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Following the breakthrough work of Tardos (Oper Res 34:250–256, 1986) in the bit-complexity model, Vavasis and Ye (Math Program 74(1):79–120, 1996) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) $$\max \, c^\top x,\, Ax = b,\, x \ge 0,\, A \in \mathbb {R}^{m \times n}$$ max c ⊤ x , A x = b , x ≥ 0 , A ∈ R m × n , Vavasis and Ye developed a primal-dual interior point method using a ‘layered least squares’ (LLS) step, and showed that $$O(n^{3.5} \log (\bar{\chi }_A+n))$$ O ( n 3.5 log ( χ ¯ A + n ) ) iterations suffice to solve (LP) exactly, where $$\bar{\chi }_A$$ χ ¯ A is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J Optim 13(4):1054–1079, 2003), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure $$\bar{\chi }^*_A$$ χ ¯ A ∗ , defined as the minimum $$\bar{\chi }_{AD}$$ χ ¯ AD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an $$O(m^2 n^2 + n^3)$$ O ( m 2 n 2 + n 3 ) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying $$\bar{\chi }_{AD} \le n(\bar{\chi }_A^*)^3$$ χ ¯ AD ≤ n ( χ ¯ A ∗ ) 3 . This algorithm also allows us to approximate the value of $$\bar{\chi }_A$$ χ ¯ A up to a factor $$n (\bar{\chi }_A^*)^2$$ n ( χ ¯ A ∗ ) 2 . This result is in surprising contrast to that of Tunçel (Math Program 86(1):219–223, 1999), who showed NP-hardness for approximating $$\bar{\chi }_A$$ χ ¯ A to within $$2^{\textrm{poly}(\textrm{rank}(A))}$$ 2 poly ( rank ( A ) ) . The key insight for our algorithm is to work with ratios $$g_i/g_j$$ g i / g j of circuits of A—i.e., minimal linear dependencies $$Ag=0$$ A g = 0 —which allow us to approximate the value of $$\bar{\chi }_A^*$$ χ ¯ A ∗ by a maximum geometric mean cycle computation in what we call the ‘circuit ratio digraph’ of A. While this resolves Monteiro and Tsuchiya’s question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved $$O(n^{2.5} \log (n)\log (\bar{\chi }^*_A+n))$$ O ( n 2.5 log ( n ) log ( χ ¯ A ∗ + n ) ) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor $$n/\log n$$ n / log n improvement on the iteration complexity bound of the original Vavasis–Ye algorithm.

Machine Learning (ML) solutions are nowadays distributed, according to the so-called server/worker architecture. One server holds the model parameters while several workers train the model. Clearly, such architecture is prone to various types of component failures, which can be all encompassed within the spectrum of a Byzantine behavior. Several approaches have been proposed recently to tolerate Byzantine workers. Yet all require trusting a central parameter server. We initiate in this paper the study of the ``general'' Byzantine-resilient distributed machine learning problem where no individual component is trusted. We show that this problem can be solved in an asynchronous system, despite the presence of $\frac{1}{3}$ Byzantine parameter servers and $\frac{1}{3}$ Byzantine workers (which is optimal). We present a new algorithm, ByzSGD, which solves the general Byzantine-resilient distributed machine learning problem by relying on three major schemes. The first, Scatter/Gather, is a communication scheme whose goal is to bound the maximum drift among models on correct servers. The second, Distributed Median Contraction (DMC), leverages the geometric properties of the median in high dimensional spaces to bring parameters within the correct servers back close to each other, ensuring learning convergence. The third, Minimum-Diameter Averaging (MDA), is a statistically-robust gradient aggregation rule whose goal is to tolerate Byzantine workers. MDA requires loose bound on the variance of non-Byzantine gradient estimates, compared to existing alternatives (e.g., Krum). Interestingly, ByzSGD ensures Byzantine resilience without adding communication rounds (on a normal path), compared to vanilla non-Byzantine alternatives. ByzSGD requires, however, a larger number of messages which, we show, can be reduced if we assume synchrony., This is a merge of arXiv:1905.03853 and arXiv:1911.07537; arXiv:1911.07537 will be retracted

The problem of uniformly placing N points onto a sphere finds applications in many areas. For example, points on the sphere correspond to unit quaternions as well as to the group of rotations SO(3) and the online version of generating uniform rotations (known as "incremental generation") plays a crucial role in a large number of engineering applications ranging from robotics and aeronautics to computer graphics. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The first online algorithm of Chen et al. was upper-bounded by 5.99 and later improved to 3.69, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. In this paper we provide a more efficient tessellation technique based on the regular icosahedron, which improves the upper-bound for the online version of this problem, decreasing it to approximately 2.84. Moreover, we show that the lower bound for the gap ratio of placing at least three points is the golden ratio, approx. 1.618, and for at least four points is no less than 1.726., Comment: 18 pages, 13 figures

