Exponential Space Algorithm Research Articles

Countdown games, and simulation on (succinct) one-counter nets

We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS 2016) for simulation preorder on the class of succinct one-counter nets (i.e., one-counter automata with no zero tests where counter increments and decrements are integers written in binary); the problem was known to be PSPACE-hard and in EXPSPACE. We show that all relations between bisimulation equivalence and simulation preorder are EXPSPACE-hard for these nets; simulation preorder is thus EXPSPACE-complete. The result is proven by a reduction from reachability games whose EXPSPACE-completeness in the case of succinct one-counter nets was shown by Hunter (RP 2015), by using other results. We also provide a direct self-contained EXPSPACE-completeness proof for a special case of such reachability games, namely for a modification of countdown games that were shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie (LMCS 2008); in our modification the initial counter value is not given but is freely chosen by the first player. We also present an alternative proof for the upper bound by Hofman et al. In particular, we give a new simplified proof of the belt theorem that yields a simple graphic presentation of simulation preorder on (non-succinct) one-counter nets and leads to a polynomial-space algorithm (which is trivially extended to an exponential-space algorithm for succinct one-counter nets).

Many-visits TSP revisited

We study the Many-Visits Traveling Salesman Problem, where given a number k(v) for each of n cities and pairwise (possibly asymmetric) integer distances, one has to find an optimal tour that visits each city v exactly k(v) times. The currently fastest algorithm is due to Berger, Kozma, Mnich and Vincze [SODA 2019, TALG 2020] and runs in time and space O⁎(5n). They also show a polynomial-space algorithm running in time O(16n+o(n)). In this work, we show three main results:•A randomized polynomial-space algorithm running in time O⁎(2nD), where D is the maximum distance between two cities. By using standard methods, this results in a (1+ϵ)-approximation running in time O⁎(2nϵ−1).•A tight analysis of Berger et al.'s exponential-space algorithm, resulting in an O⁎(4n) running time bound.•A new polynomial-space algorithm, running in time O(7.88n).

Approximate Counting of k -Paths: Simpler, Deterministic, and in Polynomial Space

Recently, Brand et al. [STOC 2018] gave a randomized mathcal O(4 k m ε -2 -time exponential-space algorithm to approximately compute the number of paths on k vertices in a graph G up to a multiplicative error of 1 ± ε based on exterior algebra. Prior to our work, this has been the state-of-the-art. In this article, we revisit the algorithm by Alon and Gutner [IWPEC 2009, TALG 2010], and obtain the following results: • We present a deterministic 4 k + O (√ k (log k +log 2 ε -1 )) m -time polynomial-space algorithm. This matches the running time of the best known deterministic polynomial-space algorithm for deciding whether a given graph G has a path on k vertices. • Additionally, we present a randomized 4 k +mathcal O(log k (log k +logε -1 )) m -time polynomial-space algorithm. Our algorithm is simple—we only make elementary use of the probabilistic method. Here, n and m are the number of vertices and the number of edges, respectively. Additionally, our approach extends to approximate counting of other patterns of small size (such as q -dimensional p -matchings).

Maximal strongly connected cliques in directed graphs: Algorithms and bounds

Finding communities in the form of cohesive subgraphs is a fundamental problem in network analysis. In domains that model networks as undirected graphs, communities are generally associated with dense subgraphs, and many community models have been proposed. Maximal cliques are arguably the most widely studied among such models, with early works dating back to the ’60s, and a continuous stream of research up to the present. In domains that model networks as directed graphs, several approaches for community detection have been proposed, but there seems to be no clear model of cohesive subgraph, i.e., of what a community should look like. We extend the fundamental model of clique to directed graphs, adding the natural constraint of strong connectivity within the clique.We consider in this paper the problem of listing all maximal strongly connected cliques in a directed graph. We first investigate the combinatorial properties of strongly connected cliques and use them to prove that every n-vertex directed graph has at most 3n/3 maximal strongly connected cliques. We then exploit these properties to produce the first algorithms with polynomial delay for enumerating maximal strongly connected cliques: a first algorithm with polynomial delay and exponential space usage, and a second one, based on reverse-search, with higher (still polynomial) delay but which uses linear space.11An extended abstract based on the exponential space algorithm appeared in Conte et al. (2017) [14].

Improved exact algorithms for mildly sparse instances of Max SAT

We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2(1−μ(c))n) for instances with n variables and m=cn clauses. In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with μ(c)=1O(clog⁡c) due to Dantsin and Wolpert (2006) [9], a randomized polynomial space algorithm with μ(c)=1O(clog3⁡c) and a deterministic polynomial space algorithm with μ(c)=1O(c2log2⁡c) due to Sakai, Seto and Tamaki (2015) [30]. Our first result is a deterministic polynomial space algorithm with μ(c)=1O(clog⁡c) that achieves the previous best time complexity without exponential space or randomization. Furthermore, this algorithm can handle instances with exponentially large weights and hard constraints.The previous algorithms and our deterministic polynomial space algorithm run super-polynomially faster than 2n only if m=O(n2). Our second results are deterministic exponential space algorithms for Max SAT with μ(c)=1O((clog⁡c)2/3) and for Max 3-SAT with μ(c)=1O(c1/2) that run super-polynomially faster than 2n when m=o(n5/2/log5/2⁡n) and m=o(n3/log2⁡n) respectively. Our third result is a randomized exponential space algorithm for Max SAT with μ(c)=1O(c1/2log3/2⁡c) that run super-polynomially faster than 2n when m=o(n3/log5⁡n).

