Exact Exponential-time Algorithms Research Articles

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.

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.

From Independent Sets and Vertex Colorings to Isotropic Spaces and Isotropic Decompositions: Another Bridge between Graphs and Alternating Matrix Spaces

In the 1970s, Lovász built a bridge between graphs and alternating matrix spaces, in the context of perfect matchings [Proceedings of FCT, 1979, pp. 565--574]. A similar connection between bipartite graphs and matrix spaces plays a key role in the recent resolutions of the noncommutative rank problem [A. Garg et al., Proceedings of FOCS, 2016, pp. 109--117; G. Ivanyos, Y. Qiao, and K. V. Subrahmanyam, Comput. Complexity, 26 (2017), pp. 717--763]. In this paper, we lay the foundation for another bridge between graphs and alternating matrix spaces, in the context of independent sets and vertex colorings. The corresponding structures in alternating matrix spaces are isotropic spaces and isotropic decompositions, both useful structures in group theory and manifold theory. We first show that the maximum independent set problem and the vertex $c$-coloring problem reduce to the maximum isotropic space problem and the isotropic $c$-decomposition problem, respectively. Next, we show that several topics and results about independent sets and vertex colorings have natural correspondences for isotropic spaces and decompositions. These include algorithmic problems, such as the maximum independent set problem for bipartite graphs, and exact exponential-time algorithms for the chromatic number, as well as mathematical questions, such as the number of maximal independent sets, and the relation between the maximum degree and the chromatic number. These connections lead to new interactions between graph theory and algebra. Some results have concrete applications to group theory and manifold theory, and we initiate a variant of these structures in the context of quantum information theory. Finally, we propose several open questions for further exploration.

Parameterized and exact algorithms for class domination coloring

A class domination coloring (also called cd-Coloring or dominated coloring) of a graph is a proper coloring in which every color class is contained in the neighborhood of some vertex. The minimum number of colors required for any cd-coloring of G, denoted by χcd(G), is called the class domination chromatic number (cd-chromatic number) of G. In this work, we consider two problems associated with the cd-coloring of a graph in the context of exact exponential-time algorithms and parameterized complexity. (1) Given a graph G on n vertices, find its cd-chromatic number. (2) Given a graph G and integers k and q, can we delete at most k vertices such that the cd-chromatic number of the resulting graph is at most q? For the first problem, we give an exact algorithm with running time O(2nn4logn). Also, we show that the problem is FPT with respect to the number q of colors as the parameter on chordal graphs. On graphs of girth at least 5, we show that the problem also admits a kernel with O(q3) vertices. For the second (deletion) problem, we show NP-hardness for each q≥2. Further, on split graphs, we show that the problem is NP-hard if q is a part of the input and FPT with respect to k and q as combined parameters. As recognizing graphs with cd-chromatic number at most q is NP-hard in general for q≥4, the deletion problem is unlikely to be FPT when parameterized by the size of the deletion set on general graphs. We show fixed parameter tractability for q∈{2,3} using the known algorithms for finding a vertex cover and an odd cycle transversal as subroutines.

Matching Cut in Graphs with Large Minimum Degree

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be {mathsf {NP}}-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is {mathsf {NP}}-complete on graphs with minimum degree two. In this paper, we show that, for any given constant c>1, Matching Cut is {mathsf {NP}}-complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant epsilon >0, Matching Cut remains {mathsf {NP}}-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree delta >n^{1-epsilon }. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree delta ge 3 in O^*(lambda ^n) time, where lambda is the positive root of the polynomial x^{delta +1}-x^{delta }-1. Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is O^*(1.0099^n) on graphs with minimum degree delta ge 469. Complementing our hardness results, we show that, for any two fixed constants 1< c <4 and c^{prime }ge 0, Matching Cut is solvable in polynomial time for graphs with large minimum degree delta ge frac{1}{c}n-c^{prime }.

