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1. C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width

   https://doi.org/10.1007/s00453-021-00839-2  

   Da Lozzo, Giordano ; Eppstein, David ; Goodrich, Michael T. ; Gupta, Siddharth ( August 2021 , Algorithmica)

   Abstract For a clustered graph , i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that (1) the subgraph induced by each cluster is drawn in the interior of the corresponding disk, (2) each edge intersects any disk at most once, and (3) the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Feng, Cohen, and Eades, Planarity for clustered graphs , ESA’95], has only been recently settled [Fulek and Tóth, Atomic Embeddability, Clustered Planarity, and Thickenability , to appear at SODA’20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs . We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT (resp., XP) algorithm for embedded flat (resp., non-flat) clustered graphs, when parameterized by the carving-width of the dual graph of the input. These are the first FPT and XP algorithms for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Tóth. In particular, our algorithm runs in quadratic time for flat instances of bounded treewidth and bounded face size. To further strengthen the relevance of this result, we show that an algorithm with running time O ( r ( n )) for flat instances whose underlying graph has pathwidth 1 would result in an algorithm with running time O ( r ( n )) for flat instances and with running time $$O(r(n^2) + n^2)$$ O ( r ( n 2 ) + n 2 ) for general, possibly non-flat, instances. 

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2. On the complexity of bidirected interleaved Dyck-reachability

   https://doi.org/10.1145/3434340  

   Li, Yuanbo ; Zhang, Qirun ; Reps, Thomas ( January 2021 , Proceedings of the ACM on Programming Languages)

   null (Ed.)

   Many program analyses need to reason about pairs of matching actions, such as call/return, lock/unlock, or set-field/get-field. The family of Dyck languages { D k }, where D k has k kinds of parenthesis pairs, can be used to model matching actions as balanced parentheses. Consequently, many program-analysis problems can be formulated as Dyck-reachability problems on edge-labeled digraphs. Interleaved Dyck-reachability (InterDyck-reachability), denoted by D k ⊙ D k -reachability, is a natural extension of Dyck-reachability that allows one to formulate program-analysis problems that involve multiple kinds of matching-action pairs. Unfortunately, the general InterDyck-reachability problem is undecidable. In this paper, we study variants of InterDyck-reachability on bidirected graphs , where for each edge ⟨ p , q ⟩ labeled by an open parenthesis ”( a ”, there is an edge ⟨ q , p ⟩ labeled by the corresponding close parenthesis ”) a ”, and vice versa . Language-reachability on a bidirected graph has proven to be useful both (1) in its own right, as a way to formalize many program-analysis problems, such as pointer analysis, and (2) as a relaxation method that uses a fast algorithm to over-approximate language-reachability on a directed graph. However, unlike its directed counterpart, the complexity of bidirected InterDyck-reachability still remains open. We establish the first decidable variant (i.e., D 1 ⊙ D 1 -reachability) of bidirected InterDyck-reachability. In D 1 ⊙ D 1 -reachability, each of the two Dyck languages is restricted to have only a single kind of parenthesis pair. In particular, we show that the bidirected D 1 ⊙ D 1 problem is in PTIME. We also show that when one extends each Dyck language to involve k different kinds of parentheses (i.e., D k ⊙ D k -reachability with k ≥ 2), the problem is NP-hard (and therefore much harder). We have implemented the polynomial-time algorithm for bidirected D 1 ⊙ D 1 -reachability. D k ⊙ D k -reachability provides a new over-approximation method for bidirected D k ⊙ D k -reachability in the sense that D k ⊙ D k -reachability can first be relaxed to bidirected D 1 ⊙ D 1 -reachability, and then the resulting bidirected D 1 ⊙ D 1 -reachability problem is solved precisely. We compare this D 1 ⊙ D 1 -reachability-based approach against another known over-approximating D k ⊙ D k -reachability algorithm. Surprisingly, we found that the over-approximation approach based on bidirected D 1 ⊙ D 1 -reachability computes more precise solutions, even though the D 1 ⊙ D 1 formalism is inherently less expressive than the D k ⊙ D k formalism. 

