arXiv subjects

Serge Gaspers

Publications and source records attributed to Serge Gaspers.

At least 55 records · Page 3Linked to original sources

arXiv2012

Don't Be Strict in Local Search!

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires n to the power of O(k) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.

cs.DS↗

arXiv2012

On Finding Optimal Polytrees

Inferring probabilistic networks from data is a notoriously difficult task. Under various goodness-of-fit measures, finding an optimal network is NP-hard, even if restricted to polytrees of bounded in-degree. Polynomial-time algorithms are known only for rare special cases, perhaps most notably for branchings, that is, polytrees in which the in-degree of every node is at most one. Here, we study the complexity of finding an optimal polytree that can be turned into a branching by deleting some number of arcs or nodes, treated as a parameter. We show that the problem can be solved via a matroid intersection formulation in polynomial time if the number of deleted arcs is bounded by a constant. The order of the polynomial time bound depends on this constant, hence the algorithm does not establish fixed-parameter tractability when parameterized by the number of deleted arcs. We show that a restricted version of the problem allows fixed-parameter tractability and hence scales well with the parameter. We contrast this positive result by showing that if we parameterize by the number of deleted nodes, a somewhat more powerful parameter, the problem is not fixed-parameter tractable, subject to a complexity-theoretic assumption.

cs.DS↗

arXiv2012

An Exponential Time 2-Approximation Algorithm for Bandwidth

The bandwidth of a graph G on n vertices is the minimum b such that the vertices of G can be labeled from 1 to n such that the labels of every pair of adjacent vertices differ by at most b. In this paper, we present a 2-approximation algorithm for the bandwidth problem that takes worst-case O(1.9797^n) time and uses polynomial space. This improves both the previous best 2- and 3-approximation algorithms of Cygan et al. which have an O(3^n) and O(2^n) worst-case time bounds, respectively. Our algorithm is based on constructing bucket decompositions of the input graph. A bucket decomposition partitions the vertex set of a graph into ordered sets (called buckets) of (almost) equal sizes such that all edges are either incident to vertices in the same bucket or to vertices in two consecutive buckets. The idea is to find the smallest bucket size for which there exists a bucket decomposition. The algorithm uses a simple divide-and-conquer strategy along with dynamic programming to achieve this improved time bound.

cs.DS↗

arXiv2012

Strong Backdoors to Bounded Treewidth SAT

There are various approaches to exploiting "hidden structure" in instances of hard combinatorial problems to allow faster algorithms than for general unstructured or random instances. For SAT and its counting version #SAT, hidden structure has been exploited in terms of decomposability and strong backdoor sets. Decomposability can be considered in terms of the treewidth of a graph that is associated with the given CNF formula, for instance by considering clauses and variables as vertices of the graph, and making a variable adjacent with all the clauses it appears in. On the other hand, a strong backdoor set of a CNF formula is a set of variables such that each possible partial assignment to this set moves the formula into a fixed class for which (#)SAT can be solved in polynomial time. In this paper we combine the two above approaches. In particular, we study the algorithmic question of finding a small strong backdoor set into the class W_t of CNF formulas whose associated graphs have treewidth at most t. The main results are positive: (1) There is a cubic-time algorithm that, given a CNF formula F and two constants k,t\ge 0, either finds a strong W_t-backdoor set of size at most 2^k, or concludes that F has no strong W_t-backdoor set of size at most k. (2) There is a cubic-time algorithm that, given a CNF formula F, computes the number of satisfying assignments of F or concludes that sb_t(F)>k, for any pair of constants k,t\ge 0. Here, sb_t(F) denotes the size of a smallest strong W_t-backdoor set of F. The significance of our results lies in the fact that they allow us to exploit algorithmically a hidden structure in formulas that is not accessible by any one of the two approaches (decomposability, backdoors) alone. Already a backdoor size 1 on top of treewidth 1 (i.e., sb_1(F)=1) entails formulas of arbitrarily large treewidth and arbitrarily large cycle cutsets.

cs.DS↗

arXiv2012

From edge-disjoint paths to independent paths

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of them contains an interior vertex of another. We prove that f(k)=k if k<3, and f(k)=3 otherwise.

