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Moustapha Diaby

Publications and source records attributed to Moustapha Diaby.

13 recordsLinked to original sources
arXiv2026

A Θ(m^9) ternary minimum-cost network flow LP model of the Assignment Problem polytope with applications to hard combinatorial optimization problems

Background: Combinatorial optimization problems (COPs) are central to Logistics and Supply Chain decision making, yet their NP-hardness prevents exact optimal solutions in reasonable time. Methods: This work addresses that limitation by developing a novel ternary network flow linear programming (LP) model of the assignment problem (AP) polytope. The model is very large scale (with Θ(m^9) variables and Θ(m^8) constraints, where m is the number of assignments). Although not intended to compete with conventional two-dimensional formulations of the AP with respect to solution procedures, it enables hard COPs to be solved exactly as "strict" (integrality requirements-free) LPs through simple transformations of their cost functions. Illustrations are given for the quadratic assignment problem (QAP) and the traveling salesman problem (TSP). Results: Because the proposed LP model is polynomial-sized and there exist polynomial-time algorithms for solving LPs, it affirms "P = NP." A separable substructure of the model shows promise for practical-scale instances due to its suitability for large-scale optimization techniques such as dantzig-Wolfe Decomposition, Column Generation, and Lagrangian Relaxation. The formulation also has greater robutness relative to standard network flow models. Conclusiuons: Overall, tyhe approach provides a systematic , modeling-barrier-free framework for representing NP-complete problems as polynomial-sized LPs, with clear theoretical interest and practical potential for medium to lrage-scale Logistics and other COP-intensive applications.

cs.DS↗
arXiv2024

On modeling NP-Complete problems as polynomial-sized linear programs: Escaping/Side-stepping the "barriers"

In view of the extended formulations (EFs) developments (e.g. "Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf [2015]. Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2"), we focus in this paper on the question of whether it is possible to model an NP-Complete problem as a polynomial-sized linear program. For the sake of simplicity of exposition, the discussions are focused on the TSP. We show that a finding that there exists no polynomial-sized extended formulation of "the TSP polytope" does not (necessarily) imply that it is "impossible" for a polynomial-sized linear program to solve the TSP optimization problem. We show that under appropriate conditions the TSP optimization problem can be solved without recourse to the traditional city-to-city ("travel leg") variables, thereby side-stepping/"escaping from" "the TSP polytope" and hence, the barriers. Some illustrative examples are discussed.

cs.CC↗
arXiv2019

On modeling hard combinatorial optimization problems as linear programs: Refutations of the "unconditional impossibility" claims

There has been a series of developments in the recent literature (by essentially a same "circle" of authors) with the absolute/unconditioned (implicit or explicit) claim that there exists no abstraction of an NP-Complete combinatorial optimization problem in which the defining combinatorial configurations (such as "tours" in the case of the traveling salesman problem (TSP) for example) can be modeled by a polynomial-sized system of linear constraints. The purpose of this paper is to provide general as well as specific refutations for these recent claims.

cs.CC↗
arXiv2016

A O(n^8) X O(n^7) Linear Programming Model of the Traveling Salesman Problem

In this paper, we present a new linear programming (LP) formulation of the Traveling Salesman Problem (TSP). The proposed model has O(n^8) variables and O(n^7) constraints, where n is the number of cities. Our numerical experimentation shows that computational times for the proposed linear program are several orders of magnitude smaller than those for the existing model [3].

cs.DM↗
arXiv2016

On "Exponential Lower Bounds for Polytopes in Combinatorial Optimization" by Fiorini et al. (2015): A Refutation For Models With Disjoint Sets of Descriptive Variables

We provide a numerical refutation of the developments of Fiorini et al. (2015)* for models with disjoint sets of descriptive variables. We also provide an insight into the meaning of the existence of a one-to-one linear map between solutions of such models. *: Fiorini, S., S. Massar, S. Pokutta, H.R. Tiwary, and R. de Wolf (2015). Exponential Lower Bounds for Polytopes in Combinatorial Optimization. Journal of the ACM 62:2, Article No. 17.

cs.CC↗
arXiv2014

The traveling salesman problem: A Linear programming formulation

In this paper, we present a polynomial-sized linear programming formulation of the Traveling Salesman Problem (TSP). The proposed linear program is a network flow-based model. Numerical implementation issues and results are discussed. (The exposition and proofs are much more detailed in an edition which I wrote in collaboration with Dr. M.H. Karwan in 2012-2014 . That edition is available at http://users.business.uconn.edu/mdiaby/P=NPProofPapers/tspPaper.pdf)

cs.CC↗
arXiv2014

Limits to the scope of applicability of extended formulations for LP models of combinatorial optimization problems: A summary

We show that new definitions of the notion of "projection" on which some of the recent "extended formulations" works (such as Kaibel (2011); Fiorini et al. (2011; 2012); Kaibel and Walter (2013); Kaibel and Weltge (2013) for example) have been based can cause those works to over-reach in their conclusions in relating polytopes to one another when the sets of the descriptive variables for those polytopes are disjoint.

cs.CC↗
arXiv2014

A O(n^8) X O(n^7) Linear Programming Model of the Quadratic Assignment Problem

This paