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A resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time f (pw) n o (pw) on n-vertex graphs of constant degree, with pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter tl + Δ , where tl is the tree-length and Δ the maximum-degree of the input graph. [ABSTRACT FROM AUTHOR]

The Minimum Circuit Size Problem (MCSP) has been the focus of intense study recently; MCSP is hard for SZK under rather powerful reductions (Allender and Das Inf. Comput. 256, 2–8, 2017), and is provably not hard under "local" reductions computable in TIME(n0.49) (Murray and Williams Theory Comput. 13(1), 1–22, 2017). The question of whether MCSP is NP-hard (or indeed, hard even for small subclasses of P) under some of the more familiar notions of reducibility (such as many-one or Turing reductions computable in polynomial time or in AC0) is closely related to many of the longstanding open questions in complexity theory (Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019; Allender et al. Comput. Complex. 26(2), 469–496, 2017; Hirahara and Santhanam 2017; Hirahara and Watanabe 2016; Hitchcock and Pavan 2015; Impagliazzo et al. 2018; Murray and Williams Theory Comput. 13(1), 1–22, 2017). All prior hardness results for MCSP hold also for computing somewhat weak approximations to the circuit complexity of a function (Allender et al. SIAM J. Comput. 35(6), 1467–1493, 2006; Allender and Das Inf. Comput. 256, 2–8, 2017; Allender et al. J. Comput. Syst. Sci. 77(1), 14–40, 2011; Hirahara and Santhanam 2017; Kabanets and Cai 2000; Rudow Inf. Process. Lett. 128, 1–4, 2017) (Subsequent to our work, a new hardness result has been announced (Ilango 2020) that relies on more exact size computations). Some of these results were proved by exploiting a connection to a notion of time-bounded Kolmogorov complexity (KT) and the corresponding decision problem (MKTP). More recently, a new approach for proving improved hardness results for MKTP was developed (Allender et al. SIAM J. Comput. 47(4), 1339–1372, 2018; Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019), but this approach establishes only hardness of extremely good approximations of the form 1 + o(1), and these improved hardness results are not yet known to hold for MCSP. In particular, it is known that MKTP is hard for the complexity class DET under nonuniform ≤ m A C 0 reductions, implying MKTP is not in AC0[p] for any prime p (Allender and Hirahara ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019). It was still open if similar circuit lower bounds hold for MCSP (But see Golovnev et al. 2019; Ilango 2020). One possible avenue for proving a similar hardness result for MCSP would be to improve the hardness of approximation for MKTP beyond 1 + o(1) to ω(1), as KT-complexity and circuit size are polynomially-related. In this paper, we show that this approach cannot succeed. More specifically, we prove that PARITY does not reduce to the problem of computing superlinear approximations to KT-complexity or circuit size via AC0-Turing reductions that make O(1) queries. This is significant, since approximating any set in P/polyAC0-reduces to just one query of a much worse approximation of circuit size or KT-complexity (Oliveira and Santhanam 2017). For weaker approximations, we also prove non-hardness under more powerful reductions. Our non-hardness results are unconditional, in contrast to conditional results presented in Allender and Hirahara (ACM Trans. Comput. Theory 11(4), 27:1–27:27, 2019) (for more powerful reductions, but for much worse approximations). This highlights obstacles that would have to be overcome by any proof that MKTP or MCSP is hard for NP under AC0 reductions. It may also be a step toward confirming a conjecture of Murray and Williams, that MCSP is not NP-complete under logtime-uniform ≤ m A C 0 reductions. [ABSTRACT FROM AUTHOR]

