Solving shortest paths efficiently on nearly acyclic directed graphs

Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn+nrlogr) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+rlogr) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bidirectional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bidirectional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m+rlogr) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn+nr²) to O(mn+r³).     

Applying Modular Decomposition to Parameterized Cluster Editing Problems

A graph G is said to be a bicluster graph if G is a disjoint union of bicliques (complete bipartite subgraphs), and a cluster graph if G is a disjoint union of cliques (complete subgraphs). In this work, we study the parameterized versions of the NP-hard Bicluster Graph Editing and Cluster Graph Editing problems. The former consists of obtaining a bicluster graph by making the minimum number of modifications in the edge set of an input bipartite graph. When at most k modifications are allowed (Bicluster(k) Graph Editing problem), this problem is FPT, and can be solved in O(4 ^k nm) time by a standard search tree algorithm. We develop an algorithm of time complexity O(4 ^k +n+m), which uses a strategy based on modular decomposition techniques; we slightly generalize the original problem as the input graph is not necessarily bipartite. The algorithm first builds a problem kernel with O(k ²) vertices in O(n+m) time, and then applies a bounded search tree. We also show how this strategy based on modular decomposition leads to a new way of obtaining a problem kernel with O(k ²) vertices for the Cluster(k) Graph Editing problem, in O(n+m) time. This problem consists of obtaining a cluster graph by modifying at most k edges in an input graph. A previous FPT algorithm of time O(1.92 ^k +n ³) for this problem was presented by Gramm et al. (Theory Comput. Syst. 38(4), 373–392, 2005, Algorithmica 39(4), 321–347, 2004). In their solution, a problem kernel with O(k ²) vertices is built in O(n ³) time.     

Efficient Graph-Theoretic Algorithms on a Linear Array with a Reconfigurable Pipelined Bus System

We present efficient algorithms for solving several fundamental graph-theoretic problems on a Linear Array with a Reconfigurable Pipelined Bus System (LARPBS), one of the recently proposed models of computation based on optical buses. Our algorithms include finding connected components, minimum spanning forest, biconnected components, bridges and articulation points for an undirected graph. We compute the connected components and minimum spanning forest of a graph in O(log n) time using O(m+n) processors where m and n are the number of edges and vertices in the graph and m=O(n ²) for a dense graph. Both the processor and time complexities of these two algorithms match the complexities of algorithms on the Arbitrary and Priority CRCW PRAM models which are two of the strongest PRAM models. The algorithms for these two problems published by Li et al. [7] have been considered to be the most efficient on the LARPBS model till now. Their algorithm [7] for these two problems require O(log n) time and O(n ³/log n) processors. Hence, our algorithms have the same time complexity but require less processors. Our algorithms for computing biconnected components, bridges and articulation points of a graph run in O(log n) time on an LARPBS with O(n ²) processors. No previous algorithm was known for these latter problems on the LARPBS.     

The Swap Edges of a Multiple-Sources Routing Tree

Let T be a spanning tree of a graph G and S⊂V(G) be a set of sources. The routing cost of T is the total distance from all sources to all vertices. For an edge e of T, the swap edge of e is the edge f minimizing the routing cost of the tree formed by replacing e with f. Given an undirected graph G and a spanning tree T of G, we investigate the problem of finding the swap edge for every tree edge. In this paper, we propose an O(mlog n+n ²)-time algorithm for the case of two sources and an O(mn)-time algorithm for the case of more than two sources, where m and n are the numbers of edges and vertices of G, respectively.     

An efficient distributed algorithm for finding all hinge vertices in networks

Let G=(V, E) be a graph with vertex set V of size n and edge set E of size m. A vertex v∈V is called a hinge vertex if there exist two vertices in V\{v} such that their distance becomes longer when v is removed. In this paper, we present a distributed algorithm that finds all hinge vertices on an arbitrary graph. The proposed algorithm works for named static asynchronous networks and achieves O(n ²) time complexity and O(m) message complexity. In particular, the total messages exchanged during the algorithm are at most 2m(log n+nlog n+1) bits.     

两元指纹向量聚类问题的复杂性与改进启发式算法

证明丢失值位数不超过2的指纹向量聚类问题为NP-Hard,并给出Figueroa等人指纹向量聚类启发式算法的改进算法.主要改进了算法的实现方法.以链表存储相容顶点集合,并以逐位扫描指纹向量的方法产生相容点集链表,可将产生相容点集的时间复杂性由O(m·n·2p)减小为O(m·(n·p 1)·2p),可使划分一个唯一极大团或最大团的时间复杂性由O(m·p·2p)减小为O(m·2p).实际测试显示,改进算法的空间复杂性平均减少为原算法的49%以下,平均可用原算法20%的时间求解与原算法相同的实例.当丢失值位数超过6时,改进算法几乎总可用不超过原算法11%的时间计算与原算法相同的实例.     

