Telikepalli Kavitha

Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and a priori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX-hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

Faster Algorithms for Approximate Distance Oracles and All-Pairs Small Stretch Paths

Let G = (V, E) be a weighted undirected graph with |V| = n and |E| = m. An estimate deltacirc(u,v) of the distance delta(u,v) in G between u,v isin V is said to be of stretch t iff delta(u,v) les deltacirc(u,v) les t middot delta(u,v). The most efficient algorithms known for computing small stretch distances in G are the approximate distance oracles of (M. Thorup and U. Zwick, 2005) and the three algorithms in (E. Cohen and U. Zwick, 2001) to compute all-pairs stretch t distances for t = 2, 7/3, and 3. We present faster algorithms for these problems. For any integer k ges 1, Thorup and Zwick (2005) gave an O(kmn1k/) algorithm to construct a data structure of size O(kn1 + 1k/) which, given a query (u,v) isin V times V, returns in O(k) time, a 2k - 1 stretch estimate of delta(u, v). But for small values of k, the time to construct the oracle is rather high. Here we present an O(n2 log n) algorithm to construct such a data structure of size O(kn1+1k/) for all integers k ges 2. Our query answering time is O(k) for k > 2 and Theta (log n) for k = 2. We use a new generic scheme for all-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for all-pairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n2 log n) time

Additive spanners and (α, β)-spanners

An (α, β)-spanner of an unweighted graph <i>G</i> is a subgraph <i>H</i> that distorts distances in <i>G</i> up to a multiplicative factor of α and an additive term β. It is well known that any graph contains a (multiplicative) (2<i>k</i>−1, 0)-spanner of size <i>O</i>(<i>n</i>^1+1/<i>k</i>) and an (additive) (1,2)-spanner of size <i>O</i>(<i>n</i>^3/2). However no other additive spanners are known to exist. In this article we develop a couple of new techniques for constructing (α, β)-spanners. Our first result is an additive (1,6)-spanner of size <i>O</i>(<i>n</i>^4/3). The construction algorithm can be understood as an economical agent that assigns <i>costs</i> and <i>values</i> to paths in the graph, <i>purchasing</i> affordable paths and ignoring expensive ones, which are intuitively well approximated by paths already purchased. We show that this <i>path buying</i> algorithm can be parameterized in different ways to yield other sparseness-distortion tradeoffs. Our second result addresses the problem of which (α, β)-spanners can be computed efficiently, ideally in linear time. We show that, for any <i>k</i>, a (<i>k</i>,<i>k</i>−1)-spanner with size <i>O</i>(<i>kn</i>^1+1/<i>k</i>) can be found in linear time, and, further, that in a distributed network the algorithm terminates in a constant number of rounds. Previous spanner constructions with similar performance had roughly twice the multiplicative distortion.

A faster algorithm for minimum cycle basis of graphs

In this paper we consider the problem of computing a minimum cycle basis in a graph G with m edges and n vertices. The edges of G have non-negative weights on them. The previous best result for this problem was an O(m ω n) algorithm, where ω is the best exponent of matrix multiplication. It is presently known that ω 0, we also design a 1+e approximation algorithm to compute a cycle basis which is at most 1+e times the weight of a minimum cycle basis. The running time of this algorithm is \(O(\frac{m^{\omega}}{\epsilon}\log(W/{\epsilon}))\) for reasonably dense graphs, where W is the largest edge weight.

Assigning Papers to Referees

Refereed conferences require every submission to be reviewed by members of a program committee (PC) in charge of selecting the conference program. There are many software packages available to manage the review process. Typically, in a bidding phase PC members express their personal preferences by ranking the submissions. This information is used by the system to compute an assignment of the papers to referees (PC members).We study the problem of assigning papers to referees. We propose to optimize a number of criteria that aim at achieving fairness among referees/papers. Some of these variants can be solved optimally in polynomial time, while others are NP-hard, in which case we design approximation algorithms. Experimental results strongly suggest that the assignments computed by our algorithms are considerably better than those computed by popular conference management software.