We revisit the maximum satisfiability problem (Max-SAT) in the data stream model. In this problem, the stream consists of m clauses that are disjunctions of literals drawn from n Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that Ω (n) space is necessary to yield a 2 / 2 + ε approximation of the optimum value; they also presented an algorithm that yields a 2 / 2 − ε approximation of the optimum value using O (ε − 2 log ⁡ n) space. In this paper, we not only focus on approximating the optimum value, but also on obtaining the corresponding Boolean assignment using sublinear o (m n) space. We present randomized single-pass algorithms that w.h.p.1 yield: • A 1 − ε approximation using O ˜ (n / ε 3) space and exponential post-processing time • A 3 / 4 − ε approximation using O ˜ (n / ε) space and polynomial post-processing time. Our ideas also extend to dynamic streams. However, we show that the streaming k -SAT problem, which asks whether one can satisfy all size- k input clauses, must use Ω (n k) space. We also consider the related minimum satisfiability problem (Min-SAT), introduced by Kohli et al. (SIAM J. Discrete Math. 1994), that asks to find an assignment that minimizes the number of satisfied clauses. For this problem, we give a O ˜ (n 2 / ε 2) space algorithm, which is sublinear when m = ω (n) , that yields an α + ε approximation where α is the approximation guarantee of the offline algorithm. If each variable appears in at most f clauses, we show that a 2 f n approximation using O ˜ (n) space is possible. Finally, for the Max-AND-SAT problem where clauses are conjunctions of literals, we show that any single-pass algorithm that approximates the optimal value up to a factor better than 1/2 with success probability at least 2/3 must use Ω (m n) space. [ABSTRACT FROM AUTHOR]

Lutz (SIAM J. Comput. 24(6), 1170–1189, 1995) proposed the following generalization of hardness: While a problem A is hard for a complexity class C if all problems in C can be reduced to A, Lutz calls a problem weakly hard if a nonnegligible part of the problems in C can be reduced to A. For the linear exponential time class E = DTIME(2lin), Lutz formalized these ideas by introducing a resource-bounded (pseudo) measure on this class and by saying that a subclass of E is negligible if it has measure 0 in E. In this paper we introduce two new weak hardness notions for E – E-nontriviality and strongly E-nontriviality. They generalize Lutz's weak hardness notion for E, but are much simpler conceptually. Namely, a set A is E-nontrivial if, for any k ≥ 1, there is a set Bk ∈E which can be reduced to A (by a polynomial time many-one reduction) and which cannot be computed in time O(2kn), and a set A is strongly E-nontrivial if the set Bk can be chosen to be almost everywhereO(2kn)-complex, i.e. if Bk can be chosen such that any algorithm that computes Bk runs for more than 2k|x| steps on all but finitely many inputs x. [ABSTRACT FROM AUTHOR]

Following a general framework of Coxeter spectral analysis of signed graphs Δ and finite posets I introduced by Simson (SIAM J. Discrete Math. 27:827–854, 2013) we present efficient numerical algorithms for the Coxeter spectral study of finite posets I = ({ 1 , ... , n } , ⪯ I) that are positive in the sense that the symmetric Gram matrix G I : = 1 2 (C I + C I t r) ∈ M n (Q) is positive definite, where C I ∈ M n (Z) is the incidence matrix of I encoding the relation ⪯ I. In the framework of scientific computing we present a complete Coxeter spectral classification of finite positive posets I of size n = | I | < 20. It extends one of the main results obtained in Ga̧siorek et al. (Eur. J. Comb. 48:127–142, 2015) for posets of size n ≤ 10. We also show that the connectivity of such posets I is determined by the complex Coxeter spectrum specc I ⊆ C ; equivalently, by the Coxeter polynomial cox I (t) ∈ Z t od I. One of the main results of the paper is a new technique to compute in a polynomial time a Z -invertible matrix B I ∈ M n (Z) such that B I t r · C I · B I = G ˇ D I , where G ˇ D I ∈ M n (Z) is a non-symmetric Gram matrix of a simply-laced Dynkin diagram D I ∈ { A n , D n , E 6 , E 7 , E 8 } associated with a finite positive poset I. [ABSTRACT FROM AUTHOR]