An algorithm for partitioning trees augmented with sibling edges

We investigate a special case of the graph partitioning problem: the partitioning of a sibling graph which is an ordered tree augmented with edges connecting consecutive nodes that share a common parent. We describe the algorithm, XS, and present a proof of its correctness.     

A tree-covering problem arising in integrity of tree-structured data

We introduce and solve a problem motivated by integrity verification in third-party data distribution: Given an undirected tree, find a minimum-cardinality set of simple paths that cover all the tree edges and, secondarily, have smallest total path lengths. We give a linear time algorithm for this problem.     

Efficient algorithms for clique problems

The k-clique problem is a cornerstone of NP-completeness and parametrized complexity. When k is a fixed constant, the asymptotically fastest known algorithm for finding a k-clique in an n-node graph runs in O(n0.792k) time (given by Nešet?il and Poljak). However, this algorithm is infamously inapplicable, as it relies on Coppersmith and Winograd's fast matrix multiplication.We present good combinatorial algorithms for solving k-clique problems. These algorithms do not require large constants in their runtime, they can be readily implemented in any reasonable random access model, and are very space-efficient compared to their algebraic counterparts. Our results are the following:

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We give an algorithm for k-clique that runs in O(nk/(εlogn)^k−1) time and O(nε) space, for all ε>0, on graphs with n nodes. This is the first algorithm to take o(nk) time and O(nc) space for c independent of k.

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Let k be even. Define a k-semiclique to be a k-node graph G that can be divided into two disjoint subgraphs U={u₁,…,uk/2} and V={v₁,…,vk/2} such that U and V are cliques, and for all i?j, the graph G contains the edge {ui,vj}. We give an time algorithm for determining if a graph has a k-semiclique. This yields an approximation algorithm for k-clique, in the following sense: if a given graph contains a k-clique, then our algorithm returns a subgraph with at least 3/4 of the edges in a k-clique.

     

Trading uninitialized space for time

The design of efficient graph algorithms usually precludes the test of edge existence, because an efficient support of that operation already requires time for the initialization of an adjacency-matrix representation. We describe an alternative representation of static directed graphs taking Θ(n+m) initialization time and using Θ(n²) space, which supports the efficient implementation of all usual operations on static graphs. The sparse graph representation allows the design of efficient graph algorithms using both iteration over all vertices adjacent with a given vertex and edge-existence operations, although at the expense of additional (uninitialized) space which may, nevertheless, be used for other purposes. To the best of our knowledge, the representation leads to the first graph algorithms with the disconcerting property that the time complexity is better than the space complexity.     

Efficient parallel algorithms for computing all pair shortest paths in directed graphs

We present parallel algorithms for computing all pair shortest paths in directed graphs. Our algorithm has time complexityO(f(n)/p+I(n)logn) on the PRAM usingp processors, whereI(n) is logn on the EREW PRAM, log logn on the CCRW PRAM,f(n) iso(n ³). On the randomized CRCW PRAM we are able to achieve time complexityO(n ³/p+logn) usingp processors. A preliminary version of this paper was presented at the 4th Annual ACM Symposium on Parallel Algorithms and Architectures, June 1992. Support by NSF Grant CCR 90-20690 and PSC CUNY Awards #661340 and #662478.     

On mimicking networks representing minimum terminal cuts

Given a capacitated undirected graph G=(V,E)$G=(V,E)$ with a set of terminals K⊂V$K\subset V$, a mimicking network   is a smaller graph H=(_VH,_EH)$H=(V_H,E_H)$ which contains the set of terminals K   and for every bipartition [U,K−U]$[U,K-U]$ of the terminals, the cost of the minimum cut separating U   from K−U$K-U$ in G is exactly equal to the cost of the minimum cut separating U   from K−U$K-U$ in H.     

The Pair Completion algorithm for the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e., a non-empty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G₁(V,E₁) and G₂(V,E₂), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V,ES), E₁⊆ES⊆E₂, which has a homogeneous set. This paper presents an algorithm that uses the concept of bias graph [S. Tang, F. Yeh, Y. Wang, An efficient algorithm for solving the homogeneous set sandwich problem, Inform. Process. Lett. 77 (2001) 17-22] to solve the problem in time, thus outperforming the other known HSSP deterministic algorithms for inputs where .     

On the longest path algorithm for reconstructing trees from distance matrices

Culberson and Rudnicki [J.C. Culberson, P. Rudnicki, A fast algorithm for constructing trees from distance matrices, Inform. Process. Lett. 30 (4) (1989) 215-220] gave an algorithm that reconstructs a degree d restricted tree from its distance matrix. According to their analysis, it runs in time O(dnlogdn) for topological trees. However, this turns out to be false; we show that the algorithm takes time in the topological case, giving tight examples.     

On the efficiency of maximum-flow algorithms on networks with small integer capacities

The performance of maximum-flow algoirthms that work in phases is studied as a function of the maximum arc capacity,C, of the network and a quantity we call thetotal potential, P, of the network, which is related to the average amount of flow that can be sent through a node. Extending results by Even and Tarjan, we derive a tightO(min{C ^1/3¦V¦^2/3,P ^1/2, ¦V¦}) upper bound on the number of phases. AnO(min{P log¦V¦,C¦V¦^3/2, ¦V¦²¦E¦}) upper bound is derived on the total length of the augmenting paths used by Dinic's algorithm. The latter quantity is useful in estimating the performance of Dinic's method on certain inputs. Our results show that on a natural class of networks, the performance of Dinic's algorithm is significantly better than would be apparent from a bound based on ¦V¦ and ¦E¦ alone. We present an application of our bounds to the maximum subgraph density problem.     

