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. Continuing our 2012 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, depth, or treewidth at most k, parameterized by k.

We give faster exponential-time randomised approximation algorithms for counting problems where polynomial-time approximation is unavailable and exact exponential-time counting remains expensive. For general n-vertex graphs, our independent-set counter runs in O^∗(1.1869^n) time, improving the previous O^∗(1.2041^n) general-graph bound. For n-variable #2-SAT, we obtain an O^∗(1.2373^n)-time approximation algorithm, narrowly below Wahlström's currently cited O^∗(1.2377^n) variable-parameter exact bound. The new algorithmic point is to take the square root after decomposition. For a single bounded unweighted self-reduction with f(x) positive leaves and recursion-compatible upper bound b(x), an enumerate-or-sample estimator gives an (ε,δ)-approximation in O^∗(√(b(x)) ε^-2log1δ) time. After preprocessing decomposes an input into many bounded cores, the combined estimator pays O^∗(√(∑_i b_i(x_i)) ε^-2log1δ), rather than estimating the cores separately at cost ∑_i √(b_i(x_i)). The same conversion improves the bases for counting maximal cliques, minimal separators, and perfect matchings in subcubic graphs. Bounded unweighted self-reductions provide the formal language; at the level of counting classes, the resulting unweighted formulation has the same Karp closure as TotP. With explicit recursion-tree access, the framework yields black-box quantum speed-ups.

Analyzing the worst-case running time of branching algorithms has traditionally focused more on designing complicated branching rules rather than developing better analysis methods for simple algorithms. In the mid-2000s, Fomin et al. (ACM 2009) introduced measure & conquer, an advanced general analysis method, sparking widespread adoption for obtaining tighter worst-case running time upper bound s for many fundamental NP-complete problems. Despite its significance, most subsequent work largely applied it without further methodological advancements and hence much potential in this direction remains untapped. Motivated by this, we present piecewise analysis, a new general method that analyzes the running time of branching algorithms. To showcase its potential, we reanalyze 2 almost 20-year-old algorithms by Fomin et al. (COCOON 2007), solving 4-Coloring and #3-Coloring, respectively. Our new analysis method improves the original running time upper bounds from O(1.7272n) and O(1.6262n) to O(1.7207n) and O(1.6225n), respectively.

We prove that computing the deterministic communication complexity of a Boolean function, given its truth table, is -complete in the standard protocol-tree-depth model, addressing a meta-complexity question raised by Yao in 1979. The reduction is from {0,1}-Vector Bin Packing and produces, in polynomial time, a communication matrix whose optimal protocol depth exhibits a one-bit gap between satisfiable and unsatisfiable instances. The main technical contribution is the relaxed-interlacing framework that makes this reduction possible. It replaces exponential-size Cartesian products with polynomial-size almost t-wise independent column sets, a pseudorandom substitute for full products, while preserving the lower-bound and protocol-control statements needed for the reduction. We develop these statements in two stages: first for classical interlacing, where projection arguments give clean lower bounds and separation statements, and then for relaxed interlacing, where a bridge lemma recovers the classical lower-bound and separation statements with controlled density loss. This leads to an extension theorem that lifts the classical lower bound to the relaxed setting and a near-exact separation theorem that lifts the corresponding protocol-control statement, with the present -completeness theorem as their main application here.

Cluster Editing, also known as Correlation Clustering, is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to k edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. However, in real-world networks, clusters are often overlapping. For example in social networks, a person might belong to several communities - e.g. those corresponding to work, school, or neighborhood. Other strong motivations come from biological network analysis and from language networks. Trying to cluster words with similar usage in the latter can be confounded by homonyms, that is, words with multiple meanings like "bat." In this paper, we introduce a new variant of Cluster Editing whereby a vertex can be split into two or more vertices. First used in the context of graph drawing, this operation allows a vertex v to be replaced by two vertices whose combined neighborhood is the neighborhood of v (and thus v can belong to more than one cluster). We call the new problem Cluster Editing with Vertex Splitting and we initiate the study of it. We show that it is NP-complete and fixed-parameter tractable when parameterized by the total number k of allowed vertex-splitting and edge-editing operations. In particular, we obtain an O(2^9k log k + n + m)-time algorithm and a 6k-vertex kernel.

This work introduces two techniques for the design and analysis of branching algorithms, illustrated through the case study of the Vertex Cover problem. First, we present a method for automatically generating branching rules through a systematic case analysis of local structures. Second, we develop a new technique for analyzing randomized branching algorithms using the Measure Conquer method, offering greater flexibility in formulating branching rules. By combining these innovations with additional techniques, we obtain the fastest known randomized algorithms in different parameters for the Vertex Cover problem on graphs with bounded degree (up to 6) and on general graphs. For example, our algorithm solves Vertex Cover on subcubic graphs in O^*(1.07625^n) time and O^*(1.13132^k) time, respectively. For graphs with maximum degree 4, we achieve running times of O^*(1.13735^n) and O^*(1.21103^k), while for general graphs we achieve O^*(1.25281^k).

