The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of solution discovery via reconfiguration for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state or configuration. Our framework integrates and formalizes different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely Vertex Cover , Independent Set , Dominating Set , and Coloring . We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.

This paper considers four problems with possible applications in network design: Given a graph G with | G | = n and an integer k ≥ 0 , does G have a DFS tree with (i) ≤ k leaves, (ii) ≥ k leaves, (iii) ≤ n − k leaves, and (iv) ≥ n − k leaves? We show that all four problems are NP-hard. When parameterized by k , we prove that while (i) is para-NP-hard and (ii) is W[1]-hard, both (iii) and (iv) admit polynomial kernels with O ( k 3 ) vertices, implying FPT algorithms running in k O ( k ) ⋅ n O ( 1 ) time. Our polynomial kernels are based on a O ( k ) -sized vertex cover structure associated with the solution of these problems. As a byproduct, we obtain polynomial kernels for these problems parameterized by the vertex cover number of the input graph.

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition.

The dynamics of real-world applications and systems require efficient methods for improving infeasible solutions or restoring corrupted ones by making modifications to the current state of a system in a restricted way. We propose a new framework of solution discovery via reconfiguration for constructing a feasible solution for a given problem by executing a sequence of small modifications starting from a given state. Our framework integrates different aspects of classical local search, reoptimization, and combinatorial reconfiguration. We exemplify our framework on a multitude of fundamental combinatorial problems, namely VERTEX COVER, INDEPENDENT SET, DOMINATING SET, and COLORING. We study the classical as well as the parameterized complexity of the solution discovery variants of those problems and explore the boundary between tractable and intractable instances.

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

This paper gives an appreciation of Edward “Ed” Blum, with accolades and stories from family and colleagues. It also gives a brief description of the situation of computer science in the 1960s, at the time Ed founded the Journal of Computer and System Sciences.

Abduction belongs to the most fundamental reasoning methods. It is a method for reverse inference, this means one is interested in explaining observed behavior by finding appropriate causes. We study logic-based abduction, where knowledge is represented by propositional formulas. The computational complexity of this problem is highly intractable in many interesting settings. In this work we therefore present an extensive parameterized complexity analysis of abduction within various fragments of propositional logic together with (combinations of) natural parameters.

In the parameterized problem MaxLin2-AA[$k$], we are given a system with variables x_1,...,x_n consisting of equations of the form Product_{i in I}x_i = b, where x_i,b in {-1, 1} and I is a nonempty subset of {1,...,n}, each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+k, where W is the total weight of all equations and k is the parameter (if k=0, the possibility is assured). We show that MaxLin2-AA[k] has a kernel with at most O(k^2 log k) variables and can be solved in time 2^{O(k log k)}(nm)^{O(1)}. This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on Max-$r$-Lin2-AA[k,r] which implies that Max-r-Lin2-AA[k,r] has a kernel with at most (2k-1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f that maps {-1,1}^n to the set of reals and whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the Edwards-Erdös bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 +(n-1)/4 edges) and obtaining a generalization.

We examine the complexity of constraint satisfaction problems that consist of a set of AllDiff constraints. Such CSPs naturally model a wide range of real-world and combinatorial problems, like scheduling, frequency allocations and graph coloring problems. As this problem is known to be NP-complete, we investigate under which further assumptions it becomes tractable. We observe that a crucial property seems to be the convexity of the variable domains and constraints. Our main contribution is an extensive study of the complexity of Multiple AllDiff CSPs for a set of natural parameters, like maximum domain size and maximum size of the constraint scopes. We show that, depending on the parameter, convexity can make the problem tractable while it is provably intractable in general

The Convex Recoloring (CR) problem measures how far a tree of characters differs from exhibiting a so-called “perfect phylogeny”. For an input consisting of a vertex-colored tree T, the problem is to determine whether recoloring at most k vertices can achieve a convex coloring, meaning by this a coloring where each color class induces a subtree. The problem was introduced by Moran and Snir (J. Comput. Syst. Sci. 73:1078–1089, 2007; J. Comput. Syst. Sci. 74:850–869, 2008) who showed that CR is NP-hard, and described a search-tree based FPT algorithm with a running time of O(k(k/log k) k n 4). The Moran and Snir result did not provide any nontrivial kernelization. In this paper, we show that CR has a kernel of size O(k 2).

