Computer Science

We study the approximability of the NP-complete \textsc{Maximum Minimal Feedback Vertex Set} problem. Informally, this natural problem seems to lie in an intermediate space between two more well-studied problems of this type: \textsc{Maximum Minimal Vertex Cover}, for which the best achievable approximation ratio is n − − √ , and \textsc{Upper Dominating Set}, which does not admit any n 1−ϵ approximation. We confirm and quantify this intuition by showing the first non-trivial polynomial time approximation for \textsc{Max Min FVS} with a ratio of O( n 2/3 ) , as well as a matching hardness of approximation bound of n 2/3−ϵ , improving the previous known hardness of n 1/2−ϵ . The approximation algorithm also gives a cubic kernel when parameterized by the solution size. Along the way, we also obtain an O(Δ) -approximation and show that this is asymptotically best possible, and we improve the bound for which the problem is NP-hard from Δ≥9 to Δ≥6 . Having settled the problem's approximability in polynomial time, we move to the context of super-polynomial time. We devise a generalization of our approximation algorithm which, for any desired approximation ratio r , produces an r -approximate solution in time n O(n/ r 3/2 ) . This time-approximation trade-off is essentially tight: we show that under the ETH, for any ratio r and ϵ>0 , no algorithm can r -approximate this problem in time n O((n/ r 3/2 ) 1−ϵ ) , hence we precisely characterize the approximability of the problem for the whole spectrum between polynomial and sub-exponential time, up to an arbitrarily small constant in the second exponent.

Consider n 2 −1 unit-square blocks in an n×n square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable -- a variation of Rush Hour with only 1×1 cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical 1×2 and horizontal 2×1 movable blocks and 4-color Subway Shuffle.

We prove that 2-Local Hamiltonian (2-LH) with Low Complexity problem is QCMA-complete by combining the results from the QMA-completeness[4] of 2-LH and QCMA-completeness of 3-LH with Low Complexity[6]. The idea is straightforward. It has been known that 2-LH is QMA-complete. By putting a low complexity constraint on the input state, we make the problem QCMA. Finally, we use similar arguments as in [4] to show that all QCMA problems can be reduced to our proposed problem.

The Multiple Travelling Salesman Problem (MTSP) is among the most interesting combinatorial optimization problems because it is widely adopted in real-life applications, including robotics, transportation, networking, etc. Although the importance of this optimization problem, there is no survey dedicated to reviewing recent MTSP contributions. In this paper, we aim to fill this gap by providing a comprehensive review of existing studies on MTSP. In this survey, we focus on MTSP's recent contributions to both classical vehicles/robots and unmanned aerial vehicles. We highlight the approaches applied to solve the MTSP as well as its application domains. We analyze the MTSP variants and propose a taxonomy and a classification of recent studies.

The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of O( N ??????) , where N is the order of the group. These algorithms require the inversion of some some group elements or rely on finding collisions, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this paper is a deterministic algorithm for solving the discrete logarithm problem in a semigroup. Specifically, let x be an element in a semigroup having finite order N x . If y?�⟨x??is given the paper provides an algorithm having time complexity O( N x ????????log N x ) to find all natural numbers m with x m =y . The paper also give an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely.

We prove a complexity dichotomy for Holant problems on the boolean domain with arbitrary sets of real-valued constraint functions. These constraint functions need not be symmetric nor do we assume any auxiliary functions as in previous results. It is proved that for every set F of real-valued constraint functions, Holant (F) is either P-time computable or #P-hard. The classification has an explicit criterion. This is the culmination of much research on this problem, and it uses previous results and techniques from many researchers. Some particularly intriguing concrete functions f 6 , f 8 and their associated families with extraordinary closure properties related to Bell states in quantum information theory play an important role in this proof.

We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z . For any ε,ζ>0 and any k≥1 , we show that Q 1 1−(1−ε ) Ω( ζ 6 k/log|Z|) ( f k )=Ω(k( ζ 5 ⋅ Q 1 ε+12ζ (f)−loglog(1/ζ))), where Q 1 ε (f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and f k denotes k parallel instances of f . As far as we are aware, this is the first direct product theorem for quantum communication. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao, and under anchored distributions due to Bavarian, Vidick and Yuen, as well as message-compression for quantum protocols due to Jain, Radhakrishnan and Sen. Our techniques also work for entangled non-local games which have input distributions anchored on any one side. In particular, we show that for any game G=(q,X×Y,A×B,V) where q is a distribution on X×Y anchored on any one side with anchoring probability ζ , then ω ∗ ( G k )= (1−(1− ω ∗ (G) ) 5 ) Ω( ζ 2 k log(|A|⋅|B|) ) where ω ∗ (G) represents the entangled value of the game G . This is a generalization of the result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem for games anchored on both sides, and potentially a simplification of their proof.

In the (binary) Distinct Vectors problem we are given a binary matrix A with pairwise different rows and want to select at most k columns such that, restricting the matrix to these columns, all rows are still pairwise different. A result by Froese et al. [JCSS] implies a 2^2^(O(k)) * poly(|A|)-time brute-force algorithm for Distinct Vectors. We show that this running time bound is essentially optimal by showing that there is a constant c such that the existence of an algorithm solving Distinct Vectors with running time 2^(O(2^(ck))) * poly(|A|) would contradict the Exponential Time Hypothesis.

We introduce an axiomatization for the notion of computation. Based on the idea of Brouwer choice sequences, we construct a model, denoted by E , which satisfies our axioms and E⊨P≠NP . In other words, regarding "effective computability" in Brouwer intuitionism viewpoint, we show P≠NP .

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.