The maximal independent set (MIS) problem is one of the most fundamental problems in the field of distributed computing. This paper focuses on the MIS problem with unreliable communication between processes in the system. We propose a relaxed notion of MIS, named almost MIS (ALMIS), and show that the loosely-stabilizing algorithm proposed in our previous work can achieve exponentially long holding time with logarithmic convergence time and space complexity regarding ALMIS, which cannot be achieved at the same time regarding MIS in our previous work.

This paper studies generalized variants of the MAXIMUM INDEPENDENT SET problem, called the MAXIMUM DISTANCE-d INDEPENDENT SET problem (MaxDdIS for short). For an integer d≥2, a distance-d independent set of an unweighted graph G=(V, E) is a subset S⊆V of vertices such that for any pair of vertices u, v∈S, the number of edges in any path between u and v is at least d in G. Given an unweighted graph G, the goal of MaxDdIS is to find a maximum-cardinality distance-d independent set of G. In this paper, we analyze the (in)approximability of the problem on r-regular graphs (r≥3) and planar graphs, as follows: (1) For every fixed integers d≥3 and r≥3, MaxDdIS on r-regular graphs is APX-hard. (2) We design polynomial-time O(rd-1)-approximation and O(rd-2/d)-approximation algorithms for MaxDdIS on r-regular graphs. (3) We sharpen the above O(rd-2/d)-approximation algorithms when restricted to d=r=3, and give a polynomial-time 2-approximation algorithm for MaxD3IS on cubic graphs. (4) Finally, we show that MaxDdIS admits a polynomial-time approximation scheme (PTAS) for planar graphs.

Some textbooks of formal languages and automata theory implicitly state the structural equality of the binary n-dimensional de Bruijn graph and the state diagram of minimum state deterministic finite automaton which accepts regular language (0+1)*1(0+1)n-1. By introducing special finite automata whose accepting states are refined with two or more colors, we extend this fact to both k-ary versions. That is, we prove that k-ary n-dimensional de Brujin graph and the state diagram for minimum state deterministic colored finite automaton which accepts the (k-1)-tuple of the regular languages (0+1+…+k-1)*1(0+1+…+k-1)n-1,...,and(0+1+…+k-1)*(k-1)(0+1+…+k-1)n-1 are isomorphic for arbitrary k more than or equal to 2. We also investigate the properties of colored finite automata themselves and give computational complexity results on three decision problems concerning color unmixedness of nondeterminisitic ones.

Let G=(V,E) be an unweighted simple graph. A distance-d independent set is a subset I ⊆ V such that dist(u, v) ≥ d for any two vertices u, v in I, where dist(u, v) is the distance between u and v. Then, Maximum Distance-d Independent Set problem requires to compute the size of a distance-d independent set with the maximum number of vertices. Even for a fixed integer d ≥ 3, this problem is NP-hard. In this paper, we design an exact exponential algorithm that calculates the size of a maximum distance-3 independent set in O(1.4143n) time.

We consider the minimum feedback vertex set problem for some bipartite graphs and degree-constrained graphs. We show that the problem is linear time solvable for bipartite permutation graphs and NP-hard for grid intersection graphs. We also show that the problem is solvable in O(n2log 6n) time for n-vertex graphs with maximum degree at most three.

The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set. However, its computational complexity is very large. In this paper, we consider cyclic codes defined by their defining set, and a new method to calculate the lower bound of the minimum distance using the discrete Fourier transform (DFT) is shown. The computational complexity of this method is compared with the shift bound's one. Moreover construction of independent set is shown.

In this paper, we propose new joint transport and MAC protocols for ad hoc wireless networks based on an optimization framework. To overcome the practical and efficiency limitations of previous research, we develop a different framework based on independent sets and propose an implementable heuristic algorithm. We address the implementation issues of the proposed algorithm. Simulation results confirm the efficiency and fairness of our protocols.