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3. Quantum computational phase transition in combinatorial problems

   https://doi.org/10.1038/s41534-022-00596-2  

   Zhang, Bingzhi ; Sone, Akira ; Zhuang, Quntao ( July 2022 , npj Quantum Information)

   Quantum Approximate Optimization algorithm (QAOA) aims to search for approximate solutions to discrete optimization problems with near-term quantum computers. As there are no algorithmic guarantee possible for QAOA to outperform classical computers, without a proof that bounded-error quantum polynomial time (BQP) ≠ nondeterministic polynomial time (NP), it is necessary to investigate the empirical advantages of QAOA. We identify a computational phase transition of QAOA when solving hard problems such as SAT—random instances are most difficult to train at a critical problem density. We connect the transition to the controllability and the complexity of QAOA circuits. Moreover, we find that the critical problem density in general deviates from the SAT-UNSAT phase transition, where the hardest instances for classical algorithms lies. Then, we show that the high problem density region, which limits QAOA’s performance in hard optimization problems (reachability deficits), is actually a good place to utilize QAOA: its approximation ratio has a much slower decay with the problem density, compared to classical approximate algorithms. Indeed, it is exactly in this region that quantum advantages of QAOA over classical approximate algorithms can be identified.

    

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4. Atomic Embeddability, Clustered Planarity, and Thickenability

   https://doi.org/10.1145/3502264  

   Fulek, Radoslav ; Tóth, Csaba D. ( April 2022 , Journal of the ACM)

   We study the atomic embeddability testing problem, which is a common generalization of clustered planarity ( c-planarity , for short) and thickenability testing, and present a polynomial-time algorithm for this problem, thereby giving the first polynomial-time algorithm for c-planarity. C-planarity was introduced in 1995 by Feng, Cohen, and Eades as a variant of graph planarity, in which the vertex set of the input graph is endowed with a hierarchical clustering and we seek an embedding (crossing free drawing) of the graph in the plane that respects the clustering in a certain natural sense. Until now, it has been an open problem whether c-planarity can be tested efficiently. The thickenability problem for simplicial complexes emerged in the topology of manifolds in the 1960s. A 2-dimensional simplicial complex is thickenable if it embeds in some orientable 3-dimensional manifold. Recently, Carmesin announced that thickenability can be tested in polynomial time. Our algorithm for atomic embeddability combines ideas from Carmesin’s work with algorithmic tools previously developed for weak embeddability testing. We express our results purely in terms of graphs on surfaces, and rely on the machinery of topological graph theory. Finally, we give a polynomial-time reduction from atomic embeddability to thickenability thereby showing that both problems are polynomially equivalent, and show that a slight generalization of atomic embeddability to the setting in which clusters are toroidal graphs is NP-complete. 

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5. A Linear Time Algorithm for Quantum 2-SAT

   De Beaudrap, Niel ; Gharibian, Sevag ( January 2016 , Conference on Computational Complexity)

   The Boolean constraint satisfaction problem 3-SAT is arguably the canonical NP-complete problem. In contrast, 2-SAT can not only be decided in polynomial time, but in fact in deterministic linear time. In 2006, Bravyi proposed a physically motivated generalization of k-SAT to the quantum setting, defining the problem "quantum k-SAT". He showed that quantum 2-SAT is also solvable in polynomial time on a classical computer, in particular in deterministic time O(n^4), assuming unit-cost arithmetic over a field extension of the rational numbers, where n is number of variables. In this paper, we present an algorithm for quantum 2-SAT which runs in linear time, i.e. deterministic time O(n+m) for n and m the number of variables and clauses, respectively. Our approach exploits the transfer matrix techniques of Laumann et al. [QIC, 2010] used in the study of phase transitions for random quantum 2-SAT, and bears similarities with both the linear time 2-SAT algorithms of Even, Itai, and Shamir (based on backtracking) [SICOMP, 1976] and Aspvall, Plass, and Tarjan (based on strongly connected components) [IPL, 1979]. 

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