the forest bound is provably higher than the optimal haplotype bound (Myers and Griffiths in Genetics 163, 375–394, 2003), a very good lower bound in practice (Song et al. in Bioinformatics 421, i413–i422, 2005). We prove that, like several other lower bounds (Bafna and Bansal in J. Comput. Biol. 13, 501–521, 2006), computing the forest bound is NP-hard. Finally, we describe an integer linear programming (ILP) formulation that computes the forest bound precisely for certain range of data. Simulation results show that the forest bound may be useful in computing lower bounds for low quality data. A preliminary version of this paper appeared in the Proceedings of COCOON 2007, LNCS, vol. 4598, pp. 16–26. The work was performed while Y. Wu was with UC Davis and supported by grants CCF-0515278 and IIS-0513910 from National Science Foundation. D. Gusfield supported by grants CCF-0515278 and IIS-0513910 from National Science Foundation.     

Strongly chordal and chordal bipartite graphs are sandwich monotone

A graph class is sandwich monotone if, for every pair of its graphs G ₁=(V,E ₁) and G ₂=(V,E ₂) with E ₁⊂E ₂, there is an ordering e ₁,…,e _k of the edges in E ₂∖E ₁ such that G=(V,E ₁∪{e ₁,…,e _i }) belongs to the class for every i between 1 and k. In this paper we show that strongly chordal graphs and chordal bipartite graphs are sandwich monotone, answering an open question by Bakonyi and Bono (Czechoslov. Math. J. 46:577–583, 1997). So far, very few classes have been proved to be sandwich monotone, and the most famous of these are chordal graphs. Sandwich monotonicity of a graph class implies that minimal completions of arbitrary graphs into that class can be recognized and computed in polynomial time. For minimal completions into strongly chordal or chordal bipartite graphs no polynomial-time algorithm has been known. With our results such algorithms follow for both classes. In addition, from our results it follows that all strongly chordal graphs and all chordal bipartite graphs with edge constraints can be listed efficiently.     

Point sets in the unit square and large areas of convex hulls of subsets of points

In this paper generalizations of Heilbronn’s triangle problem to convex hulls of j points in the unit square [0,1]² are considered. By using results on the independence number of linear hypergraphs, for fixed integers k≥3 and any integers n≥k a deterministic o(n ^6k−4) time algorithm is given, which finds distributions of n points in [0,1]² such that, simultaneously for j=3,…,k, the areas of the convex hulls determined by any j of these n points are Ω((log n)^1/(j−2)/n ^{(j−1)/(j−2)}).     

The total {k}-domatic number of a graph

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that ?_i=1^df_i(v) ￡ k\sum_{i=1}^{d}f_{i}(v)\le k for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by d_t^{k}(G)d_{t}^{\{k\}}(G). Note that d_t^{1}(G)d_{t}^{\{1\}}(G) is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for d_t^{k}(G)d_{t}^{\{k\}}(G). Many of the known bounds of d _t (G) are immediate consequences of our results.     

L(j,k)- and circular L(j,k)-labellings for the products of complete graphs

Let j and k be two positive integers with j≥k. An L(j,k)-labelling of a graph G is an assignment of nonnegative integers to the vertices of G such that the difference between labels of any two adjacent vertices is at least j, and the difference between labels of any two vertices that are at distance two apart is at least k. The minimum range of labels over all L(j,k)-labellings of a graph G is called the λ _j,k -number of G, denoted by λ _j,k (G). A σ(j,k)-circular labelling with span m of a graph G is a function f:V(G)→{0,1,…,m−1} such that |f(u)−f(v)| _m ≥j if u and v are adjacent; and |f(u)−f(v)| _m ≥k if u and v are at distance two apart, where |x| _m =min {|x|,m−|x|}. The minimum m such that there exists a σ(j,k)-circular labelling with span m for G is called the σ _j,k -number of G and denoted by σ _j,k (G). The λ _j,k -numbers of Cartesian products of two complete graphs were determined by Georges, Mauro and Stein ((2000) SIAM J Discret Math 14:28–35). This paper determines the λ _j,k -numbers of direct products of two complete graphs and the σ _j,k -numbers of direct products and Cartesian products of two complete graphs. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. This work is partially supported by FRG, Hong Kong Baptist University, Hong Kong; NSFC, China, grant 10171013; and Southeast University Science Foundation grant XJ0607230.     