On the Complexity of the Smallest Grammar Problem over Fixed Alphabets

In the smallest grammar problem, we are given a word w and we want to compute a preferably small context-free grammar G for the singleton language {w} (where the size of a grammar is the sum of the sizes of its rules, and the size of a rule is measured by the length of its right side). It is known that, for unbounded alphabets, the decision variant of this problem is NP-hard and the optimisation variant does not allow a polynomial-time approximation scheme, unless P = NP. We settle the long-standing open problem whether these hardness results also hold for the more realistic case of a constant-size alphabet. More precisely, it is shown that the smallest grammar problem remains NP-complete (and its optimisation version is APX-hard), even if the alphabet is fixed and has size of at least 17. The corresponding reduction is robust in the sense that it also works for an alternative size-measure of grammars that is commonly used in the literature (i. e., a size measure also taking the number of rules into account), and it also allows to conclude that even computing the number of rules required by a smallest grammar is a hard problem. On the other hand, if the number of nonterminals (or, equivalently, the number of rules) is bounded by a constant, then the smallest grammar problem can be solved in polynomial time, which is shown by encoding it as a problem on graphs with interval structure. However, treating the number of rules as a parameter (in terms of parameterised complexity) yields W[1]-hardness. Furthermore, we present an mathcal {O}(3^{mid {w}mid }) exact exponential-time algorithm, based on dynamic programming. These three main questions are also investigated for 1-level grammars, i. e., grammars for which only the start rule contains nonterminals on the right side; thus, investigating the impact of the “hierarchical depth” of grammars on the complexity of the smallest grammar problem. In this regard, we obtain for 1-level grammars similar, but slightly stronger results.

Evolution through segmental duplications and losses: a Super-Reconciliation approach

The classical gene and species tree reconciliation, used to infer the history of gene gain and loss explaining the evolution of gene families, assumes an independent evolution for each family. While this assumption is reasonable for genes that are far apart in the genome, it is not appropriate for genes grouped into syntenic blocks, which are more plausibly the result of a concerted evolution. Here, we introduce the Super-Reconciliation problem which consists in inferring a history of segmental duplication and loss events (involving a set of neighboring genes) leading to a set of present-day syntenies from a single ancestral one. In other words, we extend the traditional Duplication-Loss reconciliation problem of a single gene tree, to a set of trees, accounting for segmental duplications and losses. Existency of a Super-Reconciliation depends on individual gene tree consistency. In addition, ignoring rearrangements implies that existency also depends on gene order consistency. We first show that the problem of reconstructing a most parsimonious Super-Reconciliation, if any, is NP-hard and give an exact exponential-time algorithm to solve it. Alternatively, we show that accounting for rearrangements in the evolutionary model, but still only minimizing segmental duplication and loss events, leads to an exact polynomial-time algorithm. We finally assess time efficiency of the former exponential time algorithm for the Duplication-Loss model on simulated datasets, and give a proof of concept on the opioid receptor genes.

Gehrlein stability in committee selection: parameterized hardness and algorithms

In a multiwinner election based on the Condorcet criterion, we are given a set of candidates, and a set of voters with strict preference ranking over the candidates. A committee is \em weakly Gehrlein stable (WGS) if each committee member is preferred to each non-member by at least half of the voters. Recently, Aziz et al. [IJCAI 2017] studied the computational complexity of finding a WGS committee of size k. They show that this problem is \sf NP -hard in general and polynomial time solvable when the number of voters is odd. In this article, we initiate a systematic study of the problem in the realm of parameterized complexity. We first show that the problem is \sf W [1]-hard when parameterized by the size of the committee. To overcome this intractability result, we use a known reformulation of WGS as a problem on directed graphs and then use parameters that measure the "structure'' of these directed graphs. In particular, we consider the majority graph, defined as follows: there is a vertex corresponding to each candidate, and there is a directed arc from a candidate c to $c'$ if the number of voters that \em prefer c over $c'$ is more than those that prefer $c'$ over c. The problem of finding WGS committee of size k corresponds to finding a vertex subset X of size k in the majority graph with the following property: the set X contains no vertex outside the committee that has an in-neighbor in X. Observe that the polynomial time algorithm of Aziz et al. [IJCAI 2017] corresponds to solving the problem on a tournament (a complete graph with orientation on edges). Thus, natural parameters to study our problem are "closeness'' to being a tournament. We define closeness as the number of missing arcs in the given directed graph and the number of vertices we need to delete from the given directed graph such that the resulting graph is a tournament. We show that the problem is fixed parameter tractable (\sf FPT ) and admits linear kernels with respect to closeness parameters. Finally, we also design an exact exponential time algorithm running in time $OO(1.2207^nn^OO(1) )$. Here, n denotes the number of candidates.