We study the NP-hard LIST-COLORED GRAPH MOTIF problem which, given an undirected list-colored graph G = (V, E) and a multiset M of colors, asks for maximum-cardinality sets S ⊆ V and M' ⊆ M such that G[S] is connected and contains exactly (with respect to multiplicity) the colors in M'. LIST-COLORED GRAPH MOTIF has applications in the analysis of biological networks. We study LIST-COLORED GRAPH MOTIF with respect to three different parameterizations. For the parameters motif size |M| and solution size |S|, we present fixed-parameter algorithms, whereas for the parameter |V| - |M|, we show W[1]-hardness for general instances and achieve fixed-parameter tractability for a special case of LIST-COLORED GRAPH MOTIF. We implemented the fixed-parameter algorithms for parameters |M| and |S|, developed further speed-up heuristics for these algorithms, and applied them in the context of querying protein-interaction networks, demonstrating their usefulness for realistic instances. Furthermore, we show that extending the request for motif connectedness to stronger demands, such as biconnectedness or bridge-connectedness leads to W[1]-hard problems when the parameter is the motif size |M|.

The haplotype inference problem (HIP) asks to find a set of haplotypes which resolve a given set of genotypes. This problem is important in practical fields such as the investigation of diseases or other types of genetic mutations. In order to find the haplotypes which are as close as possible to the real set of haplotypes that comprise the genotypes, two models have been suggested which are by now well-studied: The perfect phylogeny model and the pure parsimony model. All known algorithms up till now for haplotype inference may find haplotypes that are not necessarily plausible, i.e., very rare haplotypes or haplotypes that were never observed in the population. In order to overcome this disadvantage, we study in this paper, a new constrained version of HIP under the above-mentioned models. In this new version, a pool of plausible haplotypes H is given together with the set of genotypes G, and the goal is to find a subset H ⊆ H that resolves G. For constrained perfect phlogeny haplotyping (CPPH), we provide initial insights and polynomial-time algorithms for some restricted cases of the problem. For constrained parsimony haplotyping (CPH), we show that the problem is fixed parameter tractable when parameterized by the size of the solution set of haplotypes.

Computing the Dodgson Score of a candidate in an election is a hard computational problem, which has been analyzed using classical and parameterized analysis. In this paper we resolve two open problems regarding the parameterized complexity of DODGSON SCORE. We show that DODGSON SCORE parameterized by the target score value $k$ does not have a polynomial kernel unless the polynomial hierarchy collapses to the third level; this complements a result of Fellows, Rosamond and Slinko who obtain a non-trivial kernel of exponential size for a generalization of this problem. We also prove that DODGSON SCORE parameterized by the number $n$ of votes is hard for $W[1]$.

The Nemhauser-Trotter local optimization theorem applies to the NP-hard \textsc{Vertex Cover} problem and has applications in approximation as well as parameterized algorithmics. We present a framework that generalizes Nemhauser and Trotter's result to vertex deletion and graph packing problems, introducing novel algorithmic strategies based on purely combinatorial arguments (not referring to linear programming as the Nemhauser-Trotter result originally did). We exhibit our framework using a generalization of \textsc{Vertex Cover}, called \textrm{\sc Bounded-Degree Deletion}, that has promise to become an important tool in the analysis of gene and other biological networks. For some fixed~$d\geq 0$, \textrm{\sc Bounded-Degree Deletion} asks to delete as few vertices as possible from a graph in order to transform it into a graph with maximum vertex degree at most~$d$. \textsc{Vertex Cover} is the special case of $d=0$. Our generalization of the Nemhauser-Trotter theorem implies that \textrm{\sc Bounded-Degree Deletion} has a problem kernel with a linear number of vertices for every constant~$d$. We also outline an application of our extremal combinatorial approach to the problem of packing stars with a bounded number of leaves. Finally, charting the border between (parameterized) tractability and intractability for \textrm{\sc Bounded-Degree Deletion}, we provide a W[2]-hardness result for \textrm{\sc Bounded-Degree Deletion} in case of unbounded $d$-values.

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in monadic second-order logic with second-order quantification on vertex sets, which includes NP-hard problems such as 3-colorability) can be solved in polynomial time for graphs of bounded clique-width. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless $P = NP$. We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NP-complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

Parameterized complexity studies a generalization of the notion of polynomial time where, in addition to the overall input size n, one also considers the effects on computational complexity of a secondary measurement, the parameter. The central notion of the field is fixed-parameter tractability (FPT), which refers to solvability in time f(k)nc, where f is some function (usually exponential) of the parameter k, and c is a constant. The subject unfolds in two basic complementary projects and associated mathematical toolkits: (1) How to design (and improve) FPT algorithms, for parameterized problems that admit them and (2) How to gather evidence that a parameterized problem probably does not admit an FPT algorithm. There are several things that one can say about the field, in a general way. ... This Special Issue of surveys of various aspects of parameterized complexity and algorithmics began on the suggestion of the Editor-in-Chief, Fionn Murtagh, who after hearing a broad account of the field at a colloquium at Royal Holloway, University of London, declared, “This is a subject that every computer scientist should know about.”

Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems) can be solved efficiently for graphs of small clique-width. It is widely believed that determining the clique-width of a graph is NP-hard; in spite of considerable efforts, no NP-hardness proof has been found so far. We give the first hardness proof. We show that the clique-width of a given graph cannot be absolutely approximated in polynomial time unless P=NP. We also show that, given a graph G and an integer k, deciding whether the clique-width of G is at most k is NPhy complete. This solves a problem that has been open since the introduction of clique-width in the early 1990s.

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time n/sup o(k)/poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t - l)-st level W[t $1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted SAT, dominating set, hitting set, set cover, and feature set, cannot be solved in time n/sup o(k)/poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W[l] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-SAT (for any fixed q /spl ges/ 2), clique, and independent set, cannot be solved in time n/sup o(k)/ unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n/sup k/ poly(m) or O(n/sup k/).

Dealing with the NP-complete Dominating Set problem on graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy-to-implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.

The parametrized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored. Concerning codes, the main results are that MAXIMUM-LIKELIHOOD DECODING and WEIGHT DISTRUBUTION are hard for the parametrized complexity class W[1]. The NP-completeness of these two problems was established by Berlekamp, McEliece, and van Tilborg in 1978 using by means of a reduction from THREE-DIMENSIONAL MATCHING. On the other hand, our proof of hardness for W[1] is based on a parametric polynomial-time transformation from PERFECT CODE in graphs. An immediate consequence of our results is that bounded-distance decoding is likely to be hard for binary linear codes. Concerning lattices, we address the THETA SERIES problem of determining for an integer lattice $\L$ %given by a set of generators, and a positive integer k whether there is a vector $x \in \L$ of Euclidean norm k. We prove here for the first time that THETA SERIES is NP-complete and show that it is also hard for W[1]. Furthermore, we prove that the NEAREST VECTOR problem for integer lattices is hard for W[1]. These problems are the counterparts of WEIGHT DISTRUBUTION and MAXIMUM-LIKELIHOOD DECODING for lattices. Relations between all these problems and combinatorial problems in graphs are discussed.

There is considerable interest in constructing large networks with given diameter and maximum degree. In certain applications, there is a natural restriction for the networks to be planar. Thus, consider the problem of determining the maximum number of nodes in a planar network with maximum degree Δ and diameter at most k. We have previously proved that this number is at most (roughly) 12kΔ⌊k/2⌋ and there is a trivial lower bound of about (Δ − 1)⌊k/2⌋. We introduce a number of general constructions which substantially improve the lower bound and yield the largest known networks. We also provide a catalog of the best-known networks for small values of Δ and k, many obtained by specialized constructions. © 1998 John Wiley & Sons, Inc. Networks 32:275–281, 1998

Many computational problems in biology involve parameters for which a small range of values cover important applications. We argue that for many problems in this setting, parameterized computational complexity rather than NP-completeness is the appropriate tool for studying apparent intractability. At issue in the theory of parameterized complexity is whether a problem can be solved in time O(n alpha) for each fixed parameter value, where alpha is a constant independent of the parameter. In addition to surveying this complexity framework, we describe a new result for the Longest Common Subsequence problem. In particular, we show that the problem is hard for W[t] for all t when parameterized by the number of strings and the size of the alphabet. Lower bounds on the complexity of this basic combinatorial problem imply lower bounds on more general sequence alignment and consensus discovery problems. We also describe a number of open problems pertaining to the parameterized complexity of problems in computational biology where small parameter values are important.

For many fixed-parameter problems that are trivially soluable in polynomial time, such as $(k\text{-})$ DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as $(k\text{-})$ FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed k the problem is soluable in time bounded by a polynomial of degree c, where c is a constant independent of k. We establish the main results of a completeness program which addresses the apparent fixed-parameter intractability of many parameterized problems. In particular, we define a hierarchy of classes of parameterized problems $FPT \subseteq W[1] \subseteq W[2] \subseteq \cdots \subseteq W[SAT] \subseteq W [P]$ and identify natural complete problems for $W[t]$ for $t \geq 2$. (In other papers we have shown many problems complete for $W[1]$.) DOMINATING SET is shown to be complete for $W[2]$, and thus is not fixed-parameter tractable unless INDEPENDENT SET, CLIQUE, IRREDUNDANT SET, and many other natural problems in $W[2]$ are also fixed-parameter tractable. We also give a compendium of currently known hardness results as an appendix.