Linear time construction of 5-phylogenetic roots for tree chordal graphs

Inspired by phylogenetic tree construction in computational biology, Lin et al. (The 11th Annual International Symposium on Algorithms and Computation (ISAAC 2000), pp. 539–551, 2000) introduced the notion of a k -phylogenetic root. A k-phylogenetic root of a graph G is a tree T such that the leaves of T are the vertices of G, two vertices are adjacent in G precisely if they are within distance k in T, and all non-leaf vertices of T have degree at least three. The k-phylogenetic root problem is to decide whether such a tree T exists for a given graph G. In addition to introducing this problem, Lin et al. designed linear time constructive algorithms for k≤4, while left the problem open for k≥5. In this paper, we partially fill this hole by giving a linear time constructive algorithm to decide whether a given tree chordal graph has a 5-phylogenetic root; this is the largest class of graphs known to have such a construction.     

Approximation algorithms for the graph orientation minimizing the maximum weighted outdegree

Given a simple, undirected graph G=(V,E) and a weight function w:E→ℤ⁺, we consider the problem of orienting all edges in E so that the maximum weighted outdegree among all vertices is minimized. It has previously been shown that the unweighted version of the problem is solvable in polynomial time while the weighted version is (weakly) NP-hard. In this paper, we strengthen these results as follows: (1) We prove that the weighted version is strongly NP-hard even if all edge weights belong to the set {1,k}, where k is any fixed integer greater than or equal to 2, and that there exists no pseudo-polynomial time approximation algorithm for this problem whose approximation ratio is smaller than (1+1/k) unless P = NP; (2) we present a new polynomial-time algorithm that approximates the general version of the problem within a ratio of (2−1/k), where k is the maximum weight of an edge in G; (3) we show how to approximate the special case in which all edge weights belong to {1,k} within a ratio of 3/2 for k=2 (note that this matches the inapproximability bound above), and (2−2/(k+1)) for any k≥3, respectively, in polynomial time.     

On backbone coloring of graphs

Let G be a graph and H a subgraph of G. A backbone-k-coloring of (G,H) is a mapping f: V(G)→{1,2,…,k} such that |f(u)−f(v)|≥2 if uv∈E(H) and |f(u)−f(v)|≥1 if uv∈E(G)\E(H). The backbone chromatic number of (G,H) is the smallest integer k such that (G,H) has a backbone-k-coloring. In this paper, we characterize the backbone chromatic number of Halin graphs G=T∪C with respect to given spanning trees T. Also we study the backbone coloring for other special graphs such as complete graphs, wheels, graphs with small maximum average degree, graphs with maximum degree 3, etc.     

Minimum entropy coloring

We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem. S. Fiorini acknowledges the support from the Fonds National de la Recherche Scientifique and GERAD-HEC Montréal. G. Joret is a F.R.S.-FNRS Research Fellow.     

A note on fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges

In the paper “Fault-free Mutually Independent Hamiltonian Cycles in Hypercubes with Faulty Edges” (J. Comb. Optim. 13:153–162, 2007), the authors claimed that an n-dimensional hypercube can be embedded with (n−1−f)-mutually independent Hamiltonian cycles when f≤n−2 faulty edges may occur accidentally. However, there are two mistakes in their proof. In this paper, we give examples to explain why the proof is deficient. Then we present a correct proof. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 95-2221-E-233-002.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Minimum ε-equivalent Circuit Size Problem

For a Boolean function f given by its truth table (of length ) and a parameter s the problem considered is whether there is a Boolean function g -equivalent to f, i.e., , and computed by a circuit of size at most s. In this paper we investigate the complexity of this problem and show that for specific values of it is unlikely to be in P/poly. Under the same assumptions we also consider the optimization variant of the problem and prove its inapproximability.     