In the realm of wireless communications and task allocations, where agents need to travel to perform tasks, a new challenge has emerged. This challenge involves efficiently distributing tasks among agents while considering their preferences and priorities. Unfortunately, many existing solutions rely on centralized systems, making them susceptible to various vulnerabilities, including privacy concerns and security threats. To address these issues, this paper introduces a novel solution named SEPP-TAP-a secure and privacy-preserving decentralized framework for multi-agent task allocation problems, harnessing the power of blockchain technology. SEPP-TAP not only protects agents' privacy and location information but also strengthens network security by mitigating threats like the Sybil attack and network flooding. Our approach employs innovative anonymization techniques to ensure that agents' identities remain concealed throughout successive task requests, breaking any associations between tasks performed by a particular agent.

Let U be a universe on n elements, let k be a positive integer, and let F be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from F, covering U with k sets from F, and packing k non-intersecting sets from F into U. Classically, these problems can be solved via inclusion-exclusion in O*(2^n) time [BjorklundHK09]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140^n) [ShimizuM22] and for Set Cover with a small number of sets with running time O(1.7274^n |F|^O(1)) [AmbainisBIKPV19]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a subset X of U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in O*(c^{|X|}) time with c<2. Our bounded-error quantum algorithm runs in O*((2+c)^(n/2)) for Set Partition, Set Cover, and Set Packing. When c<=1.147899, our algorithm is slightly faster than O*((2+c)^(n/2)); when c approaches 1, it matches the running time of [AmbainisBIKPV19] for Set Cover when |F| is subexponential in n. For Graph Coloring, we further improve the running time to O(1.7956^n) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number, we obtain a O((2-\epsilon)^n) running time for some \epsilon>0.

A weak backdoor, or simply a backdoor, for a Boolean SAT formula F into a class of SAT formulae C is a partial truth assignment T such that F[T] is in C and satisfiability is preserved. The problem of finding a backdoor from class C1 into class C2, or WB(C1,C2), can be stated as follows: Given a formula F in C1, and a natural number k, determine whether there exists a backdoor for F into C2 assigning at most k variables. The class 0-Val contains all Boolean formulae with at least one negative literal in each clause. We design a new algorithm for WB(3CNF, 0-Val) by reducing it to a local search variant of 3-SAT. We show that our algorithm runs in time O*(2.562^k), improving on the previous state-of-the-art of O*(2.85^k). Here, the O* notation is a variant of the big-O notation that allows to omit polynomial factors in the input size. Next, we look at WB(3CNF, Null), where Null is the class consisting of the empty formula. This problem was known to have a trivial running time upper bound of O*(6^k) and can easily be solved in O*(3^k) time. We use a reduction to Conflict-Free-d-Hitting-Set to prove an upper bound of O*(2.2738^k), and also prove a lower bound of 2^o(k) assuming the Exponential Time Hypothesis. Finally, Horn is the class of formulae with at most one positive literal per clause. We improve the previous O*(4.54^k) running time for WB(3CNF, Horn) problem to O*(4.17^k), by exploiting the structure of the SAT instance to give a novel proof of the non-existence of the slowest cases after a slight restructuring of the branching priorities.

We show that the `optimal' use of the parallel composition theorem corresponds to finding the size of the largest subset of queries that `overlap' on the data domain, a quantity we call the \emph{maximum overlap} of the queries. It has previously been shown that a certain instance of this problem, formulated in terms of determining the sensitivity of the queries, is NP-hard, but also that it is possible to use graph-theoretic algorithms, such as finding the maximum clique, to approximate query sensitivity. In this paper, we consider a significant generalization of the aforementioned instance which encompasses both a wider range of differentially private mechanisms and a broader class of queries. We show that for a particular class of predicate queries, determining if they are disjoint can be done in time polynomial in the number of attributes. For this class, we show that the maximum overlap problem remains NP-hard as a function of the number of queries. However, we show that efficient approximate solutions exist by relating maximum overlap to the clique and chromatic numbers of a certain graph determined by the queries. The link to chromatic number allows us to use more efficient approximate algorithms, which cannot be done for the clique number as it may underestimate the privacy budget. Our approach is defined in the general setting of $f$-differential privacy, which subsumes standard pure differential privacy and Gaussian differential privacy. We prove the parallel composition theorem for $f$-differential privacy. We evaluate our approach on synthetic and real-world data sets of queries. We show that the approach can scale to large domain sizes (up to $10^{20000}$), and that its application can reduce the noise added to query answers by up to 60\%.

In this chapter we consider extremal vertex-sets in graphs. For a property P, the extremal vertex-sets are either the inclusion-wise minimal or the inclusion-wise maximal vertex-sets with propertyP. We establish bounds on the largest number of such extremal vertex-sets that a graph may have, discuss enumeration algorithms and their use in exponential-time algorithms.

In this paper, we study the two-sided matching problem with soft diversity constraints in which each student belongs to one type and each school imposes soft targets on each type. We first identify limitations of type-specific quotas in a previous model and introduce a new general model that takes different levels of importance of types into account. Then we propose a new algorithm that yields a non-wasteful and fair outcome with respect to different levels of importance.

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.

We study the controlled school choice problem where students may belong to overlapping types and schools have soft target quotas for each type. We formalize fairness concepts for the setting that extend fairness concepts considered for restricted settings without overlapping types. Our central contribution is presenting a new class of algorithms that takes into account the representations of combinations of student types. The algorithms return matchings that are non-wasteful and satisfy fairness for same types. We further prove that the algorithms are strategyproof for the students and yield a fair outcome with respect to the induced quotas for type combinations. We experimentally compare our algorithms with two existing approaches in terms of achieving diversity goals and satisfying fairness.

We consider the two-sided stable matching setting in which there may be uncertainty about the agents’ preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model — in which for each agent, there is a probability distribution over linear preferences, (2) compact indifference model — for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model — there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

In the past few years, several new matching models have been proposed and studied that take into account complex distributional constraints. Relevant lines of work include (1) school choice with diversity constraints where students have (possibly overlapping) types and (2) hospital-doctor matching where various regional quotas are imposed. In this paper, we present a polynomial-time reduction to transform an instance of (1) to an instance of (2) and we show how the feasibility and stability of corresponding matchings are preserved under the reduction. Our reduction provides a formal connection between two important strands of work on matching with distributional constraints. We then apply the reduction in two ways. Firstly, we show that it is NP-complete to check whether a feasible and stable outcome for (1) exists. Due to our reduction, these NP-completeness results carry over to setting (2). In view of this, we help unify some of the results that have been presented in the literature. Secondly, if we have positive results for (2), then we have corresponding results for (1). One key conclusion of our results is that further developments on axiomatic and algorithmic aspects of hospital-doctor matching with regional quotas will result in corresponding results for school choice with diversity constraints.

Given two graphs H 1 and H 2 , a graph G is ( H 1 , H 2 ) ‐free if it contains no induced subgraph isomorphic to H 1 or H 2 . Let P t and C s be the path on t vertices and the cycle on s vertices, respectively. In this paper we show that for any ( P 6 , C 4 ) ‐free graph G it holds that χ ( G ) ≤ ( 3 / 2 ) ω ( G ) , where χ ( G ) and ω ( G ) are the chromatic number and clique number of G , respectively. Our bound is attained by several graphs, for instance, the 5‐cycle, the Petersen graph, the Petersen graph with an additional universal vertex, and all 4 ‐critical ( P 6 , C 4 ) ‐free graphs other than K 4 (see Hell and Huang [Discrete Appl. Math. 216 (2017), pp. 211–232]). The new result unifies previously known results on the existence of linear χ ‐binding functions for several graph classes. Our proof is based on a novel structure theorem on ( P 6 , C 4 ) ‐free graphs that do not contain clique cutsets. Using this structure theorem we also design a polynomial time 3 ∕ 2 ‐approximation algorithm for coloring ( P 6 , C 4 ) ‐free graphs. Our algorithm computes a coloring with ( 3 / 2 ) ω ( G ) colors for any ( P 6 , C 4 ) ‐free graph G in O ( n 2 m ) time.

The inverse geodesic length (IGL) is a well-known and widely used measure of network performance. It equals the sum of the inverse distances of all pairs of vertices. In network analysis, IGL of a network is often used to assess and evaluate how well heuristics perform in strengthening or weakening a network. We consider the edge-deletion problem MINIGLED. Formally, given a graph G, a budget k, and a target inverse geodesic length T, the question is whether there exists a subset of edges X with |X| ≤ ck, such that the inverse geodesic length of G − X is at most T.In this paper, we design algorithms and study the complexity of MINIGL-ED. We show that it is NP-complete and cannot be solved in subexponential time even when restricted to bipartite or split graphs assuming the Exponential Time Hypothesis. In terms of parameterized complexity, we consider the problem with respect to various parameters. We show that MINIGL-ED is fixed-parameter tractable for parameter T and vertex cover by modeling the problem as an integer quadratic program. We also provide FPT algorithms parameterized by twin cover and neighborhood diversity combined with the deletion budget k. On the negative side we show that MINIGL-ED is W[1]-hard for parameter tree-width.

In this paper, we show that every (2P(2), K-4)-free graph is 4-colorable. The bound is attained by the five-wheel and the complement of the seven-cycle. This answers an open question by Wagon [J. Combin. Theory Ser. B, 29 (1980), pp. 345{346] from the 1980s. Our result can also be viewed as a result in the study of the Vizing bound for graph classes. A major open problem in the study of computational complexity of graph coloring is whether coloring can be solved in polynomial time for (4P(1), C-4)-free graphs. Lozin and Malyshev [Discrete Appl. Math., 216 (2017), pp. 273{280] conjecture that the answer is yes. As an application of our main result, we provide the first positive evidence to the conjecture by giving a 2-approximation algorithm for coloring (4P(1), C-4)-free graphs.

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.