We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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

We say an algorithm on n by n matrices with entries in [-M, M] (or n-node graphs with edge weights from [-M, M]) is truly sub cubic if it runs in O(n^{3-\delta} \poly(\log M)) time for some \delta > 0. We define a notion of sub cubic reducibility, and show that many important problems on graphs and matrices solvable in O(n^3) time are equivalent under sub cubic reductions. Namely, the following weighted problems either all have truly sub cubic algorithms, or none of them do: - The all-pairs shortest paths problem (APSP). - Detecting if a weighted graph has a triangle of negative total edge weight. - Listing up to n^{2.99} negative triangles in an edge-weighted graph. - Finding a minimum weight cycle in a graph of non-negative edge weights. - The replacement paths problem in an edge-weighted digraph. - Finding the second shortest simple path between two nodes in an edge-weighted digraph. - Checking whether a given matrix defines a metric. - Verifying the correctness of a matrix product over the (\min, +)-semiring. Therefore, if APSP cannot be solved in n^{3-\eps} time for any \eps > 0, then many other problems also need essentially cubic time. In fact we show generic equivalences between matrix products over a large class of algebraic structures used in optimization, verifying a matrix product over the same structure, and corresponding triangle detection problems over the structure. These equivalences simplify prior work on sub cubic algorithms for all-pairs path problems, since it now suffices to give appropriate sub cubic triangle detection algorithms. Other consequences of our work are new combinatorial approaches to Boolean matrix multiplication over the (OR, AND)-semiring (abbreviated as BMM). We show that practical advances in triangle detection would imply practical BMM algorithms, among other results. Building on our techniques, we give two new BMM algorithms: a derandomization of the recent combinatorial BMM algorithm of Bansal and Williams (FOCS'09), and an improved quantum algorithm for BMM.

/pdf/subcubic-equivalences-between-path-matrix-and-triangle-1813k5gg2h.pdf

Subcubic Equivalences between Path, Matrix and Triangle Problems

The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of ~O(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] presented an algorithm that computes in ~O(m√ n + n2) time an estimate D for the diameter D, such that ⌊ 2/3 D ⌋ ≤ ^D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years.Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et. al, producing an algorithm with the same estimate but with an expected running time of ~O(m√ n). We thus show that for all sparse enough graphs, the diameter can be 3/2-approximated in o(n2) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node.We also provide strong evidence that our diameter approximation result may be hard to improve. We show that if for some constant e>0 there is an O(m2-e) time (3/2-e)-approximation algorithm for the diameter of undirected unweighted graphs, then there is an O*( (2-δ)n) time algorithm for CNF-SAT on n variables for constant δ>0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false.Motivated by this negative result, we give several improved diameter approximation algorithms for special cases. We show for instance that for unweighted graphs of constant diameter D not divisible by 3, there is an O(m2-e) time algorithm that gives a (3/2-e) approximation for constant e>0. This is interesting since the diameter approximation problem is hardest to solve for small D.

/pdf/fast-approximation-algorithms-for-the-diameter-and-radius-of-87jak6ngfp.pdf

Fast approximation algorithms for the diameter and radius of sparse graphs

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(n2 -- aepsi;) time for some aepsi; > 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 -- aepsi;)npolyn) time for some aepsi; > 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3 -- aepsi;) time for some aepsi; > 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3 -- aepsi;) time combinatorial algorithm for n × 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 pa#x0103;traa#x015F;cu [STOC 2010].

