Information Processing Letters Research Articles

A Sufficient Condition for Fractional ( 2 , b , k ) -Critical Covered Graphs

Zhou, Xu and Sun [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional ( a , b , k ) -critical covered graphs, Information Processing Letters 152(2019)105838] defined the concept of a fractional ( a , b , k ) -critical covered graph, namely, a graph G is a fractional ( a , b , k ) -critical covered graph if after removing any k vertices of G , the remaining graph of G is a fractional [ a , b ] -covered graph. In this paper, we prove that a graph G with δ ( G ) ≥ 2 + k is fractional ( 2 , b , k ) -critical covered if b i n d ( G ) > b + k b − 1 , where k ≥ 0 and b ≥ 2 + k are two integers.

On Protocols for Monotone Feasible Interpolation

Dag-like communication protocols, a generalization of the classical tree-like communication protocols, are useful objects in the realm of proof complexity (most importantly for monotone feasible interpolation) and circuit complexity. We consider three kinds of protocols in this article ( d is the degree of a protocol): — IEQ- d -dags: feasible sets of these protocols are described by inequality which means that the feasible sets are combinatorial triangles; these protocols are also called triangle-dags in the literature, — EQ- d -dags: feasible sets are described by equality, and — c -IEQ- d -dags: feasible sets are described by a conjunction of c inequalities. Garg, Göös, Kamath, and Sokolov (Theory of Computing, 2020) mentioned all these protocols, and they noted that EQ- d -dags are a special case of c -IEQ- d -dags. The exact relationship between these types of protocols is unclear. As our main contribution, we prove the following statement: EQ-2-dags can efficiently simulate c -IEQ- d -dags when c and d are constants. This implies that EQ-2-dags are at least as strong as IEQ- d -dags and that EQ-2-dags have the same strength as c -IEQ- d -dags for c ≥ 2 (because 2-IEQ-2-dags can trivially simulate EQ-2-dags). Hrubeš and Pudlák (Information Processing Letters, 2018) proved that IEQ- d -dags over the monotone Karchmer-Wigderson relation are equivalent to monotone real circuits which implies that we have exponential lower bounds for these protocols. Lower bounds for EQ-2-dags would directly imply lower bounds for the proof system R(LIN).

A degree condition for graphs being fractional (a,b,k)-critical covered

A graph G is fractional [a, b]-covered if for any e ? E(G), G possesses a fractional [a, b]-factor including e. A graph G is fractional (a, b, k)-critical covered if G ? Q is fractional [a, b]-covered for any Q ? V(G) with |Q| = k. In this paper, we verify that a graph G of order n is fractional (a, b, k)-critical covered if n ? (a+b)((2r?3)a+b+r?2)+bk+2 b , ?(G) ? (r ? 1)(a + 1) + k and max{dG(w1), dG(w2),..., dG(wr)} ? an + bk + 2 a + b for every independent vertex subset {w1,w2,... ,wr} ofG. Our main result is an improvement of the previous result [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (a, b, k)-critical covered graphs, Information Processing Letters 152(2019)105838].

Online maximum matching with recourse

We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter $k$. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most $k$ such actions per edge take place, where $k$ is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. [Information Processing Letters, 2013], whereas the special case $k=2$ was studied by Boyar et al. [WADS 2017]. In the first part of this paper we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP of Avitabile et al., by exploiting the structure of the matching problem. In addition, we show that the greedy algorithm has competitive ratio $3/2$ for every even $k$ and ratio $2$ for every odd $k$. Moreover, we present and analyze an improvement of the greedy algorithm which we call $L$-Greedy, and we show that for small values of $k$ it outperforms the algorithm AMP. In terms of lower bounds, we show that no deterministic algorithm better than $1+1/(k-1)$ exists, improving upon the known lower bound of $1+1/k$. The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of $L$-Greedy and AMP carry through in this model; moreover we show a lower bound of $(k^2-3k+6) / (k^2-4k+7)$ for all even $k \ge 4$. For $k\in\{2,3\}$, the competitive ratio is $3/2$.