Dominating set based exact algorithms for 3-coloring

We show that the 3-colorability problem can be solved in O(n^1.296) time on any n-vertex graph with minimum degree at least 15. This algorithm is obtained by constructing a dominating set of the graph greedily, enumerating all possible 3-colorings of the dominating set, and then solving the resulting 2-list coloring instances in polynomial time. We also show that a 3-coloring can be obtained in ²o(n) time for graphs having minimum degree at least ω(n) where ω(n) is any function which goes to ∞. We also show that if the lower bound on minimum degree is replaced by a constant (however large it may be), then neither a ²o(n) time nor a ²o(m) time algorithm is possible (m denotes the number of edges) for 3-colorability unless Exponential Time Hypothesis (ETH) fails. We also describe an algorithm which obtains a 4-coloring of a 3-colorable graph in O(n^1.2535) time.     

Improved deterministic approximation algorithms for Max TSP

We present an O(n³)-time approximation algorithm for the maximum traveling salesman problem whose approximation ratio is asymptotically , where n is the number of vertices in the input complete edge-weighted (undirected) graph. We also present an O(n³)-time approximation algorithm for the metric case of the problem whose approximation ratio is asymptotically . Both algorithms improve on the previous bests.     

A note on the online First-Fit algorithm for coloring k-inductive graphs

In a FOCS 1990 paper, S. Irani proved that the First-Fit online algorithm for coloring a graph uses at most O(klogn) colors for k-inductive graphs. In this note we provide a very short proof of this fact.     

Planar packing of trees and spider trees

In [A. García, C. Hernando, F. Hurtado, M. Noy, J. Tejel, Packing trees into planar graphs, J. Graph Theory (2002) 172-181] García et al. conjectured that for every two non-star trees there exists a planar graph containing them as edge-disjoint subgraphs. In this paper we prove the conjecture in the case in which one of the trees is a spider tree.     

Solving a system of difference constraints with variables restricted to a finite set

An algorithm is given for the solution of a system of difference constraints where the variables must take on values from a given finite set of real numbers.     

Approximate maximum weight branchings

We consider a special subgraph of a weighted directed graph: one comprising only the k heaviest edges incoming to each vertex. We show that the maximum weight branching in this subgraph closely approximates the maximum weight branching in the original graph. Specifically, it is within a factor of k/(k+1). Our interest in finding branchings in this subgraph is motivated by a data compression application in which calculating edge weights is expensive but estimating which are the heaviest k incoming edges is easy. An additional benefit is that since algorithms for finding branchings run in time linear in the number of edges our results imply faster algorithms although we sacrifice optimality by a small factor. We also extend our results to the case of edge-disjoint branchings of maximum weight and to maximum weight spanning forests.     

Fast algorithms for some dominating induced matching problems

We describe O(n)$O(n)$ time algorithms for finding the minimum weighted dominating induced matching of chordal, dually chordal, biconvex, and claw-free graphs. For the first three classes, we prove tight O(n)$O(n)$ bounds on the maximum number of edges that a graph having a dominating induced matching may contain. By applying these bounds, and employing existing O(n+m)$O(n+m)$ time algorithms we show that they can be reduced to O(n)$O(n)$ time. For claw-free graphs, we describe a variation of the existing algorithm for solving the unweighted version of the problem, which decreases its complexity from O(n²)$O(n^2)$ to O(n)$O(n)$, while additionally solving the weighted version. The same algorithm can be easily modified to count the number of DIM's of the given graph.     

A simple linear time certifying LBFS-based algorithm for recognizing trivially perfect graphs and their complements

We introduce a simple, linear time algorithm for recognizing trivially perfect (TP) graphs. It improves upon the algorithm of Yan et al. [J.-H. Yan, J.-J. Chen, G.J. Chang, Quasi-threshold graphs, Discrete Appl. Math. 69 (3) (1996) 247–255] in that it is certifying, producing a P₄ or a C₄ when the graph is not TP. In addition, our algorithm can be easily modified to recognize the complement of TP graphs (co-TP) in linear time as well. It is based on lexicographic BFS, and in particular the technique of partition refinement, which has been used in the recognition of many other graph classes [D.G. Corneil, Lexicographic breadth first search—a survey, in: WG 2004, in: Lecture Notes in Comput. Sci., vol. 3353, Springer, 2004, pp. 1–19].     

Simplified tight analysis of Johnson's algorithm

In their paper “Tight bound on Johnson's algorithm for maximum satisfiability” [J. Comput. System Sci. 58 (3) (1999) 622-640] Chen, Friesen and Zheng provided a tight bound on the approximation ratio of Johnson's algorithm for Maximum Satisfiability [J. Comput. System Sci. 9 (3) (1974) 256-278]. We give a simplified proof of their result and investigate to what extent it may be generalized to non-Boolean domains.