In this paper we propose an algorithm for generating maximum weight independent sets in a circle graph, that is, for putting out all maximum weight independent sets one by one without duplication. The time complexity is O(n3 + β ), where n is the number of vertices, β output size, i. e. , the sum of the cardinalities of the output sets. It is shown that the same approach can be applied for spider graphs and for circular-arc overlap graphs.

Let H=(V(H),E(H)) be a directed graph with distinguished vertices s and t. An st-path in H is a simple directed path starting from s and ending at t. Let (H) be defined as { SS is the set of vertices on an st-path in H (s and t are excluded)}. For an undirected graph G=(V(G),E(G)) with V(G) V(H)- { s,t }, if the family of maximal independent sets of G coincides with (H), we call H an MIS-diagram for G. In this paper, we provide a necessary and sufficient condition for a directed graph to be an MIS-diagram for an undirected graph. We also show that an undirected graph G has an MIS-diagram iff G is a cocomparability graph. Based on the proof of the latter result, we can construct an efficient algorithm for generating all maximal independent sets of a cocomparability graph.

The maximal linear forest problem is to find, given a graph G = (V, E), a maximal subset of V that induces a linear forest. Three parallel algorithms for this problem are presented. The first one is randomized and runs in O(log n) expected time using n2 processors on a CRCW PRAM. The second one is deterministic and runs in O(log 2n) timeusing n4 processors on an EREW PRAM. The last one is deterministic and runs in O(log 5n) time using n3 processors on an EREW PRAM. The results put the problem in the class NC.

Genetic algorithms have been shown to be very useful in a variety of search and optimization problems. In this paper we present a genetic algorithm for maximum independent set problem. We adopt a permutation encoding with a greedy decoding to solve the problem. The DIMACS benchmark graphs are used to test our algorithm. For most graphs solutions found by our algorithm are optimal, and there are also a few exceptions that solutions found by the algorithm are almost as large as maximum clique sizes. We also compare our algorithm with a hybrid genetic algorithm, called GMCA, and one of the best existing maximum clique algorithms, called CBH. The exiperimental results show that our algorithm outperformed two of the best approaches by GMCA and CBH in final solutions.

Let C{c1, , cm} be a family of subsets of a finite set S{1, , n}, a subset S of S is a co-hitting set if S contains no element of C as a subset. By using an O((log n)2) time EREW PRAM algorithm for a maximal independent set problem (MIS), we show that a maximal co-hitting set for S can be computed on an EREW PRAN in time O(αβ(log(nm))2) using O(n2 m) processors, where αmax{|cii1, , n} and βmax{|djj1, , n} with dj{ci|jci}. This implies that if αβO((log(nm))k) then the problem is solvable in NC.

A maximal l-diameter tree cover of a graph G(V,E) is a spanning subgraph C(V,EC) of G such that each connected component of C is a tree, C contains no path with more than l edges, and adding any edge in EEC to C yields either a path of length l1 or a cycle. For every function f from positive integers to positive integers, the maximal f-diameter tree cover prolem (MDTC(f) problem for short) is to find a maximal f(n)-diameter tree cover of G, given an n-node graph G. In this paper, we give two parallel algorithms for the MDTC(f) problem. The first algorithm can be implemented in time O(TMSP(n,f(n))log2n) using polynomial number of processors on an EREW PRAM, where TMSP(n,f(n) is the time needed to find a maximal set of vertex disjoint paths of length f(n) in a given n-node graph using polynomial number of processors on an EREW PRAM. We then show that if suitable restrictions are imposed on the input graph and/or on the magnitude of f, then TMSP(n,f(n))O(logkn) for some constant k and thus, for such cases, we obtain an NC algorithm for the MDTC(f) problem. The second algorithm runs in time O(n log2n/{f(n)1}) using polynomial number of processors on an EREW PRAM. Thus if f(n)Ω(n/logkn) for some kO, we obtain an NC algorithm for the MDTC(f) problem.