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1. The Complexity of Dynamic Data Race Prediction

   https://doi.org/10.1145/3373718.3394783  

   Mathur, Umang ; Pavlogiannis, Andreas ; Viswanathan, Mahesh ( January 2020 , IEEE Symposium on Logic in Computer Science)

   Writing concurrent programs is notoriously hard due to scheduling non-determinism. The most common concurrency bugs are data races, which are accesses to a shared resource that can be executed concurrently. Dynamic data-race prediction is the most standard technique for detecting data races: given an observed, data-race-free trace t, the task is to determine whether t can be reordered to a trace t* that exposes a data-race. Although the problem has received significant practical attention for over three decades, its complexity has remained elusive. In this work, we address this lacuna, identifying sources of intractability and conditions under which the problem is efficiently solvable. Given a trace t of size n over k threads, our main results are as follows. First, we establish a general O(k · n2·(k-1) upper-bound, as well as an O(nk) upper-bound when certain parameters of t are constant. In addition, we show that the problem is NP-hard and even W[1]-hard parameterized by k, and thus unlikely to be fixed-parameter tractable. Second, we study the problem over acyclic communication topologies, such as server-clients hierarchies. We establish an O(k2 · d · n2 · log n) upper-bound, where d is the number of shared variables accessed in t. In addition, we show that even for traces with k = 2 threads, the problem has no O(n2-ϵ) algorithm under the Orthogonal Vectors conjecture. Since any trace with 2 threads defines an acyclic topology, our upper-bound for this case is optimal up to polynomial improvements for up to moderate values of k and d. Finally, motivated by existing heuristics, we study a distance-bounded version of the problem, where the task is to expose a data race by a witness trace that is similar to t. We develop an algorithm that works in O(n) time when certain parameters of t are constant. 

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2. On the complexity of binary polynomial optimization over acyclic hypergraphs

   https://doi.org/10.1137/1.9781611977073.105  

   Alberto Del Pia and Silvia Di Gregorio ( January 2022 , Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   In this work we advance the understanding of the fundamental limits of computation for Binary Polynomial Optimization (BPO), which is the problem of maximizing a given polynomial function over all binary points. In our main result we provide a novel class of BPO that can be solved efficiently both from a theoretical and computational perspective. In fact, we give a strongly polynomial-time algorithm for instances whose corresponding hypergraph is β-acyclic. We note that the β-acyclicity assumption is natural in several applications including relational database schemes and the lifted multicut problem on trees. Due to the novelty of our proving technique, we obtain an algorithm which is interesting also from a practical viewpoint. This is because our algorithm is very simple to implement and the running time is a polynomial of very low degree in the number of nodes and edges of the hypergraph. Our result completely settles the computational complexity of BPO over acyclic hypergraphs, since the problem is NP-hard on α-acyclic instances. Our algorithm can also be applied to any general BPO problem that contains β-cycles. For these problems, the algorithm returns a smaller instance together with a rule to extend any optimal solution of the smaller instance to an optimal solution of the original instance. 

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3. GROUP, GRAPHS, ALGORITHMS: THE GRAPH ISOMORPHISM PROBLEM

   https://doi.org/10.1142/9789813272880_0183  

   BABAI, LÁSZLÓ ( May 2019 , Proceedings of the International Congress of Mathematicians, ICM 2018)

   Graph Isomorphism (GI) is one of a small number of natural algorithmic problems with unsettled complexity status in the P / NP theory: not expected to be NP-complete, yet not known to be solvable in polynomial time. Arguably, the GI problem boils down to filling the gap between symmetry and regularity, the former being defined in terms of automorphisms, the latter in terms of equations satisfied by numerical parameters. Recent progress on the complexity of GI relies on a combination of the asymptotic theory of permutation groups and asymptotic properties of highly regular combinatorial structures called coherent configurations. Group theory provides the tools to infer either global symmetry or global irregularity from local information, eliminating the symmetry/regularity gap in the relevant scenario; the resulting global structure is the subject of combinatorial analysis. These structural studies are melded in a divide-and-conquer algorithmic framework pioneered in the GI context by Eugene M. Luks (1980). 

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4. Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws

   Chudnovsky, M. with ( January 2022 , 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   For graphs G and H, we say that G is H-free if it does not contain H as an induced subgraph. Already in the early 1980s Alekseev observed that if H is connected, then the Max Weight Independent Set problem (MWIS) remains NP-hard in H-free graphs, unless H is a path or a subdivided claw, i.e., a graph obtained from the three-leaf star by subdividing each edge some number of times (possibly zero). Since then determining the complexity of MWIS in these remaining cases is one of the most important problems in algorithmic graph theory. A general belief is that the problem is polynomial-time solvable, which is witnessed by algorithmic results for graphs excluding some small paths or subdivided claws. A more conclusive evidence was given by the recent breakthrough result by Gartland and Lokshtanov [FOCS 2020]: They proved that MWIS can be solved in quasipolynomial time in H-free graphs, where H is any fixed path. If H is an arbitrary subdivided claw, we know much less: The problem admits a QPTAS and a subexponential-time algorithm [Chudnovsky et al., SODA 2019]. In this paper we make an important step towards solving the problem by showing that for any subdivided claw H, MWIS is polynomial-time solvable in H-free graphs of bounded degree. 

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5. Heuristic shortest hyperpaths in cell signaling hypergraphs

   https://doi.org/10.1186/s13015-022-00217-9  

   Krieger, Spencer ; Kececioglu, John ( December 2022 , Algorithms for Molecular Biology)

   Abstract Background Cell signaling pathways, which are a series of reactions that start at receptors and end at transcription factors, are basic to systems biology. Properly modeling the reactions in such pathways requires directed hypergraphs , where an edge is now directed between two sets of vertices. Inferring a pathway by the most parsimonious series of reactions corresponds to finding a shortest hyperpath in a directed hypergraph, which is NP-complete. The current state-of-the-art for shortest hyperpaths in cell signaling hypergraphs solves a mixed-integer linear program to find an optimal hyperpath that is restricted to be acyclic, and offers no efficiency guarantees. Results We present, for the first time, a heuristic for general shortest hyperpaths that properly handles cycles , and is guaranteed to be efficient . We show the heuristic finds provably optimal hyperpaths for the class of singleton-tail hypergraphs, and also give a practical algorithm for tractably generating all source-sink hyperpaths. The accuracy of the heuristic is demonstrated through comprehensive experiments on all source-sink instances from the standard NCI-PID and Reactome pathway databases, which show it finds a hyperpath that matches the state-of-the-art mixed-integer linear program on over 99% of all instances that are acyclic. On instances where only cyclic hyperpaths exist, the heuristic surpasses the state-of-the-art, which finds no solution; on every such cyclic instance, enumerating all source-sink hyperpaths shows the solution found by the heuristic was in fact optimal . Conclusions The new shortest hyperpath heuristic is both fast and accurate . This makes finding source-sink hyperpaths, which in general may contain cycles, now practical for real cell signaling networks. Availability Source code for the hyperpath heuristic in a new tool we call  (as well as for hyperpath enumeration, and all dataset instances) is available free for non-commercial use at . 

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