In a 1985 paper, Bern, Lawler, and Wong described a general method for constructing algorithms to find an optimal subgraph in a given graph. When the given graph is a member of a k-terminal recursive family of graphs ...
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In a 1985 paper, Bern, Lawler, and Wong described a general method for constructing algorithms to find an optimal subgraph in a given graph. When the given graph is a member of a k-terminal recursive family of graphs and is presented in the form of a parse tree, and when the optimal subgraph satisfies a property that is regular with respect to the family of graphs, then the method produces a linear-time algorithm. The algorithms assume the existence of multiplication tables that are specific to the regular property and to the family of graphs. In this paper we show that the general problem of computing these multiplication tables is unsolvable and provide a ''pumping'' lemma for proving that particular properties are not regular for particular k-terminal families. In contrast with these negative results, we show that all local properties, that can be verified by examining a bounded neighbourhood of each vertex in a graph, are regular with respect to all k-terminal recursive families of graphs, and we show how to automate the construction of the multiplication tables for any local property.
This paper discusses the complexity of packing k-chains (simple paths of length k) into an undirected graph;the chains packed must be either vertex-disjoint or edge-disjoint. linear-time algorithms are given for both ...
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This paper discusses the complexity of packing k-chains (simple paths of length k) into an undirected graph;the chains packed must be either vertex-disjoint or edge-disjoint. linear-time algorithms are given for both problems when the graph is a tree, and for the edge-disjoint packing problem when the graph is general and k = 2. The vertex-disjoint packing problem for general graphs is shown to be NP-complete even when the graph has maximum degree three and k = 2. Similarly the edge-disjoint packing problem is NP-complete even when the graph has maximum degree four and k = 3.
The register allocation problem for an imperative program is often modeled as the coloring problem of the interference graph of the control-flow graph of the program. The interference graph of a flow graph G is the in...
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The register allocation problem for an imperative program is often modeled as the coloring problem of the interference graph of the control-flow graph of the program. The interference graph of a flow graph G is the intersection graph of some connected subgraphs of G. These connected subgraphs represent the lives, or life times, of variables, so the coloring problem models that two variables with overlapping life times should be in different registers. For general programs with unrestricted gotos, the interference graph can be any graph, and hence we cannot in general color within a factor O(n(epsilon)) from optimality unless NP = P. It is shown that if a graph has tree width k, we can efficiently color any intersection graph of connected subgraphs within a factor ([k/2]+1) from optimality. Moreover, it is shown that structured (=goto-free) programs, including, for example, short circuit evaluations and multiple exits from loops, have tree width at most 6. Thus, for every structured program, we can do register allocation efficiently within a factor 4 from optimality, regardless of how many registers are needed. The bounded tree decomposition may be derived directly from the parsing of a structured program, and it implies that the many techniques for bounded tree width may now be applied in compiler optimization, solving problems in lineartime that are NP-hard, or even P-space hard, for general graphs. (C) 1998 Academic Press.
An ingenious graph-based watermarking scheme recently proposed by Chroni and Nikolopoulos encodes integers as a special type of reducible permutation graphs. It was claimed without proof that those graphs can withstan...
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An ingenious graph-based watermarking scheme recently proposed by Chroni and Nikolopoulos encodes integers as a special type of reducible permutation graphs. It was claimed without proof that those graphs can withstand attacks in the form of a single edge removal. We introduce a linear-time algorithm which restores the original graph after removals of k <= 2 edges, therefore proving an even stronger result. Furthermore, we prove that k <= 5 general edge modifications (removals/insertions) can always be detected in polynomial time. Both bounds are tight. Our results reinforce the interest in regarding Chroni and Nikolopoulos's scheme as a possible software watermarking solution for numerous applications. (C) 2016 Elsevier B.V. All rights reserved.
In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contr...