Subhypergraph counts in extremal and random hypergraphs and the fractional <Emphasis Type="Italic">q</Emphasis>-independence

We study the extremal parameter N(n,m,H) which is the largest number of copies of a hypergraph H that can be formed of at most n vertices and m edges. Generalizing previous work of Alon (Isr. J. Math. 38:116–130, 1981), Friedgut and Kahn (Isr. J. Math. 105:251–256, 1998) and Janson, Oleszkiewicz and the third author (Isr. J. Math. 142:61–92, 2004), we obtain an asymptotic formula for N(n,m,H) which is strongly related to the solution α _q (H) of a linear programming problem, called here the fractional q-independence number of H. We observe that α _q (H) is a piecewise linear function of q and determine it explicitly for some ranges of q and some classes of H. As an application, we derive exponential bounds on the upper tail of the distribution of the number of copies of H in a random hypergraph.     

Tree edge decomposition with an application to minimum ultrametric tree approximation

A k-decomposition of a tree is a process in which the tree is recursively partitioned into k edge-disjoint subtrees until each subtree contains only one edge. We investigated the problem how many levels it is sufficient to decompose the edges of a tree. In this paper, we show that any n-edge tree can be 2-decomposed (and 3-decomposed) within at most ⌈1.44 log n⌉ (and ⌈log n⌉ respectively) levels. Extreme trees are given to show that the bounds are asymptotically tight. Based on the result, we designed an improved approximation algorithm for the minimum ultrametric tree.     

Finding longest increasing and common subsequences in streaming data

We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet αbet has length at least k, we discuss a one-pass algorithm using O(k log αbetsize) space, with update time either O(log k) or O(log log αbetsize); for αbetsize = O(1), we can achieve O(log k) space and constant-time updates. We also prove a lower bound of Ω(k) on the space requirement for this problem for general alphabets αbet, even when the input stream is a permutation of αbet. For finding the actual LIS, we give a ⌈log (1 + 1/ɛ)-pass algorithm using O(k^1+ɛlog αbetsize) space, for any ɛ > 0. For LCS, there is a trivial Θ(1)-approximate O(log n)-space streaming algorithm when αbetsize = O(1). For general alphabets αbet, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use Ω(n/ρ²) space to approximate the LCS of two n-element streams to within a factor of ρ, even if the streams are permutations of each other. A preliminary version of this paper appears in the Proceedings of the 11th International Computing and Combinatorics Conference (COCOON'05), August 2005, pp. 263–272.     

On broadcasting in unicyclic graphs

Broadcasting is an information dissemination problem in a connected network, in which one node, called the originator, disseminates a message to all other nodes by placing a series of calls along the communication lines of the network. Finding the broadcast time of a vertex in an arbitrary graph is NP-complete. The polynomial time solvability is shown only for trees. In this paper we present a linear algorithm that determines the broadcast time of any originator in an arbitrary unicyclic graph. As a byproduct, we find a broadcast center of the unicyclic graph. We also present an O(|V|+k ²) algorithm to find the broadcast time of an arbitrary unicyclic graph, where k is the length of the cycle. In the last section we give tight lower and upper bounds on broadcast time of a spanning tree based on the broadcast time of the unicyclic graph. The results of Sects. 2, 3 and most of the proofs in Sects. 2, 3 of this paper are presented by Harutyunyan and Maraachlian (Proceedings of 13th annual COCOON, pp. 372–383, 2007). All results in Sects. 4, 5 and the complete proof of Theorem 3 are new results.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.