Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms that achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n -vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, • Independent Set and Dominating Set can be solved in time \(2^{\mathcal {O}(dk)}\cdot n^{\mathcal {O}(1)} \) using \(\mathcal {O}(dk\log n) \) space; • Max Cut can be solved in time \(n^{\mathcal {O}(dk)} \) using \(\mathcal {O}(dk\log n) \) space. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence , which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial..

In a graph where vertices have preferences over their neighbors, a matching is called popular if it does not lose a head-to-head election against any other matching when the vertices vote between the matchings. Popular matchings can be seen as an intermediate category between stable matchings and maximum-size matchings. In this article, we aim to maximize the utility of a matching that is popular but admits only a few blocking edges. We observe that, for general graphs, finding a popular matching with at most one blocking edge is already NP -complete. For bipartite instances, we study the problem of finding a maximum-utility popular matching with a bound on the number (or, more generally, the cost) of blocking edges applying a multivariate approach. We show classical and parameterized hardness results for severely restricted instances. By contrast, we design an algorithm for instances where preferences on one side admit a master list and show that this algorithm is roughly optimal.

We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown d -dimensional state. We first revisit known lower bounds [ 23 ] on quantum tomography with accuracy ε in trace distance, when the measurement choices are independent of previously observed outcomes, i.e., they are nonadaptive. We give a succinct proof of these results through the χ 2 -divergence between suitable distributions. Unlike prior work, we do not require that the measurements be given by rank-one operators. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes (e.g., two-outcome measurements). In particular, this rigorously establishes the optimality of the folklore “Pauli tomography” algorithm in terms of its sample complexity. We also derive novel bounds of \(\Omega (r^2 d/\epsilon ^2)\) and \(\Omega (r^2 d^2/\epsilon ^2)\) for learning rank r states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of unique measurement settings required (i.e., the number of different measurements used by an algorithm, each possibly with an arbitrary number of outcomes). Motivated by this consideration, we employ concentration of measure of χ 2 -divergence of suitable distributions to extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of \(\exp (O(d))\) possible measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy based on computing sample means of the given observables is optimal.

More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s -sparse polynomials, having individual degree d = O (1), is very well-studied (Shpilka, Volkovich ICALP’10; Volkovich RANDOM’17; Bhargava, Saraf and Volkovich FOCS’18, JACM’20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s poly( d ) -sparsity bound for the factors of symmetric polynomials over any field. To factor f , we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f . We prove a crucial result about convex polytopes, by introducing the concept of ‘low min-entropy’, which might also be of independent interest.

We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering that minimizes the sum of the cover times over all edges in the graph. In this work we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.0748. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work of Bansal, Batra, Farhadi, and Tetali. We also study the Minimum Sum Vertex Cover problem on regular graphs. In particular, we show that in this case the problem is hard to approximate within 1.0157. We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali, to show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.

For a well-studied family of domination-type problems, in bounded-treewidth graphs, we investigate whether it is possible to find faster algorithms. For sets σ , ρ of non-negative integers, a ( σ , ρ )-set of a graph G is a set S of vertices such that | N ( u )∩ S | ∈ σ for every u ∈ S , and | N ( v )∩ S | ∈ ρ for every \(v\not\in S \) . The problem of finding a ( σ , ρ )-set (of a certain size) unifies common problems like Independent Set , Dominating Set , Independent Dominating Set , and many others. In an accompanying paper, it is proven that, for all pairs of finite or cofinite sets ( σ , ρ ), there is an algorithm that counts ( σ , ρ )-sets in time \((c_{\sigma,\rho })^{\sf tw}\cdot n^{{\rm O}(1)} \) (if a tree decomposition of width \({\sf tw} \) is given in the input). Here, c σ , ρ is a constant with an intricate dependency on σ and ρ . Despite this intricacy, we show that the algorithms in the accompanying paper are most likely optimal, i.e., for any pair ( σ , ρ ) of finite or cofinite sets where the problem is non-trivial, and any ε > 0, a \((c_{\sigma,\rho }-\varepsilon)^{\sf tw}\cdot n^{{\rm O}(1)} \) -algorithm counting the number of ( σ , ρ )-sets would violate the Counting Strong Exponential-Time Hypothesis (#SETH). For finite sets σ and ρ , our lower bounds also extend to the decision version, showing that those algorithms are optimal in this setting as well.

Let φ be a sentence of \(\mathsf {CMSO}_2 \) (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether φ is satisfied in G . The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k , otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time \(\mathcal {O}_{\varphi,k}{(\log n)} \) . If we additionally assume that the feedback vertex number of G never exceeds k , this update time guarantee is worst-case. By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time \(\mathcal {O}_{k}{(\log n)} \) . Our data structure also works in a larger generality of relational structures over binary signatures.