Exponential Time Algorithm Research Articles

Moderate exponential-time algorithms for scheduling problems

Moderate exponential-time algorithms for scheduling problems

Counting Answers to Unions of Conjunctive Queries: Natural Tractability Criteria and Meta-Complexity

We study the problem of counting answers to unions of conjunctive queries (UCQs) under structural restrictions on the input query. Concretely, given a class C of UCQs, the problem #UCQ (C) provides as input a UCQ Ψ ∈ C and a database D and the problem is to compute the number of answers of Ψ in D. Chen and Mengel [PODS'16] have shown that for any recursively enumerable class C, the problem #UCQ (C) is either fixed-parameter tractable or hard for one of the parameterised complexity classes W[1] or #W[1]. However, their tractability criterion is unwieldy in the sense that, given any concrete class C of UCQs, it is not easy to determine how hard it is to count answers to queries in C. Moreover, given a single specific UCQ Ψ, it is not easy to determine how hard it is to count answers to Ψ. In this work, we address the question of finding a natural tractability criterion: The combined conjunctive query of a UCQ Ψ=φ 1 ∨ ... ∨ φ l is the conjunctive query ^ Ψ = φ_1 ∧ ... ∧ φ l . We show that under natural closure properties of C, the problem #UCQ (C) is fixed-parameter tractable if and only if the combined conjunctive queries of UCQs in C, and their contracts, have bounded treewidth. A contract of a conjunctive query is an augmented structure, taking into account how the quantified variables are connected to the free variables --- if all variables are free, then a conjunctive query is equal to its contract; in this special case the criterion for fixed-parameter tractability of #UCQ (C) thus simplifies to the combined queries having bounded treewidth. Finally, we give evidence that a closure property on C is necessary for obtaining a natural tractability criterion: We show that even for a single UCQ Ψ, the meta problem of deciding whether #UCQ (Ψ) can be solved in time O(|D| d ) is NP-hard for any fixed d ≥ 1. Moreover, we prove that a known exponential-time algorithm for solving the meta problem is optimal under assumptions from fine-grained complexity theory. As a corollary of our reduction, we also establish that approximating the Weisfeiler-Leman-Dimension of a UCQ is NP-hard.

A Theory of Alternating Paths and Blossoms from the Perspective of Minimum Length

The Micali–Vazirani (MV) algorithm for finding a maximum cardinality matching in general graphs, which was published in 1980, remains to this day the most efficient known algorithm for the problem. The current paper gives the first complete and correct proof of this algorithm. The MV algorithm resorts to finding minimum-length augmenting paths. However, such paths fail to satisfy an elementary property, called breadth first search honesty in this paper. In the absence of this property, an exponential time algorithm appears to be called for—just for finding one such path. On the other hand, the MV algorithm accomplishes this and additional tasks in linear time. The saving grace is the various “footholds” offered by the underlying structure, which the algorithm uses in order to perform its key tasks efficiently. The theory expounded in this paper elucidates this rich structure and yields a proof of correctness of the algorithm. It may also be of independent interest as a set of well-knit graph-theoretic facts. Funding: This work was supported in part by the National Science Foundation [Grant CCF-2230414].

On Breaking Truss-based and Core-based Communities

We introduce the general problem of identifying a smallest edge subset of a given graph whose deletion makes the graph community-free. We consider this problem under two community notions that have attracted significant attention: k -truss and k -core. We also introduce a problem variant where the identified subset contains edges incident to a given set of nodes and ensures that these nodes are not contained in any community: k -truss or k -core, in our case. These problems are directly applicable in social networks: The identified edges can be hidden by users or sanitized from the output graph; or in communication networks: the identified edges correspond to vital network connections. We present a series of theoretical and practical results. On the theoretical side, we show through non-trivial reductions that the problems we introduce are NP-hard and, in fact, hard to approximate. For the k -truss-based problems, we also show exact exponential-time algorithms, as well as a non-trivial lower bound on the size of an optimal solution. On the practical side, we develop a series of heuristics that are sped up by efficient data structures that we propose for updating the truss or core decomposition under edge deletions. In addition, we develop an algorithm to compute the lower bound. Extensive experiments on 11 real-world and synthetic graphs show that our heuristics are effective, outperforming natural baselines, and also efficient (up to two orders of magnitude faster than a natural baseline), thanks to our data structures. Furthermore, we present a case study on a co-authorship network and experiments showing that the removal of edges identified by our heuristics does not substantially affect the clustering structure of the input graph. This work extends a KDD 2021 paper, providing new theoretical results as well as introducing core-based problems and algorithms.

Decomposition and factorisation of transients in functional graphs

Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be quite large, it is interesting to decompose and factorise them into several subgraphs acting together. Polynomial equations over functional graphs provide a formal way to represent this decomposition and factorisation mechanism, and solving them validates or invalidates hypotheses on their decomposability. The current solution method breaks down a single equation into a series of basic equations of the form A×X=B (with A, X, and B being FGs) to identify the possible solutions. However, it is able to consider just FGs made of cycles only. This work proposes an algorithm for solving these basic equations for general connected FGs. By exploiting a connection with the cancellation problem, we prove that the upper bound to the number of solutions is closely related to the size of the cycle in the coefficient A of the equation. The cancellation problem is also involved in the main algorithms provided by the paper. We introduce a polynomial-time semi-decision algorithm able to provide constraints that a potential solution will have to satisfy if it exists. Then, exploiting the ideas introduced in the first algorithm, we introduce a second exponential-time algorithm capable of finding all solutions by integrating several ‘hacks’ that try to keep the exponential as tight as possible.

A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics

A Faster Exponential Time Algorithm for Bin Packing With a Constant Number of Bins via Additive Combinatorics

The computational complexity of classical knot recognition

The classical knot recognition problem is the problem of determining whether the virtual knot represented by a given diagram is classical. We prove that this problem is in NP, and we give an exponential time algorithm for the problem.

Technical Note—The Complexity of the Pricing Problem of the Set Partitioning Formulation of Vehicle Routing Problems

The pricing problem of vehicle routing problems is strongly NP hard. Many algorithms for finding optimal solutions to various vehicle routing problems rely on a subroutine to solve a so-called pricing problem of a set partitioning formulation. Only exponential time algorithms are known for these pricing problems. Therefore, the set partitioning formulation is oftentimes relaxed at the expense of weaker LP bounds, so that the resulting pricing problem can now be solved using a pseudopolynomial or polynomial time algorithm. In “The Complexity of the Pricing Problem of the Set Partitioning Formulation of Vehicle Routing Problems,” it is proven by Remy Spliet that the pricing problem is strongly NP hard for many different vehicle routing problems. This means that, unless P = NP, no pseudopolynomial or polynomial time algorithm exists for the pricing problem, justifying the common use of relaxations.

A single exponential time algorithm for homogeneous regular sequence tests

Assume that we are given a sequence [Formula: see text] of [Formula: see text] homogeneous polynomials in [Formula: see text] variables of degree at most [Formula: see text] and the ideal [Formula: see text] generated by this sequence. The aim of this paper is to present a new and effective method to determine, within the arithmetic complexity [Formula: see text], whether [Formula: see text] is regular. This algorithm has been implemented in Maple and its efficiency (compared to the classical approaches for regular sequence test) is evaluated via a set of benchmark polynomials. Furthermore, we show that if [Formula: see text] is regular then we can transform [Formula: see text] into Nœther position and at the same time compute a reduced Gröbner basis for the transformed ideal within the arithmetic complexity [Formula: see text]. Finally, under the same assumption, we establish the new upper bound [Formula: see text] for the maximum degree of the elements of any reduced Gröbner basis of [Formula: see text] in the case that [Formula: see text].

Auto-Optimizing Connection Planning Method for Chain-Type Modular Self-Reconfiguration Robots

Chain-type modular robots are capable of self-reconfiguration (SR), where the connection relationship between modules is changed according to the environment and tasks. This article focuses on the connection planning of SR based on multiple in-degree single out-degree (MISO) modules. The goal is to calculate the optimal connection planning solution: the sequence with the fewest detachment and attachment actions. To this end, we propose an auto-optimizing connection planning method that contains a polynomial-time algorithm to calculate near-optimal solutions and an exponential-time algorithm to further optimize the solutions automatically when some CPUs are idle. The method combines rapidity and optimality in the face of an NP-complete problem by using configuration pointers, strings that uniquely specify the robot's configuration. Our polynomial-time algorithm, in-degree matching (IM) uses the interchangeability of connection points to reduce reconfiguration steps. Our exponential-time algorithm, tree-based branch and bound (TBB) further optimizes the solutions to the optimum by a new branching strategy and stage cost. In the experiments, we verify the feasibility of the auto-optimizing method combining IM and TBB, and demonstrate the superiority of IM over Greedy-CM in the SR of MISO modules and the near-optimality of IM compared to the optimal solutions of TBB.