Path Contraction Faster than $2^n$

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-Contraction problem is called Path Contraction. The problem Path Contraction admits a simple algorithm running in time $2^{n}\cdot n^{{\mathcal{O}}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{{\mathcal{O}}(1)}$ bound for Path Contraction seems quite challenging. In this paper, we design an exact exponential time algorithm for Path Contraction that runs in time ${{1.99987}^n}\cdot n^{{\mathcal O}(1)}$. We also define a problem called 3-Disjoint Connected Subgraphs and design an algorithm for it that runs in time $1.88^n\cdot n^{{\mathcal O}(1)}$. The above algorithm is used as a subroutine in our algorithm for Path Contraction.

Exponential time algorithms for just-in-time scheduling problems with common due date and symmetric weights

This paper focuses on the solution, by exact exponential-time algorithms, of just-in-time scheduling problems when jobs have symmetric earliness/tardiness weights and share a non restrictive common due date. For the single machine problem, a Sort & Search algorithm is proposed with worst-case time and space complexities in $$\mathcal {O}^*(1.4143^n)$$. This algorithm relies on an original modeling of the problem. For the case of parallel machines, an algorithm integrating a dynamic programming algorithm across subsets and machines and the above Sort & Search algorithm is proposed with worst-case time and space complexities in $$\mathcal {O}^*(3^n)$$. To the best of our knowledge, these are the first worst-case complexity results obtained for non regular criteria in scheduling.

Exact Algorithms via Monotone Local Search

We give a new general approach for designing exact exponential-time algorithms for subset problems . In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the family contains at least one set. A typical example of a subset problem is W EIGHTED d -SAT. Here, the input is a CNF-formula with clauses of size at most d , and an integer W . The universe is the set of variables and the variables have integer weights. The family contains all the subsets S of variables such that the total weight of the variables in S does not exceed W and setting the variables in S to 1 and the remaining variables to 0 satisfies the formula. Our approach is based on “monotone local search,” where the goal is to extend a partial solution to a solution by adding as few elements as possible. More formally, in the extension problem, we are also given as input a subset X of the universe and an integer k . The task is to determine whether one can add at most k elements to X to obtain a set in the (implicitly defined) family. Our main result is that a c k n O(1) time algorithm for the extension problem immediately yields a randomized algorithm for finding a solution of any size with running time O ((2−1/ c ) n ). In many cases, the extension problem can be reduced to simply finding a solution of size at most k . Furthermore, efficient algorithms for finding small solutions have been extensively studied in the field of parameterized algorithms. Directly applying these algorithms, our theorem yields in one stroke significant improvements over the best known exponential-time algorithms for several well-studied problems, including d -H ITTING S ET , F EEDBACK V ERTEX S ET , N ODE U NIQUE L ABEL C OVER , and W EIGHTED d -SAT. Our results demonstrate an interesting and very concrete connection between parameterized algorithms and exact exponential-time algorithms. We also show how to derandomize our algorithms at the cost of a subexponential multiplicative factor in the running time. Our derandomization is based on an efficient construction of a new pseudo-random object that might be of independent interest. Finally, we extend our methods to establish new combinatorial upper bounds and develop enumeration algorithms.

Solving Graph Problems via Potential Maximal Cliques

The BT algorithm of Bouchitté and Todinca based on enumerating potential maximal cliques, originally proposed for the treewidth and minimum fill-in problems, yields improved exact exponential-time algorithms for various graph optimization problems related to optimal triangulations. While the BT algorithm has received significant attention in terms of theoretical analysis, less attention has been paid on engineering efficient implementations of the algorithm for different problems and thereby on empirical studies on its effectiveness in practice. In this work, we provide an experimental evaluation of an implementation of the BT algorithm, based on our second-place winning entry in the 2nd Parameterized Algorithms and Computational Experiments Challenge (PACE 2017), extended to several related graph problems: treewidth, minimum fill-in, generalized and fractional hypertreewidth, and the total table size problem. Instead of focusing on problem-specific optimization of BT for a particular problem, our focus in this work is on studying the applicability of BT more generally to a range of problems. Based on the results, we conclude that an efficient implementation of the BT algorithm yields an empirically competitive approach to each of the considered problems when compared to available implementations of alternative problem-specific algorithmic approaches.