Recent theoretical research has shown that self-adjusting and self-adaptive mechanisms can provably outperform static settings in evolutionary algorithms for binary search spaces. However, the vast majority of these studies focuses on unimodal functions which do not require the algorithm to flip several bits simultaneously to make progress. In fact, existing self-adjusting algorithms are not designed to detect local optima and do not have any obvious benefit to cross large Hamming gaps. We suggest a mechanism called stagnation detection that can be added as a module to existing evolutionary algorithms (both with and without prior self-adjusting algorithms). Added to a simple (1+1) EA, we prove an expected runtime on the well-known Jump benchmark that corresponds to an asymptotically optimal parameter setting and outperforms other mechanisms for multimodal optimization like heavy-tailed mutation. We also investigate the module in the context of a self-adjusting (1+$\lambda$) EA and show that it combines the previous benefits of this algorithm on unimodal problems with more efficient multimodal optimization. To explore the limitations of the approach, we additionally present an example where both self-adjusting mechanisms, including stagnation detection, do not help to find a beneficial setting of the mutation rate. Finally, we investigate our module for stagnation detection experimentally., Comment: 26 pages. Full version of a paper appearing at GECCO 2020

Market makers choose and design market rules to serve certain objectives, such as to maximize revenue from the sales in the case of a single seller and multiple buyers. Given such rules, market participants play against each other to maximize their utility function values on goods acquired, possibly by hiding or misrepresenting their information needed in the implementation of market rules. Today’s Internet economy has changed the information collection process and may make some of the assumptions of market rule implementation obsolete. Here we make a fresh review of works on this challenge on the Internet where new economic systems operate.

We study thetruthful facility assignmentproblem, where a set of agents with private most-preferred points on a metric space have to be assigned to facilities that lie on the metric space, under capacity constraints on the facilities. The goal is to produce such an assignment that minimizes the social cost, i.e., the total distance between the most-preferred points of the agents and their corresponding facilities in the assignment, under the constraint of truthfulness, which ensures that agents do not misreport their most-preferred points. We propose aresource augmentation framework, where a truthful mechanism is evaluated by its worst-case performance on an instance with enhanced facility capacities against the optimal mechanism on the same instance with the original capacities. We study a well-known mechanism, Serial Dictatorship, and provide an exact analysis of its performance. Among other results, we prove that Serial Dictatorship has approximation ratio$$g/(g-2)$$g/(g-2)when the capacities are multiplied by any integer$$g \ge 3$$g≥3. Our results suggest that with a limited augmentation of the resources we can achieve exponential improvements on the performance of the mechanism and in particular, the approximation ratio goes to 1 as the augmentation factor becomes large. We complement our results with bounds on the approximation ratio of Random Serial Dictatorship, the randomized version of Serial Dictatorship, when there is no resource augmentation.

We address counting and optimization variants of multicriteria global min-cut and size-constrained min-k-cut in hypergraphs. 1) For an r-rank n-vertex hypergraph endowed with t hyperedge-cost functions, we show that the number of multiobjective min-cuts is O(r2^{tr}n^{3t-1}). In particular, this shows that the number of parametric min-cuts in constant rank hypergraphs for a constant number of criteria is strongly polynomial, thus resolving an open question by Aissi, Mahjoub, McCormick, and Queyranne [Aissi et al., 2015]. In addition, we give randomized algorithms to enumerate all multiobjective min-cuts and all pareto-optimal cuts in strongly polynomial-time. 2) We also address node-budgeted multiobjective min-cuts: For an n-vertex hypergraph endowed with t vertex-weight functions, we show that the number of node-budgeted multiobjective min-cuts is O(r2^{r}n^{t+2}), where r is the rank of the hypergraph, and the number of node-budgeted b-multiobjective min-cuts for a fixed budget-vector b ∈ ℝ^t_+ is O(n²). 3) We show that min-k-cut in hypergraphs subject to constant lower bounds on part sizes is solvable in polynomial-time for constant k, thus resolving an open problem posed by Queyranne [Guinez and Queyranne, 2012]. Our technique also shows that the number of optimal solutions is polynomial. All of our results build on the random contraction approach of Karger [Karger, 1993]. Our techniques illustrate the versatility of the random contraction approach to address counting and algorithmic problems concerning multiobjective min-cuts and size-constrained k-cuts in hypergraphs.

In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time. For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity. For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018].

Let $T$ be a theory. If $T$ eliminates $\exists^\infty$, it need not follow that $T^{eq}$ eliminates $\exists^\infty$, as shown by the example of the $p$-adics. We give a criterion to determine whether $T^{eq}$ eliminates $\exists^\infty$. Specifically, we show that $T^{eq}$ eliminates $\exists^\infty$ if and only if $\exists^\infty$ is eliminated on all interpretable sets of "unary imaginaries." This criterion can be applied in cases where a full description of $T^{eq}$ is unknown. As an application, we show that $T^{eq}$ eliminates $\exists^\infty$ when $T$ is a C-minimal expansion of ACVF., Comment: 6 pages

An $H$-free editing problem asks whether we can edit at most $k$ edges to make a graph contain no induced copy of the fixed graph $H$. We obtain a polynomial kernel for this problem when $H$ is a diamond. The incompressibility dichotomy for $H$ being a 3-connected graph and the classical complexity dichotomy suggest that except for $H$ being a complete/empty graph, $H$-free editing problems admit polynomial kernels only for a few small graphs $H$. Therefore, we believe that our result is an essential step toward a complete dichotomy on the compressibility of $H$-free editing. Additionally, we give a cubic-vertex kernel for the diamond-free edge deletion problem, which is far simpler than the previous kernel of the same size for the problem.

We propose a new grid unfolding without refinement of a new sub-class of polycubes that we call Manhattan towers with an H-convex base. Such poly-cubes can be seen both as a Manhattan tower and as an orthostack. At the end of the paper we show that an extension of this algorithm to Up-and-Down Orthoterrains is also possible.

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of $d/2$ in dimension $d$, achieved by the "standard terminal simplices" and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino \& Schymura (2017) that the $d$-th covering minimum of the standard terminal $n$-simplex equals $d/2$, for every $n>d$. We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger's formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and we give strong evidence for its validity in arbitrary dimension., 44 pages, 7 figures

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN}), Comment: 10 pages, 3 figures

We study the definability of ultrafilter bases on $$\omega $$ in the sense of descriptive set theory. As a main result we show that there is no coanalytic base for a Ramsey ultrafilter, while in L we can construct $$\Pi ^1_1$$ P-point and Q-point bases. We also show that the existence of a $${\varvec{\Delta }}^1_{n+1}$$ ultrafilter is equivalent to that of a $${\varvec{\Pi }}^1_n$$ ultrafilter base, for $$n \in \omega $$ . Moreover we introduce a Borel version of the classical ultrafilter number and make some observations.

We consider the approximate minimum selection problem in presence of independent random comparison faults. This problem asks to select one of the smallest k elements in a linearly-ordered collection of n elements by only performing unreliable pairwise comparisons: whenever two elements are compared, there is a small probability that the wrong comparison outcome is observed. We design a randomized algorithm that solves this problem with a success probability of at least 1−q for q∈(0,n−kn) and any k∈[1,n−1] using O(nk⌈log1q⌉) comparisons in expectation (if k≥n or q≥n−kn the problem becomes trivial). Then, we prove that the expected number of comparisons needed by any algorithm that succeeds with probability at least 1−q must be Ω(nklog1q) whenever q is bounded away from n−kn, thus implying that the expected number of comparisons performed by our algorithm is asymptotically optimal in this range. Moreover, we show that the approximate minimum selection problem can be solved using O((nk+loglog1q)log1q) comparisons in the worst case, which is optimal when q is bounded away from n−kn and k=O(nloglog1q)., Algorithmica, 84, ISSN:0178-4617, ISSN:1432-0541

Eroders are monotonic cellular automata with a linearly ordered state set that eventually wipe out any finite island of nonzero states. One-dimensional eroders were studied by Gal'perin in the 1970s, who presented a simple combinatorial characterization of the class. The multi-dimensional case has been studied by Toom and others, but no such characterization has been found. We prove a similar characterization for those one-dimensional monotonic cellular automata that are eroders even in the presence of random noise., Comment: 32 pages, 9 figures

In network construction/restoration problems introduced in Averbakh and Pereira (IIE Trans 44(8):681–694, 2012; Eur J Oper Res 244:715–729, 2015), a server (construction crew) builds edges of a given network starting from a given vertex (the depot), with a constant construction speed. The server can travel within the already constructed part of the network with a speed that is incomparably faster than the construction speed. The recovery time of a vertex is the time when the vertex becomes connected to the depot by an already constructed path. Due dates and/or weights are associated with vertices. It is required to find an optimal construction schedule that minimizes the total weighted recovery time or the maximum lateness of the vertices. Both problems are known to be polynomially solvable on trees and NP-hard on general networks. We prove that both problems are NP-hard even on so simple extensions of trees as cactuses, and discuss some polynomially solvable cases.

We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms. We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.

Binary exponential backoff (BEB) is a decades-old algorithm for coordinating access to a shared channel. In modern networks, BEB plays an important role in WiFi (IEEE 802.11) and other wireless communication standards. Despite this track record, well-known theoretical results indicate that under bursty traffic BEB yields poor makespan, and superior algorithms are possible. To date, the degree to which these findings impact performance in wireless networks has not been examined. To address this issue, we investigate one of the strongest cases against BEB: a single burst batch of packets that simultaneously contend for access to a wireless channel. Using Network Simulator 3, we incorporate into IEEE 802.11g several newer algorithms that, while inspired by BEB, possess makespan guarantees that are theoretically superior. Surprisingly, we discover that these newer algorithms underperform BEB. Investigating further, we identify as the culprit a common abstraction regarding the cost of collisions. Our experimental results are complemented by analytical arguments that the number of collisions - and not solely makespan - is an important metric to optimize. We argue that these findings have implications for the design of backoff algorithms in wireless networks., arXiv admin note: substantial text overlap with arXiv:1705.09271

Search-based satisfiability procedures try to build a model of the input formula by simultaneously proposing candidate models and deriving new formulae implied by the input. Conflict-driven procedures perform non-trivial inferences only when resolving conflicts between formulæ and assignments representing the candidate model. CDSAT (Conflict-Driven SATisfiability) is a method for conflict-driven reasoning in unions of theories. It combines inference systems for individual theories as theory modules within a solver for the union of the theories. This article augments CDSAT with a more general lemma learning capability and with proof generation. Furthermore, theory modules for several theories of practical interest are shown to fulfill the requirements for completeness and termination of CDSAT. Proof generation is accomplished by a proof-carrying version of the CDSAT transition system that produces proof objects in memory accommodating multiple proof formats. Alternatively, one can apply to CDSAT the LCF approach to proofs from interactive theorem proving, by defining a kernel of reasoning primitives that guarantees the correctness by construction of CDSAT proofs.

Given a simple connected graph $G = (V, E)$, we seek to partition the vertex set $V$ into $k$ non-empty parts such that the subgraph induced by each part is connected, and the partition is maximally balanced in the way that the maximum cardinality of these $k$ parts is minimized. We refer this problem to as {\em min-max balanced connected graph partition} into $k$ parts and denote it as {\sc $k$-BGP}. The general vertex-weighted version of this problem on trees has been studied since about four decades ago, which admits a linear time exact algorithm; the vertex-weighted {\sc $2$-BGP} and {\sc $3$-BGP} admit a $5/4$-approximation and a $3/2$-approximation, respectively; but no approximability result exists for {\sc $k$-BGP} when $k \ge 4$, except a trivial $k$-approximation. In this paper, we present another $3/2$-approximation for our cardinality {\sc $3$-BGP} and then extend it to become a $k/2$-approximation for {\sc $k$-BGP}, for any constant $k \ge 3$. Furthermore, for {\sc $4$-BGP}, we propose an improved $24/13$-approximation. To these purposes, we have designed several local improvement operations, which could be useful for related graph partition problems., Comment: 23 pages, 7 figures, accepted for presentation at COCOA 2019 (Xiamen, China)

In this paper, we study the Sobolev extension property of Lp-quasidisks which are the generalizations of classical quasidisks. After that, we also find some applications of this property.

International audience; In this paper, we provide three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time. First, we propose an exact algorithm for in-trees, of which the complexity depends mainly on the graph height, i.e., the length of the longest chain of the precedence graph. We show that this work improves the algorithm in the literature both theoretically and experimentally. Second, we close the open problem for level-orders by showing how it is polynomially solvable. Third, we prove that preemptive scheduling in-trees is strongly NP-hard with arbitrary number of machines, of which the complexity was also open.

The work is devoted to the study quantum versions of the differential cryptanalysis based on using a combination of the quantum minimum/maximum search algorithm and the quantum counting algorithm. We have estimated the complexity and the required resources for applying the quantum differential and quantum linear cryptanalysis to searching round keys of block ciphers. It is shown that the implementation of the quantum linear method requires less logical qubits than for the implementation of the quantum differential method. The acceleration of calculations due to “quantum parallelism” in the quantum differential cryptanalysis, based on a combination of Grover’s quantum algorithms and quantum counting algorithm, is apparently absent, because the using of quantum counting as “subprogram” in the Grover algorithm eliminates quantum acceleration, as far as $$ O (\sqrt{K}) \cdot O (\sqrt{K}) \approx O (K) $$ .

Background An important initial phase of arguably most homology search and alignment methods such as required for genome alignments is seed finding. The seed finding step is crucial to curb the runtime as potential alignments are restricted to and anchored at the sequence position pairs that constitute the seed. To identify seeds, it is good practice to use sets of spaced seed patterns, a method that locally compares two sequences and requires exact matches at certain positions only. Results We introduce a new method for filtering alignment seeds that we call geometric hashing. Geometric hashing achieves a high specificity by combining non-local information from different seeds using a simple hash function that only requires a constant and small amount of additional time per spaced seed. Geometric hashing was tested on the task of finding homologous positions in the coding regions of human and mouse genome sequences. Thereby, the number of false positives was decreased about million-fold over sets of spaced seeds while maintaining a very high sensitivity. Conclusions An additional geometric hashing filtering phase could improve the run-time, accuracy or both of programs for various homology-search-and-align tasks.

The automorphism group of the Zetterberg code Z of length 17 (also a quadratic residue code) is a rank three group whose orbits on the coordinate pairs determine two strongly regular graphs equivalent to the Paley graph attached to the prime 17. As a consequence, codewords of a given weight of Z are the characteristic vectors of the blocks of a PBIBD with two associate classes of cyclic type. More generally, this construction of PBIBDs is extended to quadratic residue codes of length $$\equiv 1 \pmod {8},$$ to the adjacency codes of triangular and lattice graphs, and to the adjacency codes of various rank three graphs. A remarkable fact is the existence of 2-designs held by the quadratic residue code of length 41 for code weights 9 and 10.