/pdf/popular-conjectures-imply-strong-lower-bounds-for-dynamic-9eungloh1d.pdf

Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems

Maximum cardinality matching is a fundamental algorithmic problem with many algorithms and applications. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general n-vertex m-edge graphs) fall into two groups: there are fast (mostly randomized) algorithms that achieve a 2-approximation or worse, and there are slow algorithms with Ω([EQUATION]) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm that achieves a tradeoff between these two: a o([EQUATION]) update time and a better-than-2 approximation simultaneously. We answer this question in the affirmative. Previously, such bounds were only known for the special case of bipartite graphs.Our main result is a fully dynamic deterministic algorithm that maintains a (3/2 + e)-approximation in amortized update time O(m1/4e--2.5). In addition to achieving the trade-off described above, our algorithm manages to be polynomially faster than all existing deterministic algorithms (excluding an existing log n-approximation of Onak and Rubinfeld), while still maintaining a better-than-2 approximation.We also give stronger results for graphs whose arboricity is at most α. We show how to maintain a (1 + e)-approximate fractional matching or a (3/2 + e)-approximate integral matching in worst-case time O(α(α + log n)) for constant e. When the arboricity is constant, this bound is O(log n) and when the arboricity is polylogarithmic the update time is also polylogarithmic. Previous results for small arboricity non-bipartite graphs could only maintain a maximal matching (2-approximation).We maintain the approximate matching without explicitly using augmenting paths. We define an intermediate graph, called an EDCS and show that the EDCS H contains a large matching, and show how to maintain an EDCS in G. The EDCS was used in previous works on bipartite graphs, however the details and proofs are completely different in general graphs. The algorithm for bipartite graphs relies on ideas from flows and cuts to non-constructively prove the existence of a good matching in H, but these ideas do not seem to extend to non-bipartite graphs. In this paper we instead explicitly construct a large fractional matching in H. In some cases we can guarantee that this fractional matching is γ-restricted, which means that it only uses values either in the range [0, γ] or 1. We then combine this matching with a new structural property of maximum matchings in non-bipartite graphs, which is analogous to the cut induced by maximum matchings in bipartite graphs.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611974331.ch50

Faster fully dynamic matchings with small approximation ratios

We present an improved oracle for the distance sensitivity problem. The goal is to preprocess a directed graph G = (V,E) with non-negative edge weights to answer queries of the form: what is the length of the shortest path from x to y that does not go through some failed vertex or edge f. The previous best algorithm produces an oracle of size ~O(n2) that has an O(1) query time, and an ~O(n2√m) construction time. It was a randomized Monte Carlo algorithm that worked with high probability. Our oracle also has a constant query time and an ~O(n2) space requirement, but it has an improved construction time of ~O(mn), and it is deterministic. Note that O(1) query, O(n2) space, and O(mn) construction time is also the best known bound (up to logarithmic factors) for the simpler problem of finding all pairs shortest paths in a weighted, directed graph. Thus, barring improved solutions to the all pairs shortest path problem, our oracle is optimal up to logarithmic factors.

/pdf/a-nearly-optimal-oracle-for-avoiding-failed-vertices-and-2dgoxl1wuu.pdf

A nearly optimal oracle for avoiding failed vertices and edges

For any fixed $1 ≫ \eps ≫ 0$ we present a fully dynamic algorithm for maintaining $(2 + \eps)$-approximate all-pairs shortest paths in undirected graphs with positive edge weights. We use a randomized (Las Vegas) update algorithm (but a deterministic query procedure), so the time given is the \emph{expected} amortized update time. \\\indent Our query time $O(\log \log \log n)$. The update time is $\widetilde{O}(mn^{O(1/\sqrt{\log n})}\log(nR))$, where $R$ is the ratio between the heaviest and the lightest edge weight in the graph (so $R = 1$ in unweighted graphs). Unfortunately, the update time does have the drawback of a super-polynomial dependence on $\eps$: it grows as $(3 / \eps)^{\sqrt{\log n / \log(3/\eps)}} = n^{\sqrt{\log(3/\eps) / \log n}}$.\\\indent Our algorithm has a significantly faster update time than any other algorithm with sub-polynomial query time. For exact distances, the state of the art algorithm has an update time of $\widetilde{O}(n^2)$. For approximate distances, the best previous algorithm has a $O(kmn^{1/k})$ update time and returns $(2k - 1)$ stretch paths. Thus, it needs an update time of $O(m\sqrt{n})$ to get close to our approximation, and it has to return $O(\sqrt{\log n})$ approximate distances to match our update time.