math.CO↗

arXiv2012

Strong Backdoors to Nested Satisfiability

Knuth (1990) introduced the class of nested formulas and showed that their satisfiability can be decided in polynomial time. We show that, parameterized by the size of a smallest strong backdoor set to the target class of nested formulas, checking the satisfiability of any CNF formula is fixed-parameter tractable. Thus, for any k>0, the satisfiability problem can be solved in polynomial time for any formula F for which there exists a variable set B of size at most k such that for every truth assignment t to B, the formula F[t] is nested; moreover, the degree of the polynomial is independent of k. Our algorithm uses the grid-minor theorem of Robertson and Seymour (1986) to either find that the incidence graph of the formula has bounded treewidth - a case that is solved using model checking for monadic second order logic - or to find many vertex-disjoint obstructions in the incidence graph. For the latter case, new combinatorial arguments are used to find a small backdoor set. Combining both cases leads to an approximation algorithm producing a strong backdoor set whose size is upper bounded by a function of the optimum. Going through all assignments to this set of variables and using Knuth's algorithm, the satisfiability of the input formula is decided.

cs.DS↗

arXiv2012

A Linear Vertex Kernel for Maximum Internal Spanning Tree

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning tree with at least k internal vertices if and only if G' has a spanning tree with at least k' internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."

cs.DS↗

arXiv2012

Backdoors to Acyclic SAT

Backdoor sets, a notion introduced by Williams et al. in 2003, are certain sets of key variables of a CNF formula F that make it easy to solve the formula; by assigning truth values to the variables in a backdoor set, the formula gets reduced to one or several polynomial-time solvable formulas. More specifically, a weak backdoor set of F is a set X of variables such that there exits a truth assignment t to X that reduces F to a satisfiable formula F[t] that belongs to a polynomial-time decidable base class C. A strong backdoor set is a set X of variables such that for all assignments t to X, the reduced formula F[t] belongs to C. We study the problem of finding backdoor sets of size at most k with respect to the base class of CNF formulas with acyclic incidence graphs, taking k as the parameter. We show that 1. the detection of weak backdoor sets is W[2]-hard in general but fixed-parameter tractable for r-CNF formulas, for any fixed r>=3, and 2. the detection of strong backdoor sets is fixed-parameter approximable. Result 1 is the the first positive one for a base class that does not have a characterization with obstructions of bounded size. Result 2 is the first positive one for a base class for which strong backdoor sets are more powerful than deletion backdoor sets. Not only SAT, but also #SAT can be solved in polynomial time for CNF formulas with acyclic incidence graphs. Hence Result 2 establishes a new structural parameter that makes #SAT fixed-parameter tractable and that is incomparable with known parameters such as treewidth and clique-width. We obtain the algorithms by a combination of an algorithmic version of the Erdös-Pósa Theorem, Courcelle's model checking for monadic second order logic, and new combinatorial results on how disjoint cycles can interact with the backdoor set.

cs.DS↗

arXiv2011

k-Gap Interval Graphs

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.

cs.DS↗

arXiv2011

Backdoors to Satisfaction

A backdoor set is a set of variables of a propositional formula such that fixing the truth values of the variables in the backdoor set moves the formula into some polynomial-time decidable class. If we know a small backdoor set we can reduce the question of whether the given formula is satisfiable to the same question for one or several easy formulas that belong to the tractable class under consideration. In this survey we review parameterized complexity results for problems that arise in the context of backdoor sets, such as the problem of finding a backdoor set of size at most k, parameterized by k. We also discuss recent results on backdoor sets for problems that are beyond NP.

cs.DS↗

arXiv2011

Feedback Vertex Sets in Tournaments

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets, and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament.

cs.DM↗

arXiv2011

Complexity of Splits Reconstruction for Low-Degree Trees

Given a vertex-weighted tree T, the split of an edge xy in T is min{s_x(xy), s_y(xy)} where s_u(uv) is the sum of all weights of vertices that are closer to u than to v in T. Given a set of weighted vertices V and a multiset of splits S, we consider the problem of constructing a tree on V whose splits correspond to S. The problem is known to be NP-complete, even when all vertices have unit weight and the maximum vertex degree of T is required to be no more than 4. We show that the problem is strongly NP-complete when T is required to be a path, the problem is NP-complete when all vertices have unit weight and the maximum degree of T is required to be no more than 3, and it remains NP-complete when all vertices have unit weight and T is required to be a caterpillar with unbounded hair length and maximum degree at most 3. We also design polynomial time algorithms for the variant where T is required to be a path and the number of distinct vertex weights is constant, and the variant where all vertices have unit weight and T has a constant number of leaves. The latter algorithm is not only polynomial when the number of leaves, k, is a constant, but also fixed-parameter tractable when parameterized by k. Finally, we shortly discuss the problem when the vertex weights are not given but can be freely chosen by an algorithm. The considered problem is related to building libraries of chemical compounds used for drug design and discovery. In these inverse problems, the goal is to generate chemical compounds having desired structural properties, as there is a strong correlation between structural properties, such as the Wiener index, which is closely connected to the considered problem, and biological activity.

cs.DS↗

arXiv2011

Parameterizing by the Number of Numbers

The usefulness of parameterized algorithmics has often depended on what Niedermeier has called, "the art of problem parameterization". In this paper we introduce and explore a novel but general form of parameterization: the number of numbers. Several classic numerical problems, such as Subset Sum, Partition, 3-Partition, Numerical 3-Dimensional Matching, and Numerical Matching with Target Sums, have multisets of integers as input. We initiate the study of parameterizing these problems by the number of distinct integers in the input. We rely on an FPT result for ILPF to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. In various applied settings, problem inputs often consist in part of multisets of integers or multisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Such number-of-numbers parameterized problems often reduce to subproblems about transition systems of various kinds, parameterized by the size of the system description. We consider several core problems of this kind relevant to number-of-numbers parameterization. Our main hardness result considers the problem: given a non-deterministic Mealy machine M (a finite state automaton outputting a letter on each transition), an input word x, and a census requirement c for the output word specifying how many times each letter of the output alphabet should be written, decide whether there exists a computation of M reading x that outputs a word y that meets the requirement c. We show that this problem is hard for W[1]. If the question is whether there exists an input word x such that a computation of M on x outputs a word that meets c, the problem becomes fixed-parameter tractable.

cs.DS↗

arXiv2011

Kernels for Global Constraints

Bessiere et al. (AAAI'08) showed that several intractable global constraints can be efficiently propagated when certain natural problem parameters are small. In particular, the complete propagation of a global constraint is fixed-parameter tractable in k - the number of holes in domains - whenever bound consistency can be enforced in polynomial time; this applies to the global constraints AtMost-NValue and Extended Global Cardinality (EGC). In this paper we extend this line of research and introduce the concept of reduction to a problem kernel, a key concept of parameterized complexity, to the field of global constraints. In particular, we show that the consistency problem for AtMost-NValue constraints admits a linear time reduction to an equivalent instance on O(k^2) variables and domain values. This small kernel can be used to speed up the complete propagation of NValue constraints. We contrast this result by showing that the consistency problem for EGC constraints does not admit a reduction to a polynomial problem kernel unless the polynomial hierarchy collapses.

cs.AI↗

arXiv2010

A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set

An independent dominating set D of a graph G = (V,E) is a subset of vertices such that every vertex in V \ D has at least one neighbor in D and D is an independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum independent dominating set in a graph is an NP-hard problem. Whereas it is hard to cope with this problem using parameterized and approximation algorithms, there is a simple exact O(1.4423^n)-time algorithm solving the problem by enumerating all maximal independent sets. In this paper we improve the latter result, providing the first non trivial algorithm computing a minimum independent dominating set of a graph in time O(1.3569^n). Furthermore, we give a lower bound of Ω(1.3247^n) on the worst-case running time of this algorithm, showing that the running time analysis is almost tight.

cs.DS↗

arXiv2009

Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.

cs.DS↗

arXiv2009

A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between

In this paper we introduce "hybrid" Max 2-CSP formulas consisting of "simple clauses", namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting. We use new reductions introduced here, and recent reductions such as "clause-learning" and "2-reductions" generalized to our setting's mixture of simple and general clauses. Parametrizing an instance by the fraction p of non-simple clauses, we give an exact (exponential-time) algorithm that is the fastest known polynomial-space algorithm for p=0 (which includes the well-studied Max 2-Sat problem but also instances with arbitrary mixtures of AND and OR clauses); the only efficient algorithm for mixtures of AND, OR, and general integer-valued clauses; and tied for fastest for general Max 2-CSP (p=1). Since a pure 2-Sat input instance may be transformed to a general CSP instance in the course of being solved, the algorithm's efficiency and generality go hand in hand. Our algorithm analysis and optimization are a variation on the familiar measure-and-conquer approach, resulting in an optimizing mathematical program that is convex not merely quasi-convex, and thus can be solved efficiently and with a certificate of optimality. We produce a family of running-time upper-bound formulas, each optimized for instances with a particular value of p but valid for all instances.

cs.DS↗

arXiv2009

Exact Exponential Time Algorithms for Max Internal Spanning Tree

We consider the NP-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form O*(c^n) (c <= 3). The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of O*(1.8669^n) when analyzed with respect to the number of vertices. We also show that its running time is 2.1364^k n^O(1) when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analyses for parameterized algorithms.

cs.DS↗