Security analysis of a dynamic threshold secret sharing scheme using linear subspace method

A dealer-free and non-interactive dynamic threshold secret sharing scheme has been proposed by Harn and Hsu in Information Processing Letters in 2015. In this scheme, a (t,n) secret sharing scheme in secret sharing phase can turn into a (m,n) scheme in secret reconstruction phase, where m is the number of participating shareholders. It has been claimed that the secrecy of shares and the secrecy of the secret are unconditionally preserved if m∈(t,1+t(t+1)/2].This paper provides a security analysis of this scheme by introducing two attacks. In the first attack, we show that this scheme does not have the dynamic threshold property. In more details, any t+1 released values are sufficient to reconstruct the secret, while the updated threshold has any larger value. In the second attack, we show that any t+1 released values are sufficient to forge the released value of a non-participating shareholder. The technique enjoyed for these attacks is the linear subspace cryptanalysis. It basically measures the information leaked by the known parameters of the scheme by computing the dimension of the linear subspace spanned by these parameters. This method has shown to be capable of cryptanalysis of some secret sharing based schemes, whose security relies on keeping the coefficients of the underlying polynomial(s) secret.

Distributed Computing Column 72

As with prior December issues, this issue is devoted to a review of notable events related to distributed computing that occurred during the year. First, congratulations to Bowen Alpern and Fred Schneider, winners of the 2018 Edsger W. Dijkstra Prize in Distributed Computing for their paper \De ning Liveness"! Their paper appeared in Information Processing Letters in October 1985. The prize is jointly sponsored by ACM and EATCS, and is given alternately at PODC1 and DISC2; this year it was given at PODC. This paper formally de ned liveness properties of concurrent and distributed algorithms for the rst time and also proved that every trace property is the conjunction of a safety property and a liveness property. The full citation can be found at http://www.podc.org/dijkstra/2018-dijkstra-prize/. I am delighted to include in this column the text of the remarks that Fred and Bowen gave at PODC when the award was presented to them.

Design and Development of Animal Recognition Application Using Gamification and Sattolo Shuffle Algorithm on Android Platform

Information and communication technology has been developed rapidly and affectedvarious aspects such as propagation of information and marketing strategy of tourist attraction. KebunBinatang Ragunan is one of tourist attraction in Indonesia. Aside from recreation area, Kebun BinatangRagunan can be a place to learn animals. However, learning animals itself tends to be less attractive and less interactive. Therefore, an application was developed as one of animal learning media to increase touristmotivation. The application developed in form of quiz game by using gamification like achievement to increase their motivation in animals learning and using Sattolo Shuffle algorithm in order to make quiz more varied. After testing, the application is known affect the Behavioral Intention to Use level around 76.96% and Immersion level around 82.43% in giving motivation and attracting tourist attention to use the application. Sattolo Shuffle algorithm successfully applied in application to produce a unique sequence of each randomized quiz.
 Keywords—Achievement, Animal Recognition, Gamification, Quiz Game, Sattolo Shuffle.
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On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.

GENERALIZED CONNECTIVITY OF (n, k)-STAR GRAPHS

This paper considers Rh-connectivity of the (n, k)-star graph Sn,k, denoted by κh(Sn,k). In this note, We determine κh(Sn,k) = n+h(k−2)−1 for 2 ≤ k ≤ n−1 and 0 ≤ h ≤ n−k. The results generalize the main result in proved by W. H. Yang et al. [Information Processing Letters, 110(2010), 1007–1011] for the R1-connectivity (κ1(Sn,k)) and R2-connectivity (κ2(Sn,k)).

Fault-tolerant cycle embedding in the faulty hypercubes

The hypercube is one of the best known interconnection networks. Embedding cycles of all possible lengths in faulty hypercubes has received much attention. Let Fv (respectively, Fe) denote the set of faulty vertices (respectively, faulty edges) and fv (respectively, fe) denote the number of faulty vertices (respectively, faulty edge) in an n-dimensional hypercube Qn. Let f(e) denote the number of faulty nodes and/or faulty edges incident with the end-vertices of an edge e∈E(Qn). In this paper, we assume that each node is incident with at least three fault-free neighbors and at least three fault-free edges. Under this assumption, we show that every fault-free edge lies on a fault-free cycle of every even length from 4 to 2n−2∣Fv∣ if ∣Fv∣+∣Fe∣⩽2n−7 and f(e)⩽n−2, where n⩾5. Under our condition, our result not only improves the previously best known result of Hsieh et al. [S.-Y. Hsieh, T.-H. Shen, Edge-bipancyclicity of a hypercube with faulty vertices and edges, Discrete Applied Mathematics 156 (10) (2008) 1802–1808] where fv+fe⩽n−2 was assumed, but also extends the result of Tsai [C.-H. Tsai, Linear array and ring embedding in conditional faulty hypercubes, Theoretical Computer Science 314 (3) (2004) 431–443] where only the faulty edges were considered and Tsai [C.-H. Tsai, Cycle embedding in hypercubes with node failures, Information Processing Letters 102 (6) (2007) 242–246] where only the faulty vertices were considered.