/pdf/fully-dynamic-2-epsilon-approximate-all-pairs-shortest-paths-25ske7gbio.pdf

Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time

We present the first dynamic shortest paths algorithms that make any progress beyond a long-standing O(n) update time barrier (while maintaining a reasonable query time), although it is only progress for not-too-sparse graphs. In particular, we obtain new decremental algorithms for two approximate shortest-path problems in unweighted, undirected graphs. Both algorithms are randomized (Las Vegas).• Given a source s, we present an algorithm that maintains (1 + e)-approximate shortest paths from s with an expected total update time of O(n2+O(1/√log n)) over all deletions (so the amortized time is about O(n2/m)). The worst-case query time is constant. The best previous result goes back three decades to Even and Shiloach [16] and Dinitz [12]. They show how to decrementally maintain an exact shortest path tree with a total update time of O(mn) (amortized update time O(n)). Roditty and Zwick [22] have shown that O(mn) is actually optimal for exact paths (barring a better combinatorial algorithm for boolean matrix multiplication), unless we are willing to settle for a Ω(n) query time. In fact, until now, even approximate dynamic algorithms were not able to go beyond O(mn).• For any fixed integer k ≥ 2, we present an algorithm that decrementally maintains a distance oracle (for all pairs shortest distances) with a total expected update time of O(n2+1/k+O(1/√log n)) (amortized update time about O(n2+1/k/m). The space requirement is only O(m + n1+1/k), the stretch of the returned distances is at most 2k − 1 + e, and the worst-case query time is O(1). The best previous result of Roditty and Zwick [21] had a total update time of O(mn) and a stretch of 2k − 1. Note that our algorithm implicitly solves the decremental all-pairs shortest path problem with the same bounds; the best previous approximation algorithm of Roditty and Zwick [21] returned (1 + e) approximate distances, but used O(n2) space, and required O(mn) total update time. As with the previous problem, our algorithm is the first to make progress beyond the O(mn) total update time barrier while maintaining a small query time.We present a general framework for accelerating decremental algorithms. In particular, our main idea is to run existing decremental algorithms on a sparse subgraph (such as a spanner or emulator) of the graph rather than on the original graph G. Although this is a common approach for static approximate shortest-path problems, it has never been used in a decremental setting because maintaining the subgraph H as edges are being deleted from G might require inserting edges into H, thus ruining the "decrementality" of the setting. We overcome this by presenting an emulator whose maintenance only requires a limited number of well-behaved insertions.In other words, we present a general technique for running decremental algorithms on a sparse subgraph of the graph. Once our framework is in place, applying it to any particular decremental algorithm only requires trivial modifications; most of the work consists of showing that these algorithms as they are still work in our restricted fully dynamic setting, where we encounter not just arbitrary deletions (as in the original setting), but also restricted insertions.

http://theory.stanford.edu/~virgi/cs267/papers/rodittybernstein-apxsssp.pdf

Improved dynamic algorithms for maintaining approximate shortest paths under deletions

We present an improved algorithm for maintaining all-pairs $(1 + \epsilon)$ approximate shortest paths under deletions and weight-increases. The previous state of the art for this problem is total update time $\widetilde{O}(n^2\sqrt{m}/\epsilon)$ over all updates for directed unweighted graphs [S. Baswana, R. Hariharan, and S. Sen, J. Algorithms, 62 (2007), pp. 74--92], and $\widetilde{O}(mn/\epsilon)$ for undirected unweighted graphs [L. Roditty and U. Zwick, in Proceedings of the 45th FOCS, Rome, Italy, 2004, pp. 499--508]. Both algorithms are randomized and have constant query time. Very recently, Henzinger, Krinninger, and Nanongkai presented a deterministic version of the latter algorithm [M. Henzinger, S. Krinninger, and D. Nanongkai, in IEEE FOCS, 2013, pp. 538--547]. Note that $\widetilde{O}(mn)$ is a natural barrier because even with a $(1 + \epsilon)$ approximation, there is no $o(mn)$ combinatorial algorithm for the static all-pairs shortest path problem. Our algorithm works on directed weighte...

Aaron Bernstein

Papers

Faster fully dynamic matchings with small approximation ratios

A nearly optimal oracle for avoiding failed vertices and edges

Fully Dynamic (2 + epsilon) Approximate All-Pairs Shortest Paths with Fast Query and Close to Linear Update Time

Improved dynamic algorithms for maintaining approximate shortest paths under deletions

Maintaining Shortest Paths Under Deletions in Weighted Directed Graphs