Better lower bounds for locally decodable codes

An error-correcting code is said to be locally decodable if a randomized algorithm can recover any single bit of a message by reading only a small number of symbols of a possibly corrupted encoding of the message. Katz and Trevisan (2000) showed that any such code C: {0, 1} /spl rarr/ /spl Sigma//sup m/ with a decoding algorithm that makes at most q probes must satisfy m = /spl Omega/((n/log |/spl Sigma/|)/sup q/(q-1)/). They assumed that the decoding algorithm is non-adaptive, and left open the question of proving similar bounds for adaptive decoders. We improve the results of Katz and Trevisan (2000) in two ways. First, we give a more direct proof of their result. Second, and this is our main result, we prove that m = /spl Omega/((n/log|/spl Sigma/|)/sup q/(q-1)/) even if the decoding algorithm is adaptive. An important ingredient of our proof is a randomized method for smoothing an adaptive decoding algorithm. The main technical tool we employ is the Second Moment Method.

Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

We present two online algorithms for maintaining a topological order of a directed <i>n</i>-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles <i>m</i> arc additions in O(<i>m</i>^3/2) time. For sparse graphs (<i>m</i>/<i>n</i> = O(1)), this bound improves the best previous bound by a logarithmic factor, and is tight to within a constant factor among algorithms satisfying a natural locality property. Our second algorithm handles an arbitrary sequence of arc additions in O(<i>n</i>^5/2) time. For sufficiently dense graphs, this bound improves the best previous bound by a polynomial factor. Our bound may be far from tight: we show that the algorithm can take Ω(<i>n</i>²2^{√2 lg} <i>n</i>) time by relating its performance to a generalization of the <i>k</i>-levels problem of combinatorial geometry. A completely different algorithm running in Θ(<i>n</i>² log <i>n</i>) time was given recently by Bender, Fineman, and Gilbert. We extend both of our algorithms to the maintenance of strong components, without affecting the asymptotic time bounds.

Linear time algorithms for Abelian group isomorphism and related problems

We consider the problem of determining if two finite groups are isomorphic. The groups are assumed to be represented by their multiplication tables. We present an O(n) algorithm that determines if two Abelian groups with n elements each are isomorphic. This improves upon the previous upper bound of O(nlogn) [Narayan Vikas, An O(n) algorithm for Abelian p-group isomorphism and an O(nlogn) algorithm for Abelian group isomorphism, J. Comput. System Sci. 53 (1996) 1-9] known for this problem. We solve a more general problem of computing the orders of all the elements of any group (not necessarily Abelian) of size n in O(n) time. Our algorithm for isomorphism testing of Abelian groups follows from this result. We use the property that our order finding algorithm works for any group to design a simple O(n) algorithm for testing whether a group of size n, described by its multiplication table, is nilpotent. We also give an O(n) algorithm for determining if a group of size n, described by its multiplication table, is Abelian.

Bounded Unpopularity Matchings

We investigate the following problem: given a set of jobs and a set of people with preferences over the jobs, what is the optimal way of matching people to jobs? Here we consider the notion of popularity. A matching Mis popular if there is no matching Mi¾? such that more people prefer Mi¾? to Mthan the other way around. Determining whether a given instance admits a popular matching and, if so, finding one, was studied in [2]. If there is no popular matching, a reasonable substitute is a matching whose unpopularityis bounded. We consider two measures of unpopularity - unpopularity factordenoted by u(M) and unpopularity margindenoted by g(M). McCutchen recently showed that computing a matching Mwith the minimum value of u(M) or g(M) is NP-hard, and that if Gdoes not admit a popular matching, then we have u(M) i¾? 2 for all matchings Min G. Here we show that a matching Mthat achieves u(M) = 2 can be computed in

Faster Algorithms for All-pairs Approximate Shortest Paths in Undirected Graphs

O(m\sqrt{n})