In this work, we study the problem of testing properties of the spectrum of a mixed quantum state. Here one is given n copies of a mixed state $$\rho \in \mathbb {C}^{d\times d}$$ and the goal is to distinguish (with high probability) whether $$\rho $$ ’s spectrum satisfies some property $${\mathcal {P}}$$ or whether it is at least $$\epsilon $$ -far in $$\ell _1$$ -distance from satisfying $${\mathcal {P}}$$ . This problem was promoted in the survey of Montanaro and de Wolf (A survey of quantum property testing. Technical report, arXiv:1310.2035 , 2013) under the name of testing unitarily invariant properties of mixed states. It is the natural quantum analogue of the classical problem of testing symmetric properties of probability distributions. Unlike property testing probability distributions—where one generally hopes for algorithms with sample complexity that is sublinear in the domain size—here the hope is for algorithms with subquadratic copy complexity in the dimension d. This is because the (frequently rediscovered) “empirical Young diagram (EYD) algorithm” (Alicki et al. in J Math Phys 29(5):1158–1162, 1988; Keyl and Werner in Phys Rev A 64(5):052311, 2011; Hayashi and Matsumoto in Phys Rev A 66(2):022311, 2002; Christandl and Mitchison in Commun. Math. Phys. 261(3):789–797, 2006) can estimate the spectrum of any mixed state up to $$\epsilon $$ -accuracy using only $${\widetilde{O}}(d^2/\epsilon ^2)$$ copies. In this work, we show that given a mixed state $$\rho \in \mathbb {C}^{d \times d}$$ : Our techniques involve the asymptotic representation theory of the symmetric group; in particular Kerov’s algebra of polynomial functions on Young diagrams.

Path cover is a well-known intractable problem that finds a minimum number of vertex disjoint paths in a given graph to cover all the vertices. We show that a variant, in which the objective is to minimize the number of length-0 paths, is polynomial-time solvable. We further show that another variant, to minimize the total number of length-0 and length-1 paths, is also polynomial-time solvable. Both variants find applications in approximating the two-machine flow-shop scheduling problem in which job processing has constraints that are formulated as a conflict graph. For the unit jobs, we present a 4/3-approximation for the scheduling problem with an arbitrary conflict graph, based on the exact algorithm for the above second variant of the path cover problem. For arbitrary jobs where the conflict graph is the union of two disjoint cliques, we present a simple 3/2-approximation algorithm.

A 2-edge-coloured graph G is supereulerian if G contains a spanning closed trail in which the edges alternate in colours. We show that for general 2-edge-coloured graphs it is NP-complete to decide whether the graph is supereulerian. An eulerian factor of a 2-edge-coloured graph is a collection of vertex-disjoint induced subgraphs which cover all the vertices of G such that each of these subgraphs is supereulerian. We give a polynomial algorithm to test whether a 2-edge-coloured graph has an eulerian factor and to produce one when it exists. A 2-edge-coloured graph is (trail-)colour-connected if it contains a pair of alternating (u, v)-paths ((u, v)-trails) whose union is an alternating closed walk for every pair of distinct vertices u, v. We prove that a 2-edge-coloured complete bipartite graph is supereulerian if and only if it is colour-connected and has an eulerian factor. A 2-edge-coloured graph is M-closed if xz is an edge of G whenever some vertex u is joined to both x and z by edges of the same colour. M-closed 2-edge-coloured graphs, introduced in Contreras-Balbuena et al. (Discret Math Theoret Comput Sci 21:1, 2019), form a rich generalization of 2-edge-coloured complete graphs. We show that if G is obtained from an M-closed 2-edge-coloured graph H by substituting independent sets for the vertices of H, then G is supereulerian if and only if G is trail-colour-connected and has an eulerian factor. Finally we discuss 2-edge-coloured complete multipartite graphs and show that such a graph may not be supereulerian even if it is trail-colour-connected and has an eulerian factor.

When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{*}$-semigroups and commutative involution semigroups., Comment: 14 pages

Catalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

We study a security game over a network played between adefenderandkattackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of$$\lambda $$λnodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizingdefense-optimalnetworks which allow for the bestequilibrium defense ratio; this is the ratio ofkover the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for$$\lambda $$λconstantly close to 1 orn. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is$${\mathtt {NP}}$$NP-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for thePrice of Defensefor any$$\lambda $$λ; this is the worst equilibrium defense ratio over all graphs.

Given a graph $H$, the extremal number $\mathrm{ex}(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a number of conjectures about the extremal number of bipartite graphs. First, writing $K'_{s,t}$ for the subdivision of the bipartite graph $K_{s,t}$, we show that $\mathrm{ex}(n, K'_{s,t}) = O(n^{3/2 - \frac{1}{2s}})$. This proves a conjecture of Kang, Kim and Liu and is tight up to the implied constant for $t$ sufficiently large in terms of $s$. Second, for any integers $s, k \geq 1$, we show that $\mathrm{ex}(n, L) = \Theta(n^{1 + \frac{s}{sk+1}})$ for a particular graph $L$ depending on $s$ and $k$, answering another question of Kang, Kim and Liu. This result touches upon an old conjecture of Erd\H{o}s and Simonovits, which asserts that every rational number $r \in (1,2)$ is realisable in the sense that $\mathrm{ex}(n,H) = \Theta(n^r)$ for some appropriate graph $H$, giving infinitely many new realisable exponents and implying that $1 + 1/k$ is a limit point of realisable exponents for all $k \geq 1$. Writing $H^k$ for the $k$-subdivision of a graph $H$, this result also implies that for any bipartite graph $H$ and any $k$, there exists $\delta > 0$ such that $\mathrm{ex}(n,H^{k-1}) = O(n^{1 + 1/k - \delta})$, partially resolving a question of Conlon and Lee. Third, extending a recent result of Conlon and Lee, we show that any bipartite graph $H$ with maximum degree $r$ on one side which does not contain $C_4$ as a subgraph satisfies $\mathrm{ex}(n, H) = o(n^{2 - 1/r})$., Comment: 21 pages

We consider the following question: If a simplicial complex $$\Gamma $$ has d-homology, then does the corresponding d-cycle always induce cycles of smaller dimension that are not boundaries? We provide an answer to this question in a fixed dimension. We use the breaking of homology to show the subadditivity property for the maximal degrees of syzygies of monomial ideals in a fixed homological degree.

We extend the facial weak order from finite Coxeter groups to central hyperplane arrangements. The facial weak order extends the poset of regions of a hyperplane arrangement to all its faces. We provide four non-trivially equivalent definitions of the facial weak order of a central arrangement: (1) by exploiting the fact that the faces are intervals in the poset of regions, (2) by describing its cover relations, (3) using covectors of the corresponding oriented matroid, and (4) using certain sets of normal vectors closely related to the geometry of the corresponding zonotope. Using these equivalent descriptions, we show that when the poset of regions is a lattice, the facial weak order is a lattice. In the case of simplicial arrangements, we further show that this lattice is semidistributive and give a description of its join-irreducible elements. Finally, we determine the homotopy type of all intervals in the facial weak order., 34 pages, 12 figures

For a graph G, $$L(G)^2$$ L ( G ) 2 is the square of the line graph of G – that is, vertices of $$L(G)^2$$ L ( G ) 2 are edges of G and two edges $$e,f\in E(G)$$ e , f ∈ E ( G ) are adjacent in $$L(G)^2$$ L ( G ) 2 if at least one vertex of e is adjacent to a vertex of f and $$e\ne f$$ e ≠ f . The strong chromatic index of G, denoted by $$s'(G)$$ s ′ ( G ) , is the chromatic number of $$L(G)^2$$ L ( G ) 2 . A strong clique in G is a clique in $$L(G)^2$$ L ( G ) 2 . Finding a bound for the maximum size of a strong clique in a graph with given maximum degree is a problem connected to a famous conjecture of Erdős and Nešetřil concerning strong chromatic index of graphs. In this note we prove that a size of a strong clique in a claw-free graph with maximum degree $$\varDelta $$ Δ is at most $$\varDelta ^2 + \frac{1}{2}\varDelta $$ Δ 2 + 1 2 Δ . This result improves the only known result $$1.125\varDelta ^2+\varDelta $$ 1.125 Δ 2 + Δ , which is a bound for the strong chromatic index of claw-free graphs.

In this paper, a construction of $(n,k,\delta)$ LDPC convolutional codes over arbitrary finite fields, which generalizes the work of Robinson and Bernstein and the later work of Tong is provided. The sets of integers forming a $(k,w)$-(weak) difference triangle set are used as supports of some columns of the sliding parity-check matrix of an $(n,k,\delta)$ convolutional code, where $n\in\mathbb{N}$, $n>k$. The parameters of the convolutional code are related to the parameters of the underlying difference triangle set. In particular, a relation between the free distance of the code and $w$ is established as well as a relation between the degree of the code and the scope of the difference triangle set. Moreover, we show that some conditions on the weak difference triangle set ensure that the Tanner graph associated to the sliding parity-check matrix of the convolutional code is free from $2\ell$-cycles not satisfying the full rank condition over any finite field. Finally, we relax these conditions and provide a lower bound on the field size, depending on the parity of $\ell$, that is sufficient to still avoid $2\ell$-cycles. This is important for improving the performance of a code and avoiding the presence of low-weight codewords and absorbing sets., Comment: 22 pages, Extended version of arXiv:2001.07969

For two integers $$k>0$$ k > 0 and $$\ell $$ ℓ , a graph $$G=(V,E)$$ G = ( V , E ) is called $$(k,\ell )$$ ( k , ℓ ) -tight if $$|E|=k|V|-\ell $$ | E | = k | V | - ℓ and $$i_G(X)\le k|X|-\ell $$ i G ( X ) ≤ k | X | - ℓ for each $$X\subseteq V$$ X ⊆ V for which $$i_G(X)\ge 1$$ i G ( X ) ≥ 1 , where $$i_G(X)$$ i G ( X ) denotes the number of induced edges by X. G is called $$(k,\ell )$$ ( k , ℓ ) -redundant if $$G-e$$ G - e has a spanning $$(k,\ell )$$ ( k , ℓ ) -tight subgraph for all $$e\in E$$ e ∈ E . We consider the following augmentation problem. Given a graph $$G=(V,E)$$ G = ( V , E ) that has a $$(k,\ell )$$ ( k , ℓ ) -tight spanning subgraph, find a graph $$H=(V,F)$$ H = ( V , F ) with the minimum number of edges, such that $$G\cup H$$ G ∪ H is $$(k,\ell )$$ ( k , ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is $$(k,\ell )$$ ( k , ℓ ) -tight. For general inputs, we give a polynomial algorithm when $$k\ge \ell $$ k ≥ ℓ and show the NP-hardness of the problem when $$k k < ℓ . Since $$(k,\ell )$$ ( k , ℓ ) -tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

Stochastic combinatorial optimization problems are usually defined as planning problems, which involve purchasing and allocating resources in order to meet uncertain needs. For example, network designers need to make their best guess about the future needs of the network and purchase capabilities accordingly. Facing uncertain in the future, we either “wait and see” changes, or postpone decisions about resource allocation until the requirements or constraints become realized. Specifically, in the field of stochastic combinatorial optimization, some inputs of the problems are uncertain, but follow known probability distributions. Our goal is to find a strategy that minimizes the expected cost. In this paper, we consider the two-stage finite-scenario stochastic set cover problem and the single sink rent-or-buy problem by presenting primal-dual based approximation algorithms for these two problems with approximation ratio $$2\eta $$ and 4.39, respectively, where $$\eta $$ is the maximum frequency of the element of the ground set in the set cover problem.

Nilpotent cellular automata have the simplest possible dynamics: all initial configurations lead in bounded time into the unique fixed point of the system. We investigate nilpotency in the setup of one-dimensional non-uniform cellular automata (NUCA) where different cells may use different local rules. There are infinitely many cells in NUCA but only a finite number of different local rules. Changing the distribution of the local rules in the system may drastically change the dynamics. We prove that if the available local rules are such that every periodic distribution of the rules leads to nilpotent behavior then so do also all eventually periodic distributions. However, in some cases there may be non-periodic distributions that are not nilpotent even if all periodic distributions are nilpotent. We demonstrate such a possibility using aperiodic Wang tile sets. We also investigate temporally periodic points in NUCA. In contrast to classical uniform cellular automata, there are NUCA—even reversible equicontinuous ones—that do not have any temporally periodic points. We prove the undecidability of this property: there is no algorithm to determine if a NUCA with a given finite distribution of local rules has a periodic point.

For a regular language L, let $${{\,\mathrm{Var}\,}}(L)$$ Var ( L ) be the minimal number of nonterminals necessary to generate L by right linear grammars. Moreover, for natural numbers $$k_1,k_2,\ldots ,k_n$$ k 1 , k 2 , … , k n and an n-ary regularity preserving operation f, let $$g_f^{{{\,\mathrm{Var}\,}}}(k_1,k_2,\ldots ,k_n)$$ g f Var ( k 1 , k 2 , … , k n ) be the set of all numbers k such that there are regular languages $$L_1,L_2,\ldots , L_n$$ L 1 , L 2 , … , L n such that $${{\,\mathrm{Var}\,}}(L_i)=k_i$$ Var ( L i ) = k i for $$1\le i\le n$$ 1 ≤ i ≤ n and $${{\,\mathrm{Var}\,}}(f(L_1,L_2,\ldots , L_n))=k$$ Var ( f ( L 1 , L 2 , … , L n ) ) = k . We completely determine the sets $$g_f^{{{\,\mathrm{Var}\,}}}$$ g f Var for the operations reversal, Kleene-closures $$+$$ + and $$*$$ ∗ , and union; and we give partial results for product and intersection.

We study the concept of reversibility in connection with parallel communicating systems of finite automata (PCFA in short). We define the notion of reversibility in the case of PCFA (also covering the non-deterministic case) and discuss the relationship of the reversibility of the systems and the reversibility of its components. We show that a system can be reversible with non-reversible components, and the other way around, the reversibility of the components does not necessarily imply the reversibility of the system as a whole. We also investigate the computational power of deterministic centralized reversible PCFA. We show that these very simple types of PCFA (returning or non-returning) can recognize regular languages which cannot be accepted by reversible (deterministic) finite automata, and that they can even accept languages that are not context-free. We also separate the deterministic and non-deterministic variants in the case of systems with non-returning communication. We show that there are languages accepted by non-deterministic centralized PCFA, which cannot be recognized by any deterministic variant of the same type.