Exact algorithms for maximum independent set

We show that the maximum independent set problem on an n-vertex graph can be solved in 1.1996nnO(1) time and polynomial space, which even is faster than Robson's 1.2109nnO(1)-time exponential-space algorithm published in 1986. We also obtain improved algorithms for MIS in graphs with maximum degree 6 and 7, which run in time of 1.1893nnO(1) and 1.1970nnO(1), respectively. Our algorithms are obtained by using fast algorithms for MIS in low-degree graphs in a hierarchical way and making a careful analysis on the structure of bounded-degree graphs.

An Improved Exact Algorithm for TSP in Graphs of Maximum Degree 4

The paper presents a 1.692nnO(1)-time polynomial-space algorithm for the traveling salesman problem in an n-vertex edge-weighted graph with maximum degree 4, which improves the previous results of the 1.890nnO(1)-time polynomial-space algorithm by Eppstein and the 1.733nnO(1)-time exponential-space algorithm by Gebauer.

Parametric metric interval temporal logic

In this paper we focus on the role of parametric constants in real-time temporal logic and introduce the logic PMITL as a parametric extension of MITL. For this logic, we study decision problems which are the analogues of satisfiability, validity and model-checking problems for non-parametric temporal logic. We impose some restrictions on the use of the parameters: each parameter is used with a fixed polarity, parameters can appear only in one of the endpoints of the intervals, parametric linear expressions can be used only as right endpoints of the intervals. We show that, for parameter valuations yielding only non-singular intervals, the considered problems are all decidable and Expspace-complete, such as for the decision problems in MITL. Moreover, we show that if we relax any of the imposed restrictions, the problems become undecidable. We also investigate the computational complexity of these problems for natural fragments of PMITL, and show that for some meaningful fragments they can be solved in polynomial space and are Pspace-complete. Finally, we consider the decision problem of determining the truth of first-order queries over PMITL formulas where the parameters are used as variables that can be existentially or universally quantified. We solve this problem in several cases and exhibit an exponential-space algorithm.

A General Reduction Theorem with Applications to Pathwidth and the Complexity of MAX 2-CSP

We prove a general reduction theorem which allows us to extend bounds for certain graph parameters on cubic graphs to bounds for general graphs taking into account the individual vertex degrees. As applications, we give an algorithm for Max \(2\)-CSP whose complexity matches the algorithm of Scott and Sorkin in the case of \(d\)-regular graphs, \(d \le 5\), but is otherwise faster. It also improves on the previously fastest known algorithm in terms of the average degree, given by Golovnev and Kutzkov. Also from the general theorem, we derive a bound for the pathwidth of a general graph which equals that of Fomin et al. and Gaspers for graphs of degree at most \(6\), but is smaller otherwise, and use this to give an improved exponential-space algorithm for Max \(2\)-CSP. Finally we use the general result to give a faster algorithm for Max \(2\)-CSP on claw-free graphs.

An exact algorithm for the minimum dominating clique problem

A subset of vertices D ⊆ V of a graph G = ( V , E ) is a dominating clique if D is a dominating set and a clique of G . The existence problem ‘Given a graph G , is there a dominating clique in G ?’ is NP-complete, and thus both the Minimum and the Maximum Dominating Clique problems are NP-hard. We present an O ( 1.338 7 n ) time and polynomial space algorithm that for an input graph on n vertices either computes a minimum dominating clique or reports that the graph has no dominating clique. The algorithm uses the Branch & Reduce paradigm and its time analysis is based on the Measure & Conquer approach. We also establish a lower bound of Ω ( 1.259 9 n ) for the worst case running time of the algorithm. Finally using memorization we obtain an O ( 1.323 4 n ) time and exponential space algorithm for the same problem.

An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals, and Applications to Commutative Semigroups

It is known that the reduced Gröbner basis of general polynomial ideals can be computed in exponential space. The algorithm, obtained by Kühnle and Mayr, is, however, based on rather complex parallel computations, and, above that, makes extensive use of the parallel computation thesis. In this paper, we exhibit an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals which can be implemented without any complex parallel computations. This result is then applied to derive space optimal decision procedures for the finite enumeration and subword problems for commutative semigroups.

An Optimal Algorithm for Constructing the Reduced Gröbner Basis of Binomial Ideals

In this paper, we present an optimal, exponential space algorithm for generating the reduced Gröbner basis of binomial ideals. We make use of the close relationship between commutative semigroups and pure difference binomial ideals. Based on an optimal algorithm for the uniform word problem in commutative semigroups, we first derive an exponential space algorithm for constructing the reduced Gröbner basis of pure difference binomial ideals. In addition to some applications to finitely presented commutative semigroups, this algorithm is then extended to an exponential space algorithm for generating the reduced Gröbner basis of binomial ideals over Q in general.

On the finite-valuedness problem for sequential machines

For a set A={A1,…, Am} of n X n matrices with nonnegative integer entries, let P(A) be the set of all finite products of matrices in A. We show that there is a square space algorithm to decide, given A, whether or not P(A) is finite. As a corollary, we show that there is an exponential space algorithm to decide, given a nondeterministic Mealy sequential machine M with accepting states, whether there exists an integer d⩾0 such that on every input, the number of accepting computations of M producing distinct outputs is at most d. If d exists, the smallest such d can be computed in space exponential in size(M)+log d. The space bound reduces to polynomial for the analogous problem of ambiguity of nondeterministic finite acceptors.