Minimizing Convex Functions with Rational Minimizers

Given a separation oracle SO for a convex function f defined on ℝ n that has an integral minimizer inside a box with radius R , we show how to find an exact minimizer of f using at most O(n (n log log (n)/ log (n) + log ( R ))) calls to SO and poly ( n , log ( R )) arithmetic operations, or O(n log (nR) calls to SO and exp ( O(n) ) ⋅ poly (log (R) ) arithmetic operations. When the set of minimizers of f has integral extreme points, our algorithm outputs an integral minimizer of f . This improves upon the previously best oracle complexity of O(n 2 ( n + log ( R ))) for polynomial time algorithms and O(n 2 log ( nR ) for exponential time algorithms obtained by [Grötschel, Lovász and Schrijver, Prog. Comb. Opt. 1984, Springer 1988] over thirty years ago. Our improvement on Grötschel, Lovász and Schrijver’s result generalizes to the setting where the set of minimizers of f is a rational polyhedron with bounded vertex complexity. For the Submodular Function Minimization problem, our result immediately implies a strongly polynomial algorithm that makes at most O(n 3 log log ( n )/log ( n )) calls to an evaluation oracle, and an exponential time algorithm that makes at most O(n 2 log ( n )) calls to an evaluation oracle. These improve upon the previously best O(n 3 log 2 ( n )) oracle complexity for strongly polynomial algorithms given in [Lee, Sidford and Wong, FOCS 2015] and [Dadush, Végh and Zambelli, SODA 2018], and an exponential time algorithm with oracle complexity O(n 3 log ( n )) given in the former work. Our result is achieved via a reduction to the Shortest Vector Problem in lattices. We show how an approximately shortest vector of an auxiliary lattice can be used to effectively reduce the dimension of the problem. Our analysis of the oracle complexity is based on a potential function that simultaneously captures the size of the search set and the density of the lattice, which we analyze via tools from convex geometry and lattice theory.

Moderate exponential-time algorithms for scheduling problems

Moderate exponential-time algorithms for scheduling problems

Pattern-Avoiding Ascent Sequences of Length 3

Pattern-avoiding ascent sequences have recently been related to set-partition problems and stack-sorting problems. While the generating functions for several length-3 pattern-avoiding ascent sequences are known, those avoiding 000, 100, 110, 120 are not known. We have generated extensive series expansions for these four cases, and analysed them in order to conjecture the asymptotic behaviour.
 We provide polynomial time algorithms for the $000$ and $110$ cases, and exponential time algorithms for the $100$ and $120$ cases. We also describe how the $000$ polynomial time algorithm was detected somewhat mechanically given an exponential time algorithm.
 For 120-avoiding ascent sequences we find that the generating function has stretched-exponential behaviour and prove that the growth constant is the same as that for 201-avoiding ascent sequences, which is known.
 The other three generating functions have zero radius of convergence, which we also prove. For 000-avoiding ascent sequences we give what we believe to be the exact growth constant. We give the conjectured asymptotic behaviour for all four cases.

Exponential-time algorithms for parallel machine scheduling problems

Exponential-time algorithms for parallel machine scheduling problems

Approximating the minimal lookahead needed to win infinite games

We present an exponential-time algorithm approximating the minimal lookahead necessary to win an ω-regular delay game.

