A homomorphism from a graph G to a graph H is an edge-preserving mapping from V (G) to V (H). In the graph homomorphism problem, denoted by Hom(H), the graph H is fixed and we need to determine if there exists a homomorphism from an instance graph G to H. We study the complexity of the problem parameterized by the cutwidth of G, i.e., we assume that G is given along with a linear ordering v1, . . ., vn of V (G) such that, for each i ∈ {1, . . ., n − 1}, the number of edges with one endpoint in {v1, . . ., vi} and the other in {vi+1, . . ., vn} is at most k. We aim, for each H, for algorithms for Hom(H) running in time c^k_Hn^O(1) and matching lower bounds that exclude c^k_H^·o⁽¹⁾n^O(1) or c^k_H^(1−Ω(1))n^O(1) time algorithms under the (Strong) Exponential Time Hypothesis. In the paper we introduce a new parameter that we call mimsup(H). Our main contribution is strong evidence of a close connection between cH and mimsup(H): an information-theoretic argument that the number of states needed in a natural dynamic programming algorithm is at most mimsup(H)^k, lower bounds that show that for almost all graphs H indeed we have cH ≥ mimsup(H), assuming the (Strong) Exponential-Time Hypothesis, and an algorithm with running time exp(O(mimsup(H) · k log k))n^O(1). In the last result we do not need to assume that H is a fixed graph. Thus, as a consequence, we obtain that the problem of deciding whether G admits a homomorphism to H is fixed-parameter tractable, when parameterized by cutwidth of G and mimsup(H). The parameter mimsup(H) can be thought of as the p-th root of the maximum induced matching number in the graph obtained by multiplying p copies of H via a certain graph product, where p tends to infinity. It can also be defined as an asymptotic rank parameter of the adjacency matrix of H. Such parameters play a central role in, among others, algebraic complexity theory and additive combinatorics. Our results tightly link the parameterized complexity of a problem to such an asymptotic matrix parameter for the first time.

In distance query reconstruction, we wish to reconstruct the edge set of a hidden graph by asking as few distance queries as possible to an oracle. Given two vertices u and v, the oracle returns the shortest path distance between u and v in the graph. The length of a tree decomposition is the maximum distance between two vertices contained in the same bag. The treelength of a graph is defined as the minimum length of a tree decomposition of this graph. We present an algorithm to reconstruct an n-vertex connected graph G parameterized by maximum degree ∆ and treelength k in O_k∆(nlog² n) queries (in expectation). This is the first algorithm to achieve quasi-linear complexity for this class of graphs. The proof goes through a new lemma that could give independent insight on graphs of bounded treelength.

Let XNLP be the class of parameterized problems such that an instance of size n with parameter k can be solved nondeterministically in time f(k)n^O(1) and space f(k)log⁡(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as LIST COLORING and PRECOLORING EXTENSION with pathwidth as parameter, SCHEDULING OF JOBS WITH PRECEDENCE CONSTRAINTS, with both number of machines and partial order width as parameter, BANDWIDTH and variants of WEIGHTED CNF-SATISFIABILITY. In particular, this implies that all these problems are W[t]-hard for all t.

We study a special case of the Steiner Tree problem in which the input graph does not have a minor model of a complete graph on 4 vertices for which all branch sets contain a terminal. We show that this problem can be solved in O(n⁴) time, where n denotes the number of vertices in the input graph. This generalizes a seminal paper by Erickson et al. [Math. Oper. Res., 1987] that solves Steiner tree on planar graphs with all terminals on one face in polynomial time.

Two permutations σ and π are ℓ-similar if they can be decomposed into subpermutations σ(1), . . . ,σ(ℓ) and π(1), . . . ,π(ℓ) such that σ(i) is order-isomorphic to π(i) for all i ∈[ℓ]. Recently, Dudek, Grytczuk, and Ruciński Variations on twins in permutations, Electron. J. Combin., 28 (2021), P3.19. posed the problem of determining the minimum ℓ for which two permutations chosen independently and uniformly at random are ℓ-similar. We show that two such permutations are O(n1/3 log11/6(n))-similar with high probability, which is tight up to a polylogarithmic factor. Our result also generalizes to simultaneous decompositions of multiple permutations.

We describe a polynomial-time algorithm which, given a graph G with treewidth t, approximates the pathwidth of G to within a ratio of (t√log t) This is the first algorithm to achieve an f(t)-approximation for some function f. Our approach builds on the following key insight: every graph with large pathwidth has large treewidth or contains a subdivision of a large complete binary tree. Specifically, we show that every graph with pathwidth at least th+2 has treewidth at least t or contains a subdivision of a complete binary tree of height h+1. The bound th+2 is best possible up to a multiplicative constant. This result was motivated by, and implies (with c=2), the following conjecture of Kawarabayashi and Rossman (SODA'18): there exists a universal constant c such that every graph with pathwidth ω(kc) has treewidth at least k or contains a subdivision of a complete binary tree of height k. Our main technical algorithm takes a graph G and some (not necessarily optimal) tree decomposition of G of width t′ in the input, and it computes in polynomial time an integer h, a certificate that G has pathwidth at least h, and a path decomposition of G of width at most (t′+1)h+1. The certificate is closely related to (and implies) the existence of a subdivision of a complete binary tree of height h. The approximation algorithm for pathwidth is then obtained by combining this algorithm with the approximation algorithm of Feige, Hajiaghayi, and Lee (STOC'05) for treewidth.

An independent transversal of a graph (Formula presented.) with a vertex partition (Formula presented.) is an independent set of (Formula presented.) intersecting each block of (Formula presented.) in a single vertex. Wanless and Wood proved that if each block of (Formula presented.) has size at least (Formula presented.) and the average degree of vertices in each block is at most (Formula presented.), then an independent transversal of (Formula presented.) exists. We present a construction showing that this result is optimal: for any (Formula presented.) and sufficiently large (Formula presented.), there is a forest with a vertex partition into parts of size at least (Formula presented.) such that the average degree of vertices in each block is at most (Formula presented.), and there is no independent transversal. This unexpectedly shows that methods related to entropy compression such as the Rosenfeld–Wanless–Wood scheme or the Local Cut Lemma are tight for this problem. Further constructions are given for variants of the problem, including the hypergraph version.

A classic model to study strategic decision making in multi-agent systems is the normal-form game. This model can be generalised to allow for an infinite number of pure strategies leading to continuous games. Multi-objective normal-form games are another generalisation that model settings where players receive separate payoffs in more than one objective. We bridge the gap between the two models by providing a theoretical guarantee that a game from one setting can always be transformed to a game in the other. We extend the theoretical results to include guaranteed equivalence of Nash equilibria. The mapping makes it possible to apply algorithms from one field to the other. We demonstrate this by introducing a fictitious play algorithm for multi-objective games and subsequently applying it to two well-known continuous games. We believe the equivalence relation will lend itself to new insights by translating the theoretical guarantees from one formalism to another. Moreover, it may lead to new computational approaches for continuous games when a problem is more naturally solved in the succinct format of multi-objective games.

We investigate the parameterized complexity of Binary CSP parameterized by the vertex cover number and the treedepth of the constraint graph, as well as by a selection of related modulator-based parameters. The main findings are as follows: Binary CSP parameterized by the vertex cover number is W[3]-complete. More generally, for every positive integer d, Binary CSP parameterized by the size of a modulator to a treedepth-d graph is W[2d + 1]-complete. This provides a new family of natural problems that are complete for odd levels of the W-hierarchy. We introduce a new complexity class XSLP, defined so that Binary CSP parameterized by treedepth is complete for this class. We provide two equivalent characterizations of XSLP: the first one relates XSLP to a model of an alternating Turing machine with certain restrictions on conondeterminism and space complexity, while the second one links XSLP to the problem of model-checking first-order logic with suitably restricted universal quantification. Interestingly, the proof of the machine characterization of XSLP uses the concept of universal trees, which are prominently featured in the recent work on parity games. We describe a new complexity hierarchy sandwiched between the W-hierarchy and the A-hierarchy: For every odd t, we introduce a parameterized complexity class S[t] with W[t] ⊆ S[t] ⊆ A[t], defined using a parameter that interpolates between the vertex cover number and the treedepth. We expect that many of the studied classes will be useful in the future for pinpointing the complexity of various structural parameterizations of graph problems.

We study the parameterized complexity of computing the tree-partition-width, a graph parameter equivalent to treewidth on graphs of bounded maximum degree. On one hand, we can obtain approximations of the tree-partition-width efficiently: we show that there is an algorithm that, given an n-vertex graph G and an integer k, constructs a tree-partition of width O(k⁷) for G or reports that G has tree-partition width more than k, in time k^O(1)n². We can improve slightly on the approximation factor by sacrificing the dependence on k, or on n. On the other hand, we show the problem of computing tree-partition-width exactly is XALP-complete, which implies that it is W[t]-hard for all t. We deduce XALP-completeness of the problem of computing the domino treewidth. Finally, we adapt some known results on the parameter tree-partition-width and the topological minor relation, and use them to compare tree-partition-width to tree-cut width.

Let XNLP be the class of parameterized prob-lems such that an instance of size n with parameter k can be solved nondeterministically in time f (k) nO(1) and space f (k) log(n) (for some computable function f). We give a wide variety of XNLP-complete problems, such as List Coloringand Precoloring Extensionwith pathwidth as parameter, Scheduling Of Jobs With Precedence Constraints, with both number of machines and partial order width as parameter, Bandwidthand variants of Weighted Cnf-satisfiability and reconfiguration problems. In particular, this implies that all these problems are W[t]-hard for all t. This also answers a long standing question on the parameterized complexity of the Bandwidth problem.

We show that List Colouring can be solved on n-vertex trees by a deterministic Turing machine using O(log n) bits on the worktape. Given an n-vertex graph G = (V,E) and a list L(v) ⊆ {1, . . . , n} of available colours for each v ∈ V , a list colouring for G is a proper colouring c such that c(v) ∈ L(v) for all v.

In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k) log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on “tree-structured graphs” are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete. Besides finding a “natural home” for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k) log n space. Moreover, we introduce “tree-shaped” variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete.

We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators. Let p, q ∈ N such that p is a prime and q ≥ 3. We show: If p divides q - 1, there is a (q - 1)^ctwn^O(1) time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q - 1 - ε)^ctwn^O(1) time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH). If p does not divide q - 1, there is no ε > 0 for which there exists a (q - ε)^ctwn^O(1) time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^ctwn^O(1) time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.’18]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ε)^ctwn^O(1), assuming SETH. We also give an algorithm with matching running time for this problem. Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms’14] only excluded 2^o(tw)n^O(1) time algorithms for this problem. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain “compatibility matrix” in a non-trivial way.

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth.

The deck of a graph G is the multiset of cards {G−v:v∈V(G)}. Myrvold (1992) showed that the degree sequence of a graph on n≥7 vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree d can be reconstructed from any deck missing O(n/d³) cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.

The deck of a graph (Formula presented.) is given by the multiset of (unlabeled) subgraphs (Formula presented.). The subgraphs (Formula presented.) are referred to as the cards of (Formula presented.). Brown and Fenner recently showed that, for (Formula presented.), the number of edges of a graph (Formula presented.) can be computed from any deck missing 2 cards. We show that, for sufficiently large (Formula presented.), the number of edges can be computed from any deck missing at most (Formula presented.) cards.

We settle the parameterized complexities of several variants of independent set reconfiguration and dominating set reconfiguration, parameterized by the number of tokens. We show that both problems are XL-complete when there is no limit on the number of moves and XNL-complete when a maximum length ℓ for the sequence is given in binary in the input. The problems are known to be XNLP-complete when ℓ is given in unary instead, and W[1]- and W[2]-hard respectively when ℓ is also a parameter. We complete the picture by showing membership in those classes. Moreover, we show that for all the variants that we consider, token sliding and token jumping are equivalent under pl-reductions. We introduce partitioned variants of token jumping and token sliding, and give pl-reductions between the four variants that have precise control over the number of tokens and the length of the reconfiguration sequence.