We consider several well-studied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 
1. Is the 3SUM problem on $n$ numbers in $O(n^{2-\epsilon})$ time for some $\epsilon>0$? 
2. Can one determine the satisfiability of a CNF formula on $n$ variables in $O((2-\epsilon)^n poly n)$ time for some $\epsilon>0$? 
3. Is the All Pairs Shortest Paths problem for graphs on $n$ vertices in $O(n^{3-\epsilon})$ time for some $\epsilon>0$? 
4. Is there a linear time algorithm that detects whether a given graph contains a triangle? 
5. Is there an $O(n^{3-\epsilon})$ time combinatorial algorithm for $n\times n$ Boolean matrix multiplication? 
The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh's problem defined in a recent paper by Patrascu [STOC 2010].

Popular conjectures imply strong lower bounds for dynamic problems

Consider the following Online Boolean Matrix-Vector Multiplication problem: We are given an n x n matrix M and will receive n column-vectors of size n, denoted by v1, ..., vn, one by one. After seeing each vector vi, we have to output the product Mvi before we can see the next vector. A naive algorithm can solve this problem using O(n3) time in total, and its running time can be slightly improved to O(n3/log2 n) [Williams SODA'07]. We show that a conjecture that there is no truly subcubic (O(n3-e)) time algorithm for this problem can be used to exhibit the underlying polynomial time hardness shared by many dynamic problems. For a number of problems, such as subgraph connectivity, Pagh's problem, d-failure connectivity, decremental single-source shortest paths, and decremental transitive closure, this conjecture implies tight hardness results. Thus, proving or disproving this conjecture will be very interesting as it will either imply several tight unconditional lower bounds or break through a common barrier that blocks progress with these problems. This conjecture might also be considered as strong evidence against any further improvement for these problems since refuting it will imply a major breakthrough for combinatorial Boolean matrix multiplication and other long-standing problems if the term "combinatorial algorithms" is interpreted as "Strassen-like algorithms" [Ballard et al. SPAA'11].The conjecture also leads to hardness results for problems that were previously based on diverse problems and conjectures -- such as 3SUM, combinatorial Boolean matrix multiplication, triangle detection, and multiphase -- thus providing a uniform way to prove polynomial hardness results for dynamic algorithms; some of the new proofs are also simpler or even become trivial. The conjecture also leads to stronger and new, non-trivial, hardness results, e.g., for the fully-dynamic densest subgraph and diameter problems.

/pdf/unifying-and-strengthening-hardness-for-dynamic-problems-via-8z0ox0kv50.pdf

Unifying and Strengthening Hardness for Dynamic Problems via the Online Matrix-Vector Multiplication Conjecture

The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

Rainbow connections of graphs -- A survey

The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions.

Rainbow Connections of Graphs: A Survey

Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the APSP conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining "less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including: Tight n3-o(1) lower bounds for purely-combinatorial problems about the triangles in unweighted graphs. New n1-o(1) lower bounds for the amortized update and query times of dynamic algorithms for single-source reachability, strongly connected components, and Max-Flow. New n1.5-o(1) lower bound for computing a set of n st-maximum-flow values in a directed graph with n nodes and ~O(n) edges. There is a hierarchy of natural graph problems on n nodes with complexity nc for c ∈ (2,3).Only slightly non-trivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source-Max-Flow problem.

/pdf/matching-triangles-and-basing-hardness-on-an-extremely-4j9f8d91wx.pdf

Matching Triangles and Basing Hardness on an Extremely Popular Conjecture

We present two online algorithms for maintaining a topological order of a directed n-vertex acyclic graph as arcs are added, and detecting a cycle when one is created. Our first algorithm handles m arc additions in O(m3/2) time. For sparse graphs (m/n = 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(n5/2) 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(n22√2 lgn) time by relating its performance to a generalization of the k-levels problem of combinatorial geometry. A completely different algorithm running in Θ(n2 log n) 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.

/pdf/incremental-cycle-detection-topological-ordering-and-strong-36duidt7f2.pdf

Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

We present two online algorithms for maintaining a topological order of a directed acyclic graph as arcs are added, and detecting a cycle when one is created Our first algorithm takes O(m1/2) amortized time per arc and our second algorithm takes O(n25/m) amortized time per arc, where nis the number of vertices and mis the total number of arcs For sparse graphs, our O(m1/2) bound improves the best previous bound by a factor of lognand is tight to within a constant factor for a natural class of algorithms that includes all the existing ones Our main insight is that the two-way search method of previous algorithms does not require an ordered search, but can be more general, allowing us to avoid the use of heaps (priority queues) Instead, the deterministic version of our algorithm uses (approximate) median-finding; the randomized version of our algorithm uses uniform random sampling For dense graphs, our O(n25/m) bound improves the best previously published bound by a factor of n1/4and a recent bound obtained independently of our work by a factor of logn Our main insight is that graph search is wasteful when the graph is dense and can be avoided by searching the topological order space instead Our algorithms extend to the maintenance of strong components, in the same asymptotic time bounds

Faster Algorithms for Incremental Topological Ordering

The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges, so that every pair of vertices is connected by at least one path in which no two edges are colored the same. Our main result is that $rc(G)\leq \lceil\frac{n}{2}\rceil$ for any 2-connected graph with at least three vertices. We conjecture that $rc(G)\leq n/\kappa+C$ for a $\kappa$-connected graph $G$ of order $n$, where $C$ is a constant, and prove the conjecture for certain classes of graphs. We also prove that $rc(G)\leq(2+\varepsilon)n/\kappa+23/\varepsilon^2$ for any $\varepsilon>0$.

/pdf/rainbow-connection-number-and-connectivity-397r4r8gv0.pdf

Rainbow Connection Number and Connectivity

Rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colours needed to colour its edges, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we investigate the relationship of rainbow connection number with vertex and edge connectivity. It is already known that for a connected graph with minimum degree $\delta$, the rainbow connection number is upper bounded by $3n/(\delta + 1) + 3$ [Chandran et al., 2010]. This directly gives an upper bound of $3n/(\lambda + 1) + 3$ and $3n/(\kappa + 1) + 3$ for rainbow connection number where $\lambda$ and $\kappa$, respectively, denote the edge and vertex connectivity of the graph. We show that the above bound in terms of edge connectivity is tight up-to additive constants and show that the bound in terms of vertex connectivity can be improved to $(2 + \epsilon)n/\kappa + 23/ \epsilon^2$, for any $\epsilon > 0$. We conjecture that rainbow connection number is upper bounded by $n/\kappa + O(1)$ and show that it is true for $\kappa = 2$. We also show that the conjecture is true for chordal graphs and graphs of girth at least 7.

Rogers Mathew

Papers

Incremental Cycle Detection, Topological Ordering, and Strong Component Maintenance

Faster Algorithms for Incremental Topological Ordering

Rainbow Connection Number and Connectivity

On Rainbow Connection Number and Connectivity

Boxicity of line graphs