Inefficiency of Nash equilibria with parallel processing policy

In this paper, we revisit the coordination mechanism of parallel processing policy introduced in [L. Yu, K. She, H. Gong, C. Yu, Price of anarchy in parallel processing, Information Processing Letters 110 (8–9) (2010) 288–293]. For both the problem of minimizing makespan and the machine covering problem, we give the analysis of price of anarchy with this new mechanism. In the first problem, we point out an error in the original paper and provide a correct instance. Moreover, we show the exact PoA for identical and uniform machines. In the second problem, we obtain new results for several scheduling models.

Super Cyclically Edge Connected Half Vertex Transitive Graphs

Tian and Meng in [Y. Tian and J. Meng, λc -Optimally half vertex transitive graphs with regularity k, Information Processing Letters 109 (2009) 683 - 686] shown that a connected half vertex transitive graph with regularity k and girth g(G) ≥ 6 is cyclically optimal. In this paper, we show that a connected half vertex transitive graph G is super cyclically edge-connected if minimum degree δ(G) ≥ 6 and girth g(G) ≥ 6.

Vulnerability parameters of tensor product of complete equipartite graphs

Let \(G_{1}\) and \(G_{2}\) be two simple graphs. The tensor product of \(G_{1}\) and \(G_{2}\), denoted by \(G_{1}\times G_{2}\), has vertex set \(V(G_{1}\times G_{2})=V(G_{1})\times V(G_{2})\) and edge set \(E(G_{1}\times G_{2})=\{(u_{1},v_{1})(u_{2},v_{2}):u_{1}u_{2}\in E(G_{1})\) and \(v_{1}v_{2}\in E(G_{2})\}\). In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs \(K_{r(s)}\times K_{m(n)}\) for \(r\geq 3, m\geq 3, s\geq 1\) and \(n\geq 1,\) where \(K_{r(s)}\) denotes the complete \(r\)-partite graph in which each part has \(s\) vertices. Using the results obtained here the theorems proved in [Aygul Mamut, Elkin Vumar, Vertex Vulnerability Parameters of Kronecker Products of Complete Graphs, Information Processing Letters 106 (2008), 258-262] are obtained as corollaries.

Single-machine scheduling with past-sequence-dependent setup times and learning effects: a parametric analysis

In this article, we consider the single-machine scheduling problem with past-sequence-dependent (p-s-d) setup times and a learning effect. The setup times are proportional to the length of jobs that are already scheduled; i.e. p-s-d setup times. The learning effect reduces the actual processing time of a job because the workers are involved in doing the same job or activity repeatedly. Hence, the processing time of a job depends on its position in the sequence. In this study, we consider the total absolute difference in completion times (TADC) as the objective function. This problem is denoted as 1/LE, s psd /TADC in Kuo and Yang (2007) (‘Single Machine Scheduling with Past-sequence-dependent Setup Times and Learning Effects’, Information Processing Letters, 102, 22–26). There are two parameters a and b denoting constant learning index and normalising index, respectively. A parametric analysis of b on the 1/LE, s psd /TADC problem for a given value of a is applied in this study. In addition, a computational algorithm is also developed to obtain the number of optimal sequences and the range of b in which each of the sequences is optimal, for a given value of a. We derive two bounds b* for the normalising constant b and a* for the learning index a. We also show that, when a < a* or b > b*, the optimal sequence is obtained by arranging the longest job in the first position and the rest of the jobs in short processing time order.

