We present the first data structures that maintain near optimal maximum cardinality and maximum weighted matchings on sparse graphs in sub linear time per update. Our main result is a data structure that maintains a (1+ϵ) approximation of maximum matching under edge insertions/deletions in worst case O(√mϵ-2) time per update. This improves the 3/2 approximation given by Neiman and Solomon [20] which runs in similar time. The result is based on two ideas. The first is to re-run a static algorithm after a chosen number of updates to ensure approximation guarantees. The second is to judiciously trim the graph to a smaller equivalent one whenever possible. We also study extensions of our approach to the weighted setting, and combine it with known frameworks to obtain arbitrary approximation ratios. For a constant ϵ and for graphs with edge weights between 1 and N, we design an algorithm that maintains an (1+ϵ) approximate maximum weighted matching in O(√m log N) time per update. The only previous result for maintaining weighted matchings on dynamic graphs has an approximation ratio of 4.9108, and was shown by An and et al. [2], [3].

Fully Dynamic (1+ e)-Approximate Matchings

We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^{o(1)})} worst-case} update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen \cite{Wulff-Nilsen16a} with update time O(n^{0.5-&#x2265;ilon}) for some constant &#x2265;ilon 0 and, independently, by Nanongkai and Saranurak \cite{NanongkaiS16} with update time O(n^{0.494}) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^{0.5-&#x2265;ilon}) in \cite{Wulff-Nilsen16a} to O(n^{o(1)}) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King \cite{HenzingerK97b} and Holm et al. \cite{HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in \cite{Wulff-Nilsen16a, NanongkaiS16} which is based on Fredericksons 2-dimensional topology tree \cite{Frederickson85} and illustrates a new application to this old technique.

Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time

In this paper we consider the decremental single-source shortest paths (SSSP) problem, where given a graph G and a source node s the goal is to maintain shortest paths between s and all other nodes in G under a sequence of online adversarial edge deletions. In their seminal work, Even and Shiloach [JACM 1981] presented an exact solution to the problem with only O(mn) total update time over all edge deletions. Their classic algorithm was the best known result for the decremental SSSP problem for three decades, even when approximate shortest paths are allowed. The first improvement over the Even-Shiloach algorithm was given by Bernstein and Roditty [SODA 2011], who for the case of an unweighted and undirected graph presented an approximate (1+) algorithm with constant query time and a total update time of O(n2+O(1/√logn)). This work triggered a series of new results, culminating in a recent breakthrough of Henzinger, Krinninger and Nanongkai [FOCS 14], who presented a -approximate algorithm whose total update time is near linear O(m1+ O(1/√logn)). In this paper they posed as a major open problem the question of derandomizing their result. In fact, all known improvements over the Even-Shiloach algorithm are randomized. All these algorithms maintain some truncated shortest path trees from a small subset of nodes. While in the randomized setting it is possible to “hide” these nodes from the adversary, in the deterministic setting this is impossible: the adversary can delete all edges touching these nodes, thus forcing the algorithm to choose a new set of nodes and incur a new computation of shortest paths. In this paper we present the first deterministic decremental SSSP algorithm that breaks the Even-Shiloach bound of O(mn) total update time, for unweighted and undirected graphs. Our algorithm is (1 + є) approximate and achieves a total update time of O(n2). Our algorithm can also achieve the same bounds in the incremental setting. It is worth mentioning that for dense instances where m = Ω(n2 − 1/√log(n)), our algorithm is also faster than all existing randomized algorithms.

Deterministic decremental single source shortest paths: beyond the o(mn) bound

Graph spanners: A tutorial review

We present an algorithm for maintaining maximal matching in a graph under addition and deletion of edges. Our data structure is randomized that takes O(log n) expected amortized time for each edge update where n is the number of vertices in the graph. While there is a trivial O(n) algorithm for edge update, the previous best known result for this problem for a graph with n vertices and m edges is O({(n+ m)}^{0.7072})which is sub-linear only for a sparse graph. 
For the related problem of maximum matching, Onak and Rubinfield designed a randomized data structure that achieves O(log^2 n) amortized time for each update for maintaining a c-approximate maximum matching for some large constant c. In contrast, we can maintain a factor two approximate maximum matching in O(log n) expected time per update as a direct corollary of the maximal matching scheme. This in turn also implies a two approximate vertex cover maintenance scheme that takes O(log n) expected time per update.

Fully dynamic maximal matching in O(log n) update time

Spanner of an undirected graph G = (V,E) is a subgraph that is sparse and yet preserves all-pairs distances approximately. More formally, a spanner with stretcht ∈ N is a subgraph (V,ES), ES ⊆ E such that the distance between any two vertices in the subgraph is at most t times their distance in G. Though G is trivially a t-spanner of itself, the research as well as applications of spanners invariably deal with a t-spanner that has as small number of edges as possible.We present fully dynamic algorithms for maintaining spanners in centralized as well as synchronized distributed environments. These algorithms are designed for undirected unweighted graphs and use randomization in a crucial manner.Our algorithms significantly improve the existing fully dynamic algorithms for graph spanners. The expected size (number of edges) of a t-spanner maintained at each stage by our algorithms matches, up to a polylogarithmic factor, the worst case optimal size of a t-spanner. The expected amortized time (or messages communicated in distributed environment) to process a single insertion/deletion of an edge by our algorithms is close to optimal.

/pdf/fully-dynamic-randomized-algorithms-for-graph-spanners-1offvml987.pdf

Fully dynamic randomized algorithms for graph spanners

Purpose: This paper aims to explore ten years of data on net premiums received and incurred claims of select health insurance companies and compare it with public sector total health insurance. 
Design/methodology/approach: It is an exploratory paper comparing year-on-year premiums received and incurred claims of health during the period 2011-12 to 2020-21. ( Indian financial year start from April to March). Data has been shown in charts with trend lines without constants. Further, the Compounded Annual Growth Rate ( CAGR) has been calculated using log-linear regression. 
Findings:  The private sector is in business to earn profits, while the public sector is liberal in remitting the claims. Private-sector businesses earn more premiums than the public sector. However, the private sector’s incurred claims are comparatively much less than the public sector’s. The private sector is increasing business, but public sector growth is falling. The difference between year-on-year net premiums and incurred claims of the public sector is decreasing at 6.95%, but the same is increasing in the private sector at 19.63%. However, it cannot be generalised that the private sector is taking over the public sector business. The public is playing its social responsibility with accountability. However, the private sector aims to earn more earnings.

/pdf/analytical-study-of-incurred-claims-ratio-of-private-health-1e4h2zhu.pdf

Sumeet Khurana

Papers

Fully dynamic randomized algorithms for graph spanners

Analytical Study of Incurred Claims Ratio of Private Health Insurers