Parallel PROFIT/COST algorithms through fast derandomization

We present parallel algorithms for the PROFIT/COST problem with time complexity using O(m+n) processors. The design of these algorithms employ both the derandomization technique and the pipeline technique. They can be used to partition the vertices of a graph into two sets such that the number of edges incident with vertices in both sets is at least half of the total number of edges in the graph. Parallel algorithms for the PROFIT/COST problem have known applications in the design of parallel algorithms for several graph problems. Received: 18 March 1992 / 8 January 1999     

Piecemeal Graph Exploration by a Mobile Robot

We study how a mobile robot can learn an unknown environment in a piecemeal manner. The robot's goal is to learn a complete map of its environment, while satisfying the constraint that it must return every so often to its starting position (for refueling, say). The environment is modeled as an arbitrary, undirected graph, which is initially unknown to the robot. We assume that the robot can distinguish vertices and edges that it has already explored. We present a surprisingly efficient algorithm for piecemeal learning an unknown undirected graph G=(V, E) in which the robot explores every vertex and edge in the graph by traversing at most O(E+V^1+o(1)) edges. This nearly linear algorithm improves on the best previous algorithm, in which the robot traverses at most O(E+V²) edges. We also give an application of piecemeal learning to the problem of searching a graph for a “treasure.”     

An $\tilde{O}(m^{2}n)$ Algorithm for Minimum Cycle Basis of Graphs

We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ^ω n), where ω is the best exponent of matrix multiplication. It is presently known that ω<2.376. We exhibit an O(m ² n+mn ²log n) algorithm. When the edge weights are integers, we have an O(m ² n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ^ω ) time. For any ε>0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ^ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight. A preliminary version of this paper appeared in Kavitha et al. (31st International Colloquium on Automata, Languages and Programming (ICALP), pp. 846–857, 2004). T. Kavitha and K.E. Paluch were in Max-Planck-Institut für Informatik, Saarbrücken, Germany, while this work was done.     

On the Minimum Feedback Vertex Set Problem: Exact and Enumeration Algorithms

We present a time algorithm finding a minimum feedback vertex set in an undirected graph on n vertices. We also prove that a graph on n vertices can contain at most 1.8638 ⁿ minimal feedback vertex sets and that there exist graphs having 105 ^n/10≈1.5926 ⁿ minimal feedback vertex sets. Preliminary extended abstracts of this paper appeared in the proceedings of SWAT’06 [29] and IWPEC’06 [18]. Additional support of F.V. Fomin, S. Gaspers and A.V. Pyatkin by the Research Council of Norway. The work of A.V. Pyatkin was partially supported by grants of the Russian Foundation for Basic Research (project code 05-01-00395), INTAS (project code 04–77–7173). I. Razgon is supported by Science Foundation Ireland (Grant Number 05/IN/I886).     

A Technique to Speed Up Parallel Fully Dynamic Algorithms for MST

We provide a new EREW PRAM algorithm to maintain the minimum spanning tree (MST) of an undirected weighted graph. Our approach combines the sparsification data structure with a novel parallel technique which efficiently treats single edge deletions. The proposed parallel algorithm requires O(log n) time and O(n^2/3 log(m/n)) work, where n and m are respectively the number of nodes and edges of the given graph.     

Finding a dominating set on bipartite graphs

Finding a dominating set of minimum cardinality is an NP-hard graph problem, even when the graph is bipartite. In this paper we are interested in solving the problem on graphs having a large independent set. Given a graph G with an independent set of size z, we show that the problem can be solved in time O^∗(²n−z), where n is the number of vertices of G. As a consequence, our algorithm is able to solve the dominating set problem on bipartite graphs in time O^∗(²n/2). Another implication is an algorithm for general graphs whose running time is O(n^1.7088).     

An Approximation Algorithm for the Minimum Co-Path Set Problem

We present an approximation algorithm for the problem of finding a minimum set of edges in a given graph G whose removal from G leaves a graph in which each connected component is a path. It achieves a ratio of \frac 107\frac {10}{7} and runs in O(n ^1.5) time, where n is the number of vertices in the input graph. The previously best approximation algorithm for this problem achieves a ratio of 2 and runs in O(n ²) time.     

A refined search tree technique for Dominating Set on planar graphs

We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8^kn) and O(8^kk+n³), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time data-reduction rules for a more general “annotated” problem on black/white graphs that asks for a set of k vertices (black or white) that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.     

