Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

In this paper we present a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity $\tilde O(n^{5/4})$, where $n$ denotes the number of vertices in the graph. This improves the previous upper bound $O(n^{9/7})=O(n^{1.285...})$ recently obtained by Lee, Magniez and Santha. Our result shows, for the first time, that in the quantum query complexity setting unweighted triangle finding is easier than its edge-weighted version, since for finding an edge-weighted triangle Belovs and Rosmanis proved that any quantum algorithm requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries. Our result also illustrates some limitations of the non-adaptive learning graph approach used to obtain the previous $O(n^{9/7})$ upper bound since, even over unweighted graphs, any quantum algorithm for triangle finding obtained using this approach requires $\Omega(n^{9/7}/\sqrt{\log n})$ queries as well. To bypass the obstacles characterized by these lower bounds, our quantum algorithm uses combinatorial ideas exploiting the graph-theoretic properties of triangle finding, which cannot be used when considering edge-weighted graphs or the non-adaptive learning graph approach.Comment: 17 pages, to appear in FOCS'14; v2: minor correction

Improved Time Bounds for All Pairs Non-decreasing Paths in General Digraphs

We present improved algorithms for solving the All Pairs Non-decreasing Paths (APNP) problem on weighted digraphs. Currently, the best upper bound on APNP is O~(n^{(9+omega)/4})=O(n^{2.844}), obtained by Vassilevska Williams [TALG 2010 and SODA\u2708], where omega<2.373 is the usual exponent of matrix multiplication. Our first algorithm improves the time bound to O~(n^{2+omega/3})=O(n^{2.791}). The algorithm determines, for every pair of vertices s, t, the minimum last edge weight on a non-decreasing path from s to t, where a non-decreasing path is a path on which the edge weights form a non-decreasing sequence. The algorithm proposed uses the combinatorial properties of non-decreasing paths. Also a slightly improved algorithm with running time O(n^{2.78}) is presented

Finding a heaviest vertex-weighted triangle is not harder than matrix multiplication

We show that a maximum-weight triangle in an undirected graph with n vertices and real weights assigned to vertices can be found in time O(n(omega) + n(2+o(1))), where omega is the exponent of the fastest matrix multiplication algorithm. By the currently best bound on omega, the running time of our algorithm is O(n(2.376)). Our algorithm substantially improves the previous time-bounds for this problem, and its asymptotic time complexity matches that of the fastest known algorithm for finding any triangle (not necessarily a maximum-weight one) in a graph. We can extend our algorithm to improve the upper bounds on finding a maximum-weight triangle in a sparse graph and on finding a maximum-weight subgraph isomorphic to a fixed graph. We can find a maximum-weight triangle in a vertex-weighted graph with m edges in asymptotic time required by the fastest algorithm for finding any triangle in a graph with m edges, i.e., in time O(m(1.41)). Our algorithms for a maximum-weight fixed subgraph (in particular any clique of constant size) are asymptotically as fast as the fastest known algorithms for a fixed subgraph

The NFA Acceptance Hypothesis: Non-Combinatorial and Dynamic Lower Bounds

We pose the fine-grained hardness hypothesis that the textbook algorithm for the NFA Acceptance problem is optimal up to subpolynomial factors, even for dense NFAs and fixed alphabets. We show that this barrier appears in many variations throughout the algorithmic literature by introducing a framework of Colored Walk problems. These yield fine-grained equivalent formulations of the NFA Acceptance problem as problems concerning detection of an $s$-$t$-walk with a prescribed color sequence in a given edge- or node-colored graph. For NFA Acceptance on sparse NFAs (or equivalently, Colored Walk in sparse graphs), a tight lower bound under the Strong Exponential Time Hypothesis has been rediscovered several times in recent years. We show that our hardness hypothesis, which concerns dense NFAs, has several interesting implications: - It gives a tight lower bound for Context-Free Language Reachability. This proves conditional optimality for the class of 2NPDA-complete problems, explaining the cubic bottleneck of interprocedural program analysis. - It gives a tight $(n+nm^{1/3})^{1-o(1)}$ lower bound for the Word Break problem on strings of length $n$ and dictionaries of total size $m$. - It implies the popular OMv hypothesis. Since the NFA acceptance problem is a static (i.e., non-dynamic) problem, this provides a static reason for the hardness of many dynamic problems. Thus, a proof of the NFA Acceptance hypothesis would resolve several interesting barriers. Conversely, a refutation of the NFA Acceptance hypothesis may lead the way to attacking the current barriers observed for Context-Free Language Reachability, the Word Break problem and the growing list of dynamic problems proven hard under the OMv hypothesis.Comment: 31 pages, Accepted at ITC

Open Problems in (Hyper)Graph Decomposition

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation

Even faster elastic-degenerate string matching via fast matrix multiplication

An elastic-degenerate (ED) string is a sequence of n sets of strings of total length N, which was recently proposed to model a set of similar sequences. The ED string matching (EDSM) problem is to find all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention in the combinatorial pattern matching community, and an O(nm1.5 √(log m) + N)-time algorithm is known [Aoyama et al., CPM 2018]. The standard assumption in the prior work on this question is that N is substantially larger than both n and m, and thus we would like to have a linear dependency on the former. Under this assumption, the natural open problem is whether we can decrease the 1.5 exponent in the time complexity, similarly as in the related (but, to the best of our knowledge, not equivalent) word break problem [Backurs and Indyk, FOCS 2016].Our starting point is a conditional lower bound for the EDSM problem. We use the popular combinatorial Boolean matrix multiplication (BMM) conjecture stating that there is no truly subcubic combinatorial algorithm for BMM [Abboud and Williams, FOCS 2014]. By designing an appropriate reduction we show that a combinatorial algorithm solving the EDSM problem in O(nm1.5−∊ + N) time, for any ∊ > 0, refutes this conjecture. Of course, the notion of combinatorial algorithms is not clearly defined, so our reduction should be understood as an indication that decreasing the exponent requires fast matrix multiplication.Two standard tools used in algorithms on strings are string periodicity and fast Fourier transform. Our main technical contribution is that we successfully combine these tools with fast matrix multiplication to design a non-combinatorial O(nm1.381 + N)-time algorithm for EDSM. To the best of our knowledge, we are the first to do so.</p

Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing

Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM