scholarly journals Definable inapproximability: new challenges for duplicator

2019 ◽  
Vol 29 (8) ◽  
pp. 1185-1210 ◽  
Author(s):  
Albert Atserias ◽  
Anuj Dawar
Keyword(s):  
Approximate Solution ◽  
Lower Bounds ◽  
Optimization Problems ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Constant Factor ◽  
Point Of View ◽  
Hardness Of Approximation ◽  
Fixed Constant ◽  
Algorithmic Techniques

Abstract We consider the hardness of approximation of optimization problems from the point of view of definability. For many $\textrm{NP}$-hard optimization problems it is known that, unless $\textrm{P} = \textrm{NP} $, no polynomial-time algorithm can give an approximate solution guaranteed to be within a fixed constant factor of the optimum. We show, in several such instances and without any complexity theoretic assumption, that no algorithm that is expressible in fixed-point logic with counting (FPC) can compute an approximate solution. Since important algorithmic techniques for approximation algorithms (such as linear or semidefinite programming) are expressible in FPC, this yields lower bounds on what can be achieved by such methods. The results are established by showing lower bounds on the number of variables required in first-order logic with counting to separate instances with a high optimum from those with a low optimum for fixed-size instances.

scholarly journals Unary Integer Linear Programming with Structural Restrictions

2018 ◽  
Author(s):  
Eduard Eiben ◽  
Robert Ganian ◽  
Dušan Knop ◽  
Sebastian Ordyniak
Keyword(s):  
Linear Programming ◽  
Lower Bounds ◽  
Integer Linear Programming ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Graph Representation ◽  
Input Size ◽  
Variable Domains ◽  
Fixed Constant ◽  
Variable Interactions

Recently a number of algorithmic results have appeared which show the tractability of Integer Linear Programming (ILP) instances under strong restrictions on variable domains and/or coefficients (AAAI 2016, AAAI 2017, IJCAI 2017). In this paper, we target ILPs where neither the variable domains nor the coefficients are restricted by a fixed constant or parameter; instead, we only require that our instances can be encoded in unary. We provide new algorithms and lower bounds for such ILPs by exploiting the structure of their variable interactions, represented as a graph. Our first set of results focuses on solving ILP instances through the use of a graph parameter called clique-width, which can be seen as an extension of treewidth which also captures well-structured dense graphs. In particular, we obtain a polynomial-time algorithm for instances of bounded clique-width whose domain and coefficients are polynomially bounded by the input size, and we complement this positive result by a number of algorithmic lower bounds. Afterwards, we turn our attention to ILPs with acyclic variable interactions. In this setting, we obtain a complexity map for the problem with respect to the graph representation used and restrictions on the encoding.

scholarly journals Graphs and complete intersection toric ideals

2015 ◽  
Vol 14 (09) ◽  
pp. 1540011 ◽  
Author(s):  
I. Bermejo ◽  
I. García-Marco ◽  
E. Reyes
Keyword(s):  
Complete Intersection ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Toric Ideals ◽  
Induced Subgraphs ◽  
Toric Ideal ◽  
Vertex Set ◽  
Combinatorial Point ◽  
Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

Loading Deep Networks Is Hard

1994 ◽  
Vol 6 (5) ◽  
pp. 842-850 ◽  
Author(s):  
Jiří Šíma
Keyword(s):  
Architectural Design ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Network Architectures ◽  
Deep Networks ◽  
General Network ◽  
Loading Problem ◽  
Np Complete ◽  
Connectionist Learning

The loading problem formulated by J. S. Judd seems to be a relevant model for supervised connectionist learning of the feedforward networks from the complexity point of view. It is known that loading general network architectures is NP-complete (intractable) when the (training) tasks are also general. Many strong restrictions on architectural design and/or on the tasks do not help to avoid the intractability of loading. Judd concentrated on the width expanding architectures with constant depth and found a polynomial time algorithm for loading restricted shallow architectures. He suppressed the effect of depth on loading complexity and left as an open prototypical computational problem the loading of easy regular triangular architectures that might capture the crux of depth difficulties. We have proven this problem to be NP-complete. This result does not give much hope for the existence of an efficient algorithm for loading deep networks.

scholarly journals Computing the Partition Function for Perfect Matchings in a Hypergraph

2011 ◽  
Vol 20 (6) ◽  
pp. 815-835 ◽  
Author(s):  
ALEXANDER BARVINOK ◽  
ALEX SAMORODNITSKY
Keyword(s):  
Partition Function ◽  
Polynomial Time ◽  
Pairwise Disjoint ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Perfect Matchings ◽  
Constant Factor ◽  
Higher Dimensional ◽  
Element Set

Given non-negative weightswSon thek-subsetsSof akm-element setV, we consider the sum of the productswS1⋅⋅⋅wSmover all partitionsV=S1∪ ⋅⋅⋅ ∪Sminto pairwise disjointk-subsetsSi. When the weightswSare positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial inmfactor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman–Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.