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In this paper we present an algorithm for determining the number of spanning trees of a graph G which takes advantage of the structure of the modular decomposition tree of G. Specifically, our algorithm works by contracting the modular decomposition tree of the input graph G in a bottom-up fashion until it becomes a single node;then, the number of spanning trees of G is computed as the product of a collection of values which are associated with the vertices of G and are updated during the contraction process. In particular, when applied on a (q, q - 4)-graph for fixed q, a P-4-tidy graph, or a tree-cograph, our algorithm computes the number of its spanning trees in timelinear in the size of the graph, where the complexity of arithmetic operations is measured under the uniform-cost criterion. Therefore we give the first linear-time algorithm for the counting problem in the considered graph classes. (C) 2014 Elsevier B.V. All rights reserved.
Digital watermarks have been regarded as a promising way to fight copyright violations in the software industry. In some graph-based watermarking schemes, identification data is disguised as control-flow graphs of dum...
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Digital watermarks have been regarded as a promising way to fight copyright violations in the software industry. In some graph-based watermarking schemes, identification data is disguised as control-flow graphs of dummy code. Recently, Chroni and Nikolopoulos proposed an ingenious such scheme whereby an integer is encoded into a particular kind of permutation graph. We give a formal characterization of the class of graphs generated by their encoding function, which we call canonical reducible permutation graphs. A linear-time recognition algorithm is also given, setting the basis for a polynomial-time algorithm to restore watermarks that underwent the malicious removal of some edges. Finally, we give a simpler decoding algorithm for Chroni and Nikolopoulos' watermarks.
We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to...
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We study a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short. Given a graph G and a subset T of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality;note that, if T is empty (or, equivalently, k = 0), the stated problem coincides with the classical path cover problem. The kPC problem generalizes some path cover related problems, such as the 1HP and 2HP problems, which have been proved to be NP-complete. Note that the complexity status for both 1HP and 2HP problems on interval graphs remains an open question (Damaschke ( 1993)[9]). In this paper, we show that the kPC problem can be solved in lineartime on the class of proper interval graphs, that is, in O(n + m) time on a proper interval graph on n vertices and m edges. The proposed algorithm is simple, requires linear space. and also enables us to solve the 1HP and 2HP problems on proper interval graphs within the same time and space complexity. (C) 2009 Elsevier B.V. All rights reserved.
The minimum-weight spanning tree problem is one of the most typical and well-known problems of combinatorial optimisation. Efficient solution techniques had been known for many years. However, in the last two decades ...
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The minimum-weight spanning tree problem is one of the most typical and well-known problems of combinatorial optimisation. Efficient solution techniques had been known for many years. However, in the last two decades asymptotically faster algorithms have been invented. Each new algorithm brought the time bound one step closer to linearity and finally Karger, Klein and Tarjan proposed the only known expected linear-time method. Modern algorithms make use of more advanced data structures and appear to be more complicated to implement. Most authors and practitioners refer to these but still use the classical ones, which are considerably simpler but asymptotically slower. The paper first presents a survey of the classical methods and the more recent algorithmic developments. Modern algorithms are then compared with the classical ones and their relative performance is evaluated through extensive empirical tests, using reasonably large-size problem instances. Randomly generated problem instances used in the tests range from small networks having 512 nodes and 1024 edges to quite large ones with 16384 nodes and 524288 edges. The purpose of the comparative study is to investigate the conjecture that modern algorithms are also easy to apply and have constants of proportionality small enough to make them competitive in practice with the older ones.
We characterize interval graphs in terms of a certain linear order on their vertex set. It turns out that with this linear order, the well-known greedy heuristic "always use the smallest available color" yie...
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We characterize interval graphs in terms of a certain linear order on their vertex set. It turns out that with this linear order, the well-known greedy heuristic "always use the smallest available color" yields an exact coloring algorithm for interval graphs. In addition, the heuristic can be implemented to run in lineartime.
We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path co...
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We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T of k vertices of V (G), a k-fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices of T are all endpoints of the paths in P. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality;note that, if T is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112: 4964, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.
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