We present approximation algorithms for the Fault-tolerant k -Supplier with Outliers ( F k SO ) problem. This is a common generalization of two known problems – k -Supplier with Outliers, and Fault-tolerant k -Supplier – each of which generalize the well-known k -Supplier problem. In the k -Supplier problem the goal is to serve n clients C , by opening k facilities from a set of possible facilities F ; the objective function is the farthest that any client must travel to access an open facility. In F k SO , each client v has a fault-tolerance ℓ v , and now desires ℓ v facilities to serve it; so each client v ’s contribution to the objective function is now its distance to the ℓ v th closest open facility. Furthermore, we are allowed to choose m clients that we will serve, and only those clients contribute to the objective function, while the remaining n − m are considered outliers. Our main result is a (4 t − 1)-approximation for the F k SO problem, where t is the number of distinct values of ℓ v that appear in the instance. At t = 1, i.e. in the case where the ℓ v ’s are uniformly some ℓ, this yields a 3-approximation, improving upon the 11-approximation given for the uniform case by Inamdar and Varadarajan [2020], who also introduced the problem. Our result for the uniform case matches tight 3-approximations that exist for k -Supplier, k -Supplier with Outliers, and Fault-tolerant k -Supplier. Our key technical contribution is an application of the round-or-cut schema to F k SO . Guided by an LP relaxation, we reduce to a simpler optimization problem, which we can solve to obtain distance bounds for the “round” step, and valid inequalities for the “cut” step. By varying how we reduce to the simpler problem, we get varying distance bounds – we include a variant that gives a (2 t + 1)-approximation, which is better for t ∈ {2, 3}. In addition, for t = 1, we give a more straightforward application of round-or-cut, yielding a 3-approximation that is much simpler than our general algorithm.

We present a simple algorithm based on the inclusion-exclusion principle for computing the sum of the coefficients of all multilinear monomials in a polynomial computed by an arithmetic circuit over any field F . This problem is formally denoted as (k,n) -MLC and is defined as follows: — Given an arithmetic circuit computing a homogeneous polynomial \(f \in \mathbb {F}[x_1,\ldots ,x_n]\) of degree k , where \(\mathbb {F}\) is any field, compute the sum of the coefficients of all multilinear monomials in f . We provide a deterministic algorithm for the ( k,n )-MLC problem with a running time of \(O(n^{k/2} \cdot 2^{O(k)} \cdot s^{O(1)})\) , where s is the size of the circuit. Our algorithm is also polynomial space-bounded. The ( k,n )-MLC problem was first studied by Koutis and Williams [ 15 , 16 , 17 , 22 ] and later studied by Arvind et al., [ 3 ], where they obtain a deterministic algorithm with a running time of \(O^*({n\atopwithdelims (){\downarrow k/2}}\cdot s^{O(\log k)})\) 1 \(^,\) 2 using Hadamard product of polynomials. Pratt [ 18 ] proposed a deterministic algorithm that runs in \(O^*(n^{k/2})\) time, but this approach relies on a non-trivial Waring decomposition of symmetric polynomials over the rationals, which does not extend to small finite fields. Our method is notable for its simplicity and general applicability to all fields. In contrast to existing works, we introduce the complement of the ( k,n )-MLC problem, which we call the ( k,n )-NMLC problem. This approach looks at finding the sum of the coefficients of all non-multilinear monomials, which gives us a new way to think about the ( k,n )-MLC problem. This new method results in a simple algorithm and helps explain where the leading exponent \(k/2\) comes from. We develop a deterministic algorithm for the ( k,n )-NMLC problem that runs in \(O(n^{k/2} \cdot 2^{O(k)} \cdot s)\) time and uses polynomial space. The algorithm for the ( k,n )-MLC problem is straightforward: it subtracts the sum of the coefficients of all non-multilinear monomials (the result from the ( k,n )-NMLC problem) from the total sum of all monomial coefficients in the polynomial, which includes both multilinear and non-multilinear monomials. We apply these results to obtain a deterministic algorithm for the exact counting version of the r -simple k -path problem, as introduced in References [ 1 , 13 ], and also to a generalization of the ( k,n )-MLC problem, which we call the \((r,k,n)\) -MLC problem. This involves computing the sum of the coefficients of all monomials in the polynomial produced by the given arithmetic circuit, where the degree of each variable is at most r .