Minimum-Cost Heterogeneous Node Placement in Wireless Sensor Networks

The operational cost-effectiveness of wireless sensor networks depends on the placement of heterogeneous nodes: base stations, sensor nodes, and relay nodes. To achieve the global optimality of minimum-cost heterogeneous node placement, we find the locations of base stations, sensor nodes, and relay nodes, simultaneously. The objective is to minimize the sum of node production and placement costs and transmission outage probabilities in the routing tree. First, we formulate this minimum-cost heterogeneous node placement problem as a new NP-hard Steiner tree problem. Then, to solve this problem for both small and large instances, we propose an exponential-time exact algorithm, a polynomial-time heuristic algorithm, and several post-processing algorithms. On a personal computer, our exact algorithm is sufficiently fast to produce optimal solutions for small instances with dozens of vertices, while our heuristic and post-processing algorithms can, respectively, produce and improve suboptimal solutions for large instances with 100 000 vertices and 1 000 000 edges within 6 s. We also compare the heuristic and optimal solutions for small instances with dozens of vertices and show that the ratios between the respective solution costs are generally below 1.35, and our post-processing algorithms can improve this number to 1.1. This indicates that our heuristic and post-processing algorithms are likely to produce near-optimal solutions in practice and are useful for minimum-cost heterogeneous node placement when computational resources are scarce.

Harmonious coloring: Parameterized algorithms and upper bounds

A harmonious coloring of a graph is a partitioning of its vertex set into parts such that, there are no edges inside each part, and there is at most one edge between any pair of parts. It is known that finding a minimum harmonious coloring number is NP-hard even in special classes of graphs like trees and split graphs.We initiate a study of parameterized and exact exponential time complexity of harmonious coloring. We consider various parameterizations like by solution size, by above or below known guaranteed bounds and by the vertex cover number of the graph. While the problem has a simple quadratic kernel when parameterized by the solution size, our main result is that the problem is fixed-parameter tractable when parameterized by the size of a vertex cover of the graph. This is shown by reducing the problem to multiple instances of fixed variable integer linear programming.We also observe that it is W[1]-hard to determine whether at most n−k or Δ+1+k colors are sufficient in a harmonious coloring of an n-vertex graph G, where Δ is the maximum degree of G and k is the parameter. Concerning exact exponential time algorithms, we develop a 2nnO(1) algorithm for finding a minimum harmonious coloring in split graphs improving on the naive 2O(nlog⁡n) algorithm.

The approximability of maximum rooted triplets consistency with fan triplets and forbidden triplets

The NP-hard maximum rooted resolved triplets consistency problem (MRTC) takes as input a set S of leaf labels and a set R of resolved triplets over S and asks for a rooted phylogenetic tree that is consistent with the maximum number of elements in R. This article studies the approximability of a generalization of the problem called the maximum rooted triplets consistency problem (MTC) where in addition to resolved triplets, the input may contain fan triplets, forbidden resolved triplets, and forbidden fan triplets. To begin with, we observe that MTC admits a 1∕4-approximation in polynomial time. Next, we generalize Wu’s exact exponential-time algorithm for MRTC (Wu, 2004) to MTC. Forcing the algorithm to always output a rooted k-ary phylogenetic tree for any specified k≥2 subsequently leads to an exponential-time approximation scheme (ETAS) for MTC. We then present a polynomial-time approximation scheme (PTAS) for complete instances of MTC (meaning that for every S′⊆S with |S′|=3, R contains at least one rooted triplet involving the leaf labels in S′), based on the techniques introduced by Jiang et al. (2001) for a related problem. We also study the computational complexity of MTC restricted to fan triplets and forbidden fan triplets. Finally, extensions to weighted instances are considered.

HybridHAM: A Novel Hybrid Heuristic for Finding Hamiltonian Cycle

Hamiltonian Cycle Problem is one of the most explored combinatorial problems. Being an NP-complete problem, heuristic approaches are found to be more powerful than exponential time exact algorithms. This paper presents an efficient hybrid heuristic that sits in between the complex reliable approaches and simple faster approaches. The proposed algorithm is a combination of greedy, rotational transformation and unreachable vertex heuristics that works in three phases. In the first phase, an initial path is created by using greedy depth first search. This initial path is then extended to a Hamiltonian path in second phase by using rotational transformation and greedy depth first search. Third phase converts the Hamiltonian path into a Hamiltonian cycle by using rotational transformation. The proposed approach could find Hamiltonian cycles from a set of hard graphs collected from the literature, all the Hamiltonian instances (1000 to 5000 vertices) given in TSPLIB, and some instances of FHCP Challenge Set. Moreover, the algorithm has O(n3) worst case time complexity. The performance of the algorithm has been compared with the state-of-the-art algorithms and it was found that HybridHAM outperforms others in terms of running time.