Geodesic pancyclicity and balanced pancyclicity of the generalized base-[formula omitted] hypercube

Recently, Chan et al. introduced geodesic-pancyclic graphs [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, Discrete Applied Mathematics 155 (15) (2007) 1971–1978] and weakly geodesic pancyclicity [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes, Applied Mathematics and Computation 207 (2009) 333–339]. Hsu et al. proposed a new cycle-embedding property called balanced pancyclicity [H.C. Hsu, P.L. Lai, C.H. Tsai, Geodesic pancyclicity and balanced pancyclicity of augmented cubes, Information Processing Letters 101 (2007) 227–232]. For a graph G(V,E) and any two vertices x and y of V, a cycle R containing x and y can be divided into two paths, Pt1 and Pt2, joining x and y such that len(Pt1)≤len(Pt2), where len(λ) denotes the length of the path λ. A geodesic cycle contains Pt1, which is the shortest path joining x and y in G, whereas, in a balanced cycle of an even (respectively, odd) length, len(Pt1)=len(Pt2) (respectively, len(Pt1)=len(Pt2)−1). A graph is weakly geodesic pancyclic (respectively, balanced pancyclic) if every two vertices x and y are contained in a geodesic cycle (respectively, balanced cycle) from Max(3,2Dist(x,y)) to N, where N is the order of the graph. The interconnection network considered in this paper is the generalized base-b hypercube, which is an attractive variant of the well-known hypercube. In fact, the generalized base-b hypercube is the Cartesian product of complete graphs with b vertices. The generalized base-b hypercube is superior to the hypercube in many criteria, such as diameter, connectivity, and fault diameter. In this paper, we study weakly geodesic pancyclicity and balanced pancyclicity of the generalized base-b hypercube. We show that the generalized base-b hypercube is weakly geodesic pancyclic for b≥3 and balanced pancyclic for b≥4.

Two types of path structures of graphs

In 2005, Rahman and Kaykobad proved that if G is a 2-connected graph with n vertices and d(u) + d(v ) + (u,v)⩾n+1 for each pair of distinct non-adjacent vertices u,v in G, then G is traceable [Information Processing Letters, 94 (2005), 1, 37–41]. In this present paper, we prove that if G is a 2-connected graph with n vertices and d(u)+d(v)+(u,v)⩾n for each pair of distinct non-adjacent vertices u,v in G, then G is traceable or G belongs to a class of wellstructured exceptional graphs

Edge-fault-tolerant edge-bipancyclicity of bubble-sort graphs

The bubble-sort graph B n is a bipartite graph. Kikuchi and Araki [Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs. Information Processing Letters, 100, 52–59 (2006)] have proved that B n is edge-bipancyclic for n ≥ 5 and B n − F is bipancyclic when n ≥ 4 and |F| ≤ n − 3. In this paper, we improve this result by showing that for any edge set F of B n with |F| ≤ n − 3, every edge of B n −F lies on a cycle of every even length from 6 to n! for n ≥ 5 and every edge of B n − F lies on a cycle of every even length from 8 to n! for n = 4.

Contributors

Contributors

Erratum and addendum to “A linear time algorithm for finding all hinge vertices of a permutation graph” [Information Processing Letters 59 (2) (1996) 103–107

This note points out and corrects an error in the algorithm proposed in [Ting-Yem Ho, Yue-Li Wang and Ming-Tsan Juan, A linear time algorithm for finding all hinge vertices of a permutation graph, Information Processing Letters 59 (2) (1996) 103–107].

Running time analysis of Ant Colony Optimization for shortest path problems

Ant Colony Optimization (ACO) is a modern and very popular optimization paradigm inspired by the ability of ant colonies to find shortest paths between their nest and a food source. Despite its popularity, the theory of ACO is still in its infancy and a solid theoretical foundation is needed. We present bounds on the running time of different ACO systems for shortest path problems. First, we improve previous results by Attiratanasunthron and Fakcharoenphol [Information Processing Letters 105 (3) (2008) 88–92] for single-destination shortest paths and extend their results from DAGs to arbitrary directed graphs. Our upper bound is asymptotically tight for large evaporation factors, holds with high probability, and transfers to the all-pairs shortest paths problem. There, a simple mechanism for exchanging information between ants with different destinations yields a significant improvement. A comparison with evolutionary and genetic approaches indicates that ACO is among the best known metaheuristics for the all-pairs shortest paths problem.