The parameterized computational complexity of a collection of well-known problems including: BAND-WIDTH, PRECEDENCE CONSTRAINED MULTIPROCES-SOR SCHEDULING, LONGEST COMMON SUBSEQUENCE, DNA PHYSICAL MAPPING (or INTERNALIZING COL-ORED GRAPHS), PERFECT PHYLOGENY (or TRIANGU-LATING COLORED GRAPHS), COLORED CUTWIDTH, and FEASIBLE REGISTER ASSIGNMENT is explored.It is shown that these problems are hard for various levels of the W hierarchy.In the case of PRECEDENCE CONSTRAINED MULTIPROCESSOR SCHEDULING the results can be interpreted as providing substantial new complexity lower bounds on the outcome of [OPEN 8] of the Garey and Johnson list.

Let S be a set of vertices in a graph $G = ( V,E )$. The authors state that a vertex u in S has a private neighbor (relative to S) if either u is not adjacent to any vertex in S or u is adjacent to a vertex w that is not adjacent to any other vertex in S. Based on the notion of private neighbors, a set of eight graph theoretic parameters can be defined whose inequality relationships can be described by a three-dimensional cube. Most of these parameters have already been studied independently. This paper unifies this study and helps to form a cohesive theory of private neighbors in graphs. Theoretical and algorithmic properties of this private neighbor cube are investigated, and many open questions are raised.

We describe three applications in computational learning theory

The existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems is nonconstructively proved. Some were not previously known to be in $\mathcal{P}$ at all; others were only known to be in $\mathcal{P}$ by way of brute force or dynamic programming formulations with unboundedly high-degree polynomial running times. The methods applied include the recent Robertson–Seymour theorems on the well-partial-ordering of graphs under both the minor and immersion orders. The complexity of search versions of these problems is also briefly addressed.

A report is presented on a collection of constructions of symmetric networks that provide the largest known values for the number of nodes that can be placed in a network of a given degree and diameter. Some of the constructions are in the range of current potential engineering significance. The constructions are Cayley graphs of linear groups obtained by experimental computation.>

This paper studies graph properties of the following forms: For every partition of the vertex set that satisfies an upper (or lower) bound on the number of elements in each partition class, there is a transversal of the partition that is an independent (or dominating) set. A possible application to fault-tolerant data storage is discussed, and bounds for the parameters that are functions of minimum and maximum degree are established. The complexity of associated decision problems is also addressed.

The authors address the question of why some fixed-parameter problem families solvable in polynomial time seem to be harder than others with respect to fixed-parameter tractability: whether there is a constant alpha such that all problems in the family are solvable in time O(n/sup alpha /). The question is modeled by considering a class of polynomially indexed relations. The main results show that (1) this setting supports notions of completeness that can be used to explain the apparent hardness of certain problems with respect to fixed-parameter tractability, and (2) some natural problems are complete.>

A theorem that is a graph-theoretic analog of the Myhill-Nerode characterization of regular languages is proved. The theorem is used to establish that for many applications obstruction sets are computable by known algorithms. The focus is exclusively on what is computable (by a known algorithm) in principle, as opposed to what is computable in practice.>

Recent advances in well-partial-order theory, especially the seminal contributions of Robertson and Seymour, have troubling consequences for those who would equate tractability with polynomial-time decidability. Specifically:

Recent advances in graph theory and graph algorithms dramatically alter the traditional view of concrete complexity theory, in which a decision problem is generally shown to be in P by producing an efficient algorithm to solve an optimization version of the problem. Nonconstructive tools are now available for classifying problems as decidable in polynomial time by guaranteeing only the existence of polynomial-time decision algorithms. In this paper these new methods are employed to prove membership in P for a number of problems whose complexities are not otherwise known. Powerful consequences of these techniques are pointed out and their utility is illustrated. A type of partially ordered set that supports this general approach is defined and explored.

The authors study the problem of assigning program fragments to a system of processing elements in which low-level operations are performed in parallel. Such a system is said to be linearly connected if each processing element can only communicate directly with its two nearest neighbors. They show that the problem of determining whether a perfect assignment exists is NP-complete but can be solved in linear time if the number of processing elements is fixed. They demonstrate that the related problem of determining whether any assignment exists which can be performed in a given number of machine cycles is NP-complete. For this problem, the objective of which corresponds to minimizing the program fragment's execution time, the authors also investigate the behavior of classes of near-optimal heuristic algorithms. This present evidence to indicate that guaranteeing acceptable worst-case performance is a very difficult problem as well.>