Scenario reduction revisited: fundamental limits and guarantees

The goal of scenario reduction is to approximate a given discrete distribution with another discrete distribution that has fewer atoms. We distinguish continuous scenario reduction, where the new atoms may be chosen freely, and discrete scenario reduction, where the new atoms must be chosen from among the existing ones. Using the Wasserstein distance as measure of proximity between distributions, we identify those n-point distributions on the unit ball that are least susceptible to scenario reduction, i.e., that have maximum Wasserstein distance to their closest m-point distributions for some prescribed $$m<n$$ . We also provide sharp bounds on the added benefit of continuous over discrete scenario reduction. Finally, to our best knowledge, we propose the first polynomial-time constant-factor approximations for both discrete and continuous scenario reduction as well as the first exact exponential-time algorithms for continuous scenario reduction.

On distance-preserving elimination orderings in graphs: Complexity and algorithms

For every connected graph G, a subgraph H of G is isometric if the distance between any two vertices in H is the same in H as in G. A distance-preserving elimination ordering of G is a total ordering of its vertex-set V(G), denoted (v1,v2,…,vn), such that any subgraph Gi=G∖(v1,v2,…,vi) with 1≤i<n is isometric. This kind of ordering has been introduced by Chepoi in his study on weakly modular graphs (Chepoi, 1998). We prove that it is NP-complete to decide whether such ordering exists for a given graph — even if it has diameter at most 2. Then, we prove on the positive side that the problem of computing a distance-preserving ordering when there exists one is fixed-parameter-tractable in the treewidth. Lastly, we describe a heuristic in order to compute a distance-preserving ordering when there exists one that we compare to an exact exponential time algorithm and to an ILP formulation for the problem.

Stable Matching Games: Manipulation via Subgraph Isomorphism

In this paper we consider a problem that arises from a strategic issue in the stable matching model (with complete preference lists) from the viewpoint of exact-exponential time algorithms. Specifically, we study the Stable Extension of Partial Matching (SEOPM) problem, where the input consists of the complete preference lists of men, and a partial matching. The objective is to find (if one exists) a set of preference lists of women, such that the men-optimal Gale–Shapley algorithm outputs a perfect matching that contains the given partial matching. Kobayashi and Matsui (Algorithmica 58:151–169, 2010) proved this problem is NP-complete. In this article, we give an exact-exponential algorithm for SEOPM running in time $$2^{{\mathcal {O}} (n)}$$ , where n denotes the number of men/women. We complement our algorithmic finding by showing that unless Exponential Time Hypothesis (ETH) fails, our algorithm is asymptotically optimal. That is, unless ETH fails, there is no algorithm for SEOPM running in time $$2^{o(n)}$$ . Our algorithm is a non-trivial combination of a parameterized algorithm for Subgraph Isomorphism, a relationship between stable matching and finding an out-branching in an appropriate graph and enumerating all possible non-isomorphic out-branchings. Our results cover both the cases when the preference lists are strict and complete, and when they are strict but possibly incomplete.

Finding Consensus Strings with Small Length Difference between Input and Solution Strings

The C losest S ubstring P roblem is to decide, for given strings s 1 , … , s k of length at most ℓ and numbers m and d , whether there is a length- m string s and length- m substrings s ′ i of s i , such that s has a Hamming distance of at most d from each s ′ i . If we instead require the sum of all the Hamming distances between s and each s ′ i to be bounded by d , then it is called the C onsensus P atterns P roblem . We contribute to the parameterised complexity analysis of these classical NP-hard string problems by investigating the parameter (ℓ − m ), i.e., the length difference between input and solution strings. For most combinations of (ℓ − m ) and one of the classical parameters ( m , ℓ, k , or d ), we obtain fixed-parameter tractability. However, even for constant (ℓ − m ) and constant alphabet size, both problems remain NP-hard. While this follows from known results with respect to the C losest S ubstring , we need a new reduction in the case of the C onsensus P atterns . As a by-product of this reduction, we obtain an exact exponential-time algorithm for both problems, which is based on an alphabet reduction.