A catalytic Turing machine is a model of computation that is created by equipping a Turing machine with an additional auxiliary tape, which is initially filled with arbitrary content; the machine can read or write on the auxiliary tape during the computation, but it is constrained to halt with the same content in the auxiliary tape as it initially had. This article, studies the power of some natural variants of catalytic Turing machines with \(O(\log n)\) -size work tape and a polynomial-size auxiliary tape. We first define the notion of unambiguous catalytic Turing machines and prove that under a standard derandomization assumption, the class of problems solved by unambiguous catalytic Turing machines is the same as the class of problems solved by nondeterministic catalytic Turing machines. We then introduce the notion of randomized catalytic Turing machines and show that the resulting complexity class \(\mathsf {CBPL}\) is contained in the class \(\mathsf {ZPP}\) . We also explore the notion of symmetricity in the context of catalytic computation and prove that, under the same assumption as before, randomized catalytic Turing machines, symmetric catalytic Turing machines, and deterministic catalytic Turing machines that run in polynomial time are equally powerful.

A variety of constructive manipulation, control, and bribery problems for approval-based multiwinner voting have been extensively studied recently. However, their destructive counterparts seem to be less explored. This article investigates the complexity of several destructive bribery problems under five prestigious approval-based multiwinner voting rules—approval voting, satisfaction approval voting, net-satisfaction approval voting, Chamberlin-Courant approval voting, and proportional approval voting. Broadly, these problems aim to determine if a number of given candidates can be excluded from any winning committees by performing a limited number of modification operations. We offer a complete landscape of the complexity of the problems. For NP -hard problems, we study their parameterized complexity with respect to meaningful parameters.

We establish that effective continued fraction dimension originally defined using s -gales [ 21 ] is robust, but surprisingly, that the effective continued fraction dimension and effective (base- b ) Hausdorff dimension of the same real can be unequal in general. We initially provide an equivalent characterization of continued fraction dimension using Kolmogorov complexity. We also prove new bounds on the Lebesgue measure of continued fraction cylinders, which may be of independent interest. We apply these bounds to reveal an unexpected behavior of continued fraction dimension. It is known that effective dimension is invariant with respect to base conversion [ 8 ]. We also know that Martin-Löf randomness and computable randomness are invariant not only with respect to base conversion, but also with respect to the continued fraction representation [ 21 ]. In contrast, for any \(0 \lt \varepsilon \lt 0.5\) , we prove the existence of a real whose effective Hausdorff dimension is less than \(\varepsilon\) but whose effective continued fraction dimension is greater than or equal to 0.5. This phenomenon is related to the “non-faithfulness” of certain families of covers [ 1 , 23 ]. We also establish that, for any real number, the effective continued fraction dimension of the real number is always greater than or equal to its effective Hausdorff dimension.

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 by 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.

Cumulative memory—the sum of space used per step over the duration of a computation—is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many instances of an algorithm are interleaved in parallel. We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight. Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions. Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly( n ) requires cumulative memory \(\tilde{\Omega }(n^2) \) , any classical matrix multiplication algorithm requires cumulative memory Ω ( n 6 / T ), any quantum sorting circuit requires cumulative memory Ω ( n 3 / T ), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω ( k 3 n / T 2 ).

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.

Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k -types, for a sufficiently large k . We give sufficient conditions when this method can be applied and apply these conditions to obtain a new complexity dichotomy for CSPs of first-order expansions of the basic relations of the well-studied spatial reasoning formalism RCC5. We also classify which of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory; we prove that RCC5 has a Ramsey order expansion.

In this article, we study the classic Weighted 3-Set k -Packing problem: given a universe U , a family \( {\mathcal {S}}\) of subsets of size 3 of U , a weight function \(w : {\mathcal {S}} \rightarrow \mathbb {R}\) , \(W \in \mathbb {R}\) , and a parameter \(k \in \mathbb {N}\) , the objective is to decide if there is a subfamily \( {\mathcal {S}}\) ′ ⊆ \( {\mathcal {S}}\) of k disjoint sets and total weight at least W . We present a deterministic parameterized algorithm for this problem that runs in time O * (8.097 k ), where O * hides factors polynomial in the input size. This substantially improves upon the previously best deterministic algorithm for Weighted 3-Set k -Packing , which runs in time O * (12.155 k ) SIDMA [ 18 ], and was also the best deterministic algorithm for the unweighted version of this problem. Our algorithm is based on a novel application of the method of representative sets that might be of independent interest.