Simple and Improved Parameterized Algorithms for Multiterminal Cuts

Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min?(n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: $O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m))Given a graph G=(V,E) with n vertices and m edges, and a subset T of k vertices called terminals, the Edge (respectively, Vertex) Multiterminal Cut problem is to find a set of at most l edges (non-terminal vertices), whose removal from G separates each terminal from all the others. These two problems are NP-hard for k≥3 but well-known to be polynomial-time solvable for k=2 by the flow technique. In this paper, based on a notion farthest minimum isolating cut, we design several simple and improved algorithms for Multiterminal Cut. We show that Edge Multiterminal Cut can be solved in O(2 ^l kT(n,m)) time and Vertex Multiterminal Cut can be solved in O(k ^l T(n,m)) time, where T(n,m)=O(min (n ^2/3,m ^1/2)m) is the running time of finding a minimum (s,t) cut in an unweighted graph. Furthermore, the running time bounds of our algorithms can be further reduced for small values of k: Edge 3-Terminal Cut can be solved in O(1.415 ^l T(n,m)) time, and Vertex {3,4,5,6}-Terminal Cuts can be solved in O(2.059 ^l T(n,m)), O(2.772 ^l T(n,m)), O(3.349 ^l T(n,m)) and O(3.857 ^l T(n,m)) time respectively. Our results on Multiterminal Cut can also be used to obtain faster algorithms for Multicut: O((min(?{2k},l)+1)^2k2^lT(n,m))O((\min(\sqrt{2k},l)+1)^{2k}2^{l}T(n,m)) -time algorithm for Edge Multicut and O((2k) ^k+l/2 T(n,m))-time algorithm for Vertex Multicut.     

Multicolorings of Series-Parallel Graphs

Let G be a graph, and let each vertex v of G have a positive integer weight ω(v). A multicoloring of G is to assign each vertex v a set of ω(v) colors so that any pair of adjacent vertices receive disjoint sets of colors. This paper presents an algorithm to find a multicoloring of a given series-parallel graph G with the minimum number of colors in time O(n W), where n is the number of vertices and W is the maximum weight of vertices in G.     

Exact Algorithms for <Emphasis Type="Italic">L</Emphasis>(2,1)-Labeling of Graphs

The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O ^*((k+1) ⁿ ) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006 ⁿ ). Furthermore we show that dynamic programming can be used to establish an O(3.8730 ⁿ ) algorithm to compute an optimal L(2,1)-labeling.     

Reduced constants for simple cycle graph separation

 If G is an n vertex maximal planar graph and δ≤1 3, then the vertex set of G can be partitioned into three sets A, B, C such that neither A nor B contains more than (1−δ)n vertices, no edge from G connects a vertex in A to a vertex in B, and C is a cycle in G containing no more than (√2δ+√2−2δ)√n+O(1) vertices. Specifically, when δ=1 3, the separator C is of size (√2/3+√4/3)√n+O(1), which is roughly 1.97√n. The constant 1.97 is an improvement over the best known so far result of Miller 2√2≈2.82. If non-negative weights adding to at most 1 are associated with the vertices of G, then the vertex set of G can be partitioned into three sets A, B, C such that neither A nor B has weight exceeding 1−δ, no edge from G connects a vertex in A to a vertex in B, and C is a simple cycle with no more than 2√n+O(1) vertices. Received: 8 September 1993/11 December 1995     

Efficient Algorithms for the Problems of Enumerating Cuts by Non-decreasing Weights

In this paper, we study the problems of enumerating cuts of a graph by non-decreasing weights. There are four problems, depending on whether the graph is directed or undirected, and on whether we consider all cuts of the graph or only s-t cuts for a given pair of vertices s,t. Efficient algorithms for these problems with [(O)\tilde](n²m)\tilde{O}(n^{2}m) delay between two successive outputs have been known since 1992, due to Vazirani and Yannakakis. In this paper, improved algorithms are presented. The delays of the presented algorithms are O(nmlog (n ²/m)). Vazirani and Yannakakis’s algorithms have been used as basic subroutines in the solutions of many problems. Therefore, our improvement immediately reduces the running time of these solutions. For example, for the minimum k-cut problem, the upper bound is immediately reduced by a factor of [(O)\tilde](n)\tilde{O}(n) for k=3,4,5,6.     

Strong-Diameter Decompositions of Minor Free Graphs

We provide the first sparse covers and probabilistic partitions for graphs excluding a fixed minor that have strong diameter bounds; i.e. each set of the cover/partition has a small diameter as an induced sub-graph. Using these results we provide improved distributed name-independent routing schemes. Specifically, given a graph excluding a minor on r vertices and a parameter ρ>0 we obtain the following results: (1) a polynomial algorithm that constructs a set of clusters such that each cluster has a strong-diameter of O(r ² ρ) and each vertex belongs to 2 ^O(r) r! clusters; (2) a name-independent routing scheme with a stretch of O(r ²), headers of O(log n+rlog r) bits, and tables of size 2 ^O(r) r! log ⁴ n/log log n bits; (3) a randomized algorithm that partitions the graph such that each cluster has strong-diameter O(r6 ^r ρ) and the probability an edge (u,v) is cut is O(r  d(u,v)/ρ).