Exact Algorithms for Intervalizing Coloured Graphs

In the Intervalizing Coloured Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses \(2^{\mathcal {O}(n/\log n)}\) time. We also give an \(\mathcal {O}^{\ast }(2^{n})\) algorithm for the case that the number of colors is not fixed.     

A Note on Exact Algorithms for Vertex Ordering Problems on Graphs

In this note, we give a proof that several vertex ordering problems can be solved in O ^∗(2 ⁿ ) time and O ^∗(2 ⁿ ) space, or in O ^∗(4 ⁿ ) time and polynomial space. The algorithms generalize algorithms for the Travelling Salesman Problem by Held and Karp (J. Soc. Ind. Appl. Math. 10:196–210, 1962) and Gurevich and Shelah (SIAM J. Comput. 16:486–502, 1987). We survey a number of vertex ordering problems to which the results apply.     

任意图支配集精确算法回顾

该文综述了任意图支配集精确算法分析和设计的新进展.支配集问题是经典NP完全问题,很多问题都能与它相联系.我们针对最小支配集、最大独立集、最小独立支配集、最小连通支配集、最小加权支配集问题提供了详尽算法描述和实例说明,以使文章自包含方便阅读.文中还讨论了诸如分支简化策略、复杂度分析、测度分析、记忆等技术.自Claude Berge首次准确阐述现代图支配概念后,经过很长一段时期的沉寂,关于指数时间精确算法设计的研究热情在过去五年中显著增涨.除回顾这些最新成果之外,作者还盼望国内研究团体能更加重视这个快速发展的研究领域.     

Space-Efficient Algorithms for Longest Increasing Subsequence

Theory of Computing Systems - Given a sequence of integers, we want to find a longest increasing subsequence of the sequence. It is known that this problem can be solved in $Oleft (n log nright...     

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k ^O(dk) n time algorithm for finding a dominating set of size at most k in a d-degenerated graph with n vertices. This proves that the dominating set problem is fixed-parameter tractable for degenerated graphs. For graphs that do not contain K _h as a topological minor, we give an improved algorithm for the problem with running time (O(h)) ^hk n. For graphs which are K _h -minor-free, the running time is further reduced to (O(log h)) ^hk/2 n. Fixed-parameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an H-minor-free graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(nlog n) worst-case time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs. A preliminary version of this paper appeared in the Proceedings of the 13th Annual International Computing and Combinatorics Conference (COCOON), Banff, Alberta, Canada (2007), pp. 394–405. N. Alon research supported in part by a grant from the Israel Science Foundation, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. This paper forms part of a Ph.D. thesis written by S. Gutner under the supervision of Prof. N. Alon and Prof. Y. Azar in Tel Aviv University.     

Efficient Exact Algorithms on Planar Graphs: Exploiting Sphere Cut Decompositions

We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  $O(2^{6.903\sqrt{n}})We present a general framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour and Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. To exemplify our approach we show how to obtain an  O(2^6.903?n)O(2^{6.903\sqrt{n}}) time algorithm solving weighted Hamiltonian Cycle on an n-vertex planar graph. Similar technique solves Planar Graph Travelling Salesman Problem with n cities in time O(2^9.8594?n)O(2^{9.8594\sqrt{n}}) . Our approach can be used to design parameterized algorithms as well. For example, we give an algorithm that for a given k decides if a planar graph on n vertices has a cycle of length at least k in time O(2^13.6?kn+n³)O(2^{13.6\sqrt{k}}n+n^{3}) .     

Exact Algorithms for MAX-SAT

The maximum satisfiability problem (MAX-SAT) is stated as follows: Given a boolean formula in CNF, find a truth assignment that satisfies the maximum possible number of its clauses. MAX-SAT is MAX-SNP-complete and received much attention recently. One of the challenges posed by Alber, Gramm and Niedermeier in a recent survey paper asks: Can MAX-SAT be solved in less than 2ⁿ “steps”? Here, n is the number of distinct variables in the formula and a step may take polynomial time of the input. We answered this challenge positively by showing that a popular algorithm based on branch-and-bound is bounded by O(2ⁿ) in time, where n is the maximum number of occurrences of any variable in the input.When the input formula is in 2-CNF, that is, each clause has at most two literals, MAX-SAT becomes MAX-2-SAT and the decision version of MAX-2-SAT is still NP-complete. The best bound of the known algorithms for MAX-2-SAT is O(m2^m/5), where m is the number of clauses. We propose an efficient decision algorithm for MAX-2-SAT whose time complexity is bound by O(n2ⁿ). This result is substantially better than the previously known results. Experimental results also show that our algorithm outperforms any algorithm we know on MAX-2-SAT.     

Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.     

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

The Induced Graph Matching problem asks to find \(k\) disjoint induced subgraphs isomorphic to a given graph  \(H\) in a given graph \(G\) such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in \(k\) on claw-free graphs when \(H\) is a fixed connected graph, and even admits a polynomial kernel when  \(H\) is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove \(\mathsf {W}[1]\) -hardness of Induced Graph Matching on graphs excluding \(K_{1,4}\) as an induced subgraph, for any fixed complete graph \(H\) . In particular, we show that Independent Set is \(\mathsf {W}[1]\) -hard on \(K_{1,4}\) -free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or \(\mathsf {NP}\) -complete, depending on the connectivity of \(H\) and the structure of \(G\) .     