The cost of a cycle is a square

2002 ◽  
Vol 67 (1) ◽  
pp. 35-60 ◽  
Author(s):  
A. Carbone
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Heuristic Algorithms ◽  
Sequent Calculus ◽  
Propositional Calculus ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Proof Search ◽  
The Cost ◽  
Flow Graphs

AbstractThe logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with (kn+1) lines. In particular, there is a polynomial time algorithm which eliminates cycles from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.

scholarly journals Interchangeability of Relevant Cycles in Graphs

10.37236/1494 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Petra M. Gleiss ◽  
Josef Leydold ◽  
Peter F. Stadler
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper And Lower Bounds ◽  
Cycle Basis ◽  
Cycle Bases

The set ${\cal R}$ of relevant cycles of a graph $G$ is the union of its minimum cycle bases. We introduce a partition of ${\cal R}$ such that each cycle in a class ${\cal W}$ can be expressed as a sum of other cycles in ${\cal W}$ and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class ${\cal W}$. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.

scholarly journals On the Max-Min Fair Stochastic Allocation of Indivisible Goods

2020 ◽  
Vol 34 (02) ◽  
pp. 2070-2078
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita
Keyword(s):  
Approximate Solution ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Indivisible Goods ◽  
Additive Utility ◽  
Ex Ante ◽  
Stochastic Setting

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.

scholarly journals Using Robinson-Foulds Supertrees in Divide-and-Conquer Phylogeny Estimation

2021 ◽  
Author(s):  
Xilin Yu ◽  
Thien Le ◽  
Sarah A. Christensen ◽  
Erin K. Molloy ◽  
Tandy Warnow
Keyword(s):  
Optimization Problems ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Tree Of Life ◽  
Divide And Conquer ◽  
Greedy Heuristic ◽  
Mcmc Methods ◽  
Np Hard ◽  
Phylogeny Estimation ◽  
Source Form

Abstract One of the Grand Challenges in Science is the construction of the Tree of Life , an evolutionary tree containing several million species, spanning all life on earth. However, the construction of the Tree of Life is enormously computationally challenging, as all the current most accurate methods are either heuristics for NP -hard optimization problems or Bayesian MCMC methods that sample from tree space. One of the most promising approaches for improving scalability and accuracy for phylogeny estimation uses divide-and-conquer: a set of species is divided into overlapping subsets, trees are constructed on the subsets, and then merged together using a ``supertree method". Here, we present Exact-RFS-2, the first polynomial-time algorithm to find an optimal supertree of two trees, using the Robinson-Foulds Supertree (RFS) criterion (a major approach in supertree estimation that is related to maximum likelihood supertrees), and we prove that finding the RFS of three input trees is NP -hard. We also present GreedyRFS (a greedy heuristic that operates by repeatedly using Exact-RFS-2 on pairs of trees, until all the trees are merged into a single supertree). We evaluate Exact-RFS-2 and GreedyRFS, and show that they have better accuracy than the current leading heuristic for RFS. Exact-RFS-2 and GreedyRFS are available in open source form on Github at github.com/yuxilin51/GreedyRFS

scholarly journals Stabilizing Weighted Graphs

2020 ◽  
Vol 45 (4) ◽  
pp. 1318-1341
Author(s):  
Zhuan Khye Koh ◽  
Laura Sanità
Keyword(s):  
Approximation Algorithm ◽  
Weighted Graph ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Constant Factor ◽  
Maximum Weight ◽  
Minimum Cardinality ◽  
Matching Games ◽  
Maximum Weight Matching ◽  
Network Bargaining

An edge-weighted graph [Formula: see text] is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching. Stable graphs play an important role in network bargaining games and cooperative matching games, because they characterize instances that admit stable outcomes. We give the first polynomial-time algorithm to find a minimum cardinality subset of vertices whose removal from G yields a stable graph, for any weighted graph G. The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest. In contrast, we show that the problem of finding a minimum cardinality subset of edges whose removal from a weighted graph G yields a stable graph, does not admit any constant-factor approximation algorithm, unless P = NP. In this setting, we develop an O(Δ)-approximation algorithm for the problem, where Δ is the maximum degree of a node in G.

THE BENFORD-NEWCOMB DISTRIBUTION AND UNAMBIGUOUS CONTEXT-FREE LANGUAGES

2008 ◽  
Vol 19 (03) ◽  
pp. 717-727
Author(s):  
BALA RAVIKUMAR
Keyword(s):  
Positive Integer ◽  
Polynomial Time ◽  
Formal Language ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Language L ◽  
Combinatorial Technique ◽  
Pumping Lemma ◽  
Context Free

For a string w ∈ {0,1, 2,…, d-1}*, let vald(w) denote the integer whose base d representation is the string w and let MSDd(x) denote the most significant or the leading digit of a positive integer x when x is written as a base d integer. Hirvensalo and Karhumäki [9] studied the problem of computing the leading digit in the ternary representation of 2x ans showed that this problem has a polynomial time algorithm. In [16], some applications are presented for the problem of computing the leading digit of AB for given inputs A and B. In this paper, we study this problem from a formal language point of view. Formally, we consider the language Lb,d,j = {w|w ∈ {0,1, 2,…, d-1}*, 1 ≤ j ≤ 9, MSDb(dvalb(w))) = j} (and some related classes of languages) and address the question of whether this and some related languages are context-free. Standard pumping lemma proofs seem to be very difficult for these languages. We present a unified and simple combinatorial technique that shows that these languages are not unambiguous context-free languages. The Benford-Newcomb distribution plays a central role in our proofs.

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