Constructing depth-optimum circuits for adders and And-Or paths

We examine the fundamental problem of constructing depth-optimum circuits for binary addition. More precisely, as in literature, we consider the following problem: Given auxiliary inputs t0,…,tm−1, the so-called generate and propagate signals, construct a depth-optimum circuit over the basis {And2,Or2} computing all n carry bits of an n-bit adder, where m=2n−1. In fact, carry bits are And-Or paths, i.e., Boolean functions of the form t0∨(t1∧(t2∨(…tm−1)…)). Classical approaches construct so-called prefix circuits which do not achieve a competitive depth. For instance, the popular construction by Kogge and Stone (1973) is only a 2-approximation. A lower bound on the depth of any prefix circuit is 1.44log2m+const, while recent non-prefix circuits have a depth of log2m+log2log2m+const. However, it is unknown whether any of these polynomial-time approaches achieves the optimum depth for all m∈N.We present a new exponential-time algorithm solving the problem optimally. The previously best exact algorithm by Hegerfeld (2018) with a running time of O(2.45m) is viable only for m≤29. Our algorithm is significantly faster: We achieve a theoretical running time of O(2.02m) and apply sophisticated pruning strategies to improve practical running times dramatically. This allows us to compute optimum circuits for all m≤64. Combining these computational results with new theoretical insights, we derive the optimum depths for the computation of all carry bits of 2k-bit adder circuits for all k≤13, previously known only for k≤4.In fact, we solve a more general problem, namely delay optimization of generalized And-Or paths, which originates from late-stage logic optimization in VLSI design. Delay is a natural extension of circuit depth to prescribed input arrival times; and generalized And-Or paths are a generalization of And-Or paths where And and Or do not necessarily alternate. Our algorithm arises from our new structure theorem which characterizes delay-optimum generalized And-Or path circuits.

Bag Query Containment and Information Theory

The query containment problem is a fundamental algorithmic problem in data management. While this problem is well understood under set semantics, it is by far less understood under bag semantics. In particular, it is a long-standing open question whether or not the conjunctive query containment problem under bag semantics is decidable. We unveil tight connections between information theory and the conjunctive query containment under bag semantics. These connections are established using information inequalities, which are considered to be the laws of information theory. Our first main result asserts that deciding the validity of a generalization of information inequalities is many-one equivalent to the restricted case of conjunctive query containment in which the containing query is acyclic; thus, either both these problems are decidable or both are undecidable. Our second main result identifies a new decidable case of the conjunctive query containment problem under bag semantics. Specifically, we give an exponential-time algorithm for conjunctive query containment under bag semantics, provided the containing query is chordal and admits a simple junction tree.

Faster exponential-time algorithms for approximately counting independent sets

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance ε is at most O(20.2680n) times a polynomial in 1/ε. On bipartite graphs, the exponential term in the running time is improved to O(20.2372n). Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, Štefankovič and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless P=NP.

The exponential-time hypothesis and the relative complexity of optimization and logical reasoning problems

Obtaining lower bounds for NP-hard problems has for a long time been an active area of research. Algebraic techniques introduced by Jonsson et al. (2017) [4] show that the fine-grained time complexity of the parameterized ▪ problem correlates to the lattice of strong partial clones. With this ordering they isolated a relation R such that ▪ can be solved at least as fast as any other NP-hard ▪ problem. In this paper we extend this method and show that such languages also exist for the surjective SAT problem, the max ones problem, the propositional abduction problem, and the Boolean valued constraint satisfaction problem over finite-valued constraint languages. These languages may be interesting when investigating the borderline between polynomial time, subexponential time and exponential-time algorithms since they in a precise sense can be regarded as NP-hard problems with minimum time complexity. Indeed, with the help of these languages we relate all of the above problems to the exponential time hypothesis (ETH) in several different ways.

Induced star partition of graphs

Given a graph G , we call a partition ( V 1 , V 2 , … , V k ) of its vertex set an induced star partition of G if each set in the partition induces a star. In this paper, we consider the problem of finding an induced star partition of a graph of minimum size and its decision versions. This problem may be viewed as an amalgamation of the well-known dominating set and coloring problems. We obtain the following main results: (1) Deciding whether a graph can be partitioned into k induced stars is NP-complete for each fixed k ≥ 3 and has a polynomial time algorithm for each k ≤ 2 . (2) It is NP-hard to approximate the minimum induced star partition size within n 1 2 − ϵ for all ϵ > 0 . (3) The decision version of the induced star partition problem is NP-complete for subcubic bipartite planar graphs, line graphs (a subclass of K 1 , r -free graphs, r ≥ 3 ), K 1 , 5 -free split graphs and co-tripartite graphs. (4) The minimum induced star partition problem has an r 2 -approximation algorithm for K 1 , r -free graphs ( r ≥ 2 ) and a 2-approximation algorithms for split graphs. We also identify some fixed parameter tractable and exact exponential time algorithms that follow from the literature.