Exact Algorithms for Graph Homomorphisms

Graph homomorphism, also called H-coloring, is a natural generalization of graph coloring: There is a homomorphism from a graph G to a complete graph on k vertices if and only if G is k-colorable. During recent years the topic of exact (exponential-time) algorithms for NP-hard problems in general, and for graph coloring in particular, has led to extensive research. Consequently, it is natural to ask how the techniques developed for exact graph coloring algorithms can be extended to graph homomorphisms. By the celebrated result of Hell and Nesetril, for each fixed simple graph H, deciding whether a given simple graph G has a homomorphism to H is polynomial-time solvable if H is a bipartite graph, and NP-complete otherwise. The case where H is the cycle of length 5, is the first NP-hard case different from graph coloring. We show that for an odd integer , whether an input graph G with n vertices is homomorphic to the cycle of length k, can be decided in time . We extend the results obtained for cycles, which are graphs of treewidth two, to graphs of bounded treewidth as follows: if H is of treewidth at most t, then whether input graph G with n vertices is homomorphic to H can be decided in time .     

Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of planar straight-line representation has not been solved completely. In this paper we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.     

Analysis of Evolutionary Algorithms for the Longest Common Subsequence Problem

In the longest common subsequence problem the task is to find the longest sequence of letters that can be found as a subsequence in all members of a given finite set of sequences. The problem is one of the fundamental problems in computer science with the task of finding a given pattern in a text as an important special case. It has applications in bioinformatics; problem-specific algorithms and facts about its complexity are known. Motivated by reports about good performance of evolutionary algorithms for some instances of this problem a theoretical analysis of a generic evolutionary algorithm is performed. The general algorithmic framework encompasses EAs as different as steady state GAs with uniform crossover and randomized hill-climbers. For all these algorithms it is proved that even rather simple special cases of the longest common subsequence problem can neither be solved to optimality nor approximately be solved up to an approximation factor arbitrarily close to 2.     

Exact and Approximation Algorithms for Clustering

In this paper we present an n^ O(k ^1-1/d ) -time algorithm for solving the k -center problem in \reals ^d , under L _∈ fty - and L ₂ -metrics. The algorithm extends to other metrics, and to the discrete k -center problem. We also describe a simple (1+ɛ) -approximation algorithm for the k -center problem, with running time O(nlog k) + (k/ɛ)^ O(k ^1-1/d ) . Finally, we present an n^ O(k ^1-1/d ) -time algorithm for solving the L -capacitated k -center problem, provided that L=Ω(n/k ^1-1/d ) or L=O(1) . Received July 25, 2000; revised April 6, 2001.     

Exact Algorithms for Exact Satisfiability and Number of Perfect Matchings

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 ^m l ^O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2 ⁿ n ^O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 ⁿ ) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number.     

Set Graphs. III. Proof Pearl: Claw-Free Graphs Mirrored into Transitive Hereditarily Finite Sets

We report on the formalization of two classical results about claw-free graphs, which have been verified correct by Jacob T. Schwartz’s proof-checker Referee. We have proved formally that every connected claw-free graph admits (1) a near-perfect matching, (2) Hamiltonian cycles in its square. To take advantage of the set-theoretic foundation of Referee, we exploited set equivalents of the graph-theoretic notions involved in our experiment: edge, source, square, etc. To ease some proofs, we have often resorted to weak counterparts of well-established notions such as cycle, claw-freeness, longest directed path, etc.     

A Quadratic Algorithm for Finding Next-to-Shortest Paths in Graphs

Given an edge-weighted undirected graph G and two prescribed vertices u and v, a next-to-shortest (u,v)-path is a shortest (u,v)-path amongst all (u,v)-paths having length strictly greater than the length of a shortest (u,v)-path. In this paper, we deal with the problem of computing a next-to-shortest (u,v)-path. We propose an O(n²){\mathcal{O}}(n^{2}) time algorithm for solving this problem, which significantly improves the bound of a previous one in O(n³){\mathcal{O}}(n^{3}) time where n is the number of vertices in G.     

Reduction Algorithms for Graphs of Small Treewidth

This paper presents a number of new ideas and results on graph reduction applied to graphs of bounded treewidth. S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese (J. Assoc. Comput. Mach.40, 1134–1164 (1993)) have shown that many decision problems on graphs can be solved in linear time on graphs of bounded treewidth, using a finite set of reduction rules. These algorithms can be used to solve problems on graphs of bounded treewidth without the need to obtain a tree decomposition of the input graph first. We show that the reduction method can be extended to solve the construction variants of many decision problems on graphs of bounded treewidth, including all problems definable in monadic second order logic. We also show that a variant of these reduction algorithms can be used to solve (constructive) optimization problems in O(n) time. For example, optimization and construction variants of I S and H C N can be solved in this way on graphs of small treewidth. Additionally, we show that the results of H. L. Bodlaender and T. Hagerup (SIAM J. Comput.27, 1725–1746 (1998)) can be applied to our reduction algorithms, which results in parallel reduction algorithms that use O(n) operations and O(log n log* n) time on an EREW PRAM, or O(log n) time on a CRCW PRAM.     

The Longest Path Problem Is Polynomial on Cocomparability Graphs

The longest path problem is the problem of finding a path of maximum length in a graph. As a generalization of the Hamiltonian path problem, it is NP-complete on general graphs and, in fact, on every class of graphs that the Hamiltonian path problem is NP-complete. Polynomial solutions for the longest path problem have recently been proposed for weighted trees, Ptolemaic graphs, bipartite permutation graphs, interval graphs, and some small classes of graphs. Although the Hamiltonian path problem on cocomparability graphs was proved to be polynomial almost two decades ago, the complexity status of the longest path problem on cocomparability graphs has remained open; actually, the complexity status of the problem has remained open even on the smaller class of permutation graphs. In this paper, we present a polynomial-time algorithm for solving the longest path problem on the class of cocomparability graphs. Our result resolves the open question for the complexity of the problem on such graphs, and since cocomparability graphs form a superclass of both interval and permutation graphs, extends the polynomial solution of the longest path problem on interval graphs and provides polynomial solution to the class of permutation graphs.