Hepatitis is a therapeutic disorder caused by the inflammation/infection of liver and regarded as the existence of cells causing inflammation in the tissues of body parts. Hepatitis is deliberated as a lethal disease worldwide, especially in developing countries mainly due to contaminated drinking water, unhygienic sanitary conditions and careless blood transfusion. This infection is basically considered as viral infection even though this sort of liver infection can also take place due to autoimmune, toxin, medications, unprotected physical relations, drugs and alcohol. Many approaches of identifying viral hepatitis have been sought so for, which include physical inspection, liver function tests (LFTs), liver surgery (biopsy), imaging studies such as sonogram or CT scan, ultrasound, blood tests, viral serology panel, DNA test, and viral antibody testing. In this article, we propose mathematical analysis of viral hepatitis types using Pythagorean fuzzy soft sets (PFSSs) via TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution), VIKOR (Vise Kriterijumska Optimizacija Kompromisno Resenje) and generalized aggregation operators models and show that all the three techniques render the same optimal choice. We also present a commentary yielding comparison between the three techniques considering their structure of evaluation.

We first introduce the class of bipartite absolute retracts with respect to tree obstructions with at most k leaves. Then, using the theory of homomorphism duality, we show that this class of absolute retracts coincides exactly with the bipartite graphs which admit a $$(k+1)$$ -ary near-unanimity polymorphism. This result mirrors the case for reflexive graphs and generalizes a known result for bipartite graphs which admit a 3-ary near-unanimity polymorphism.

The efficiency of state-of-the-art algorithms for the dueling bandits problem is essentially due to a clever exploitation of (stochastic) transitivity properties of pairwise comparisons: If one arm is likely to beat a second one, which in turn is likely to beat a third one, then the first is also likely to beat the third one. By now, however, there is no way to test the validity of corresponding assumptions, although this would be a key prerequisite to guarantee the meaningfulness of the results produced by an algorithm. In this paper, we investigate the problem of testing different forms of stochastic transitivity in an online manner. We derive lower bounds on the expected sample complexity of any sequential hypothesis testing algorithm for various forms of stochastic transitivity, thereby providing additional motivation to focus on weak stochastic transitivity. To this end, we introduce an algorithmic framework for the dueling bandits problem, in which the statistical validity of weak stochastic transitivity can be tested, either actively or passively, based on a multiple binomial hypothesis test. Moreover, by exploiting a connection between weak stochastic transitivity and graph theory, we suggest an enhancement to further improve the efficiency of the testing algorithm. In the active setting, both variants achieve an expected sample complexity that is optimal up to a logarithmic factor.

We calculate the spectrum of the diagram for each finite 3-transposition group. Such graphs with a given minimum eigenvalue have occurred in the context of compact Griess subalgebras of vertex operator algebras.

We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}$ { ( x , F ( x ) ) : x ∈ F 2 n } where F is a vectorial Boolean function mapping ${\mathbb {F}}_{2^{n}}$ F 2 n to ${\mathbb {F}}_{2^{n}}$ F 2 n .

The critical relations are the building blocks of the relational clone of a relational structure with respect to the relational operations intersection and direct product. In this paper we describe the critical relations of crowns. As a consequence, we obtain that the subpower membership problem for any crown is polynomial-time solvable.

In an earlier paper by three of the present authors and Csaba Schneider, it was shown that, for $$m\ge 2$$ m ≥ 2 , a set of $$m+1$$ m + 1 partitions of a set $$\Omega $$ Ω , any m of which are the minimal non-trivial elements of a Cartesian lattice, either form a Latin square (if $$m=2$$ m = 2 ), or generate a join-semilattice of dimension m associated with a diagonal group over a base group G. In this paper we investigate what happens if we have $$m+r$$ m + r partitions with $$r\ge 2$$ r ≥ 2 , any m of which are minimal elements of a Cartesian lattice. If $$m=2$$ m = 2 , this is just a set of mutually orthogonal Latin squares. We consider the case where all these squares are isotopic to Cayley tables of groups, and give an example to show the groups need not be all isomorphic. For $$m>2$$ m > 2 , things are more restricted. Any $$m+1$$ m + 1 of the partitions generate a join-semilattice admitting a diagonal group over a group G. It may be that the groups are all isomorphic, though we cannot prove this. Under an extra hypothesis, we show that G must be abelian and must have three fixed-point-free automorphisms whose product is the identity. (We describe explicitly all abelian groups having such automorphisms.) Under this hypothesis, the structure gives an orthogonal array, and conversely in some cases. If the group is cyclic of prime order p, then the structure corresponds exactly to an arc of cardinality $$m+r$$ m + r in the $$(m-1)$$ ( m - 1 ) -dimensional projective space over the field with p elements, so all known results about arcs are applicable. More generally, arcs over a finite field of order q give examples where G is the elementary abelian group of order q. These examples can be lifted to non-elementary abelian groups using p-adic techniques.