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- W1994217103 abstract "We introduce a structural parameter of a graph, the longest directed path length (LDPL), and express the boundary of the difficulty of the problems along this parameter. The parallel complexity of the lexicographically first maximal independent set (LFMIS) problem gradually increases as the value of LDPL grows; the LFMIS problem on a graph of LDPL O( log k n) can be solved in O( log k n) time in parallel; and the problem is P-complete on a graph of LDPL Θ(n ∊ ). Computing the LDPL itself is in NC 2 . This is important in the sense that a measure is valid only if measuring the complexity of a problem is easier than solving it. On the other hand, we also show the limits of the measure. We reduce the LFMIS problem to a kind of the lexicographically first maximal subgraph (LFMS) problems on a graph of LDPL 1. This implies that, even on a graph of LDPL 1, the LFMS problem is P-complete. Finally we discuss the probability that a random graph has LDPL l and show that a random graph of which each edge exists with probability p has LDPL Θ(np) with high probability. This implies the limit of the LDPL on the average case, because the LFMIS problem on a random graph can be efficiently solved in parallel." @default.
- W1994217103 created "2016-06-24" @default.
- W1994217103 creator A5054829031 @default.
- W1994217103 date "1999-12-01" @default.
- W1994217103 modified "2023-10-14" @default.
- W1994217103 title "A MEASURE FOR THE LEXICOGRAPHICALLY FIRST MAXIMAL INDEPENDENT SET PROBLEM AND ITS LIMITS" @default.
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- W1994217103 doi "https://doi.org/10.1142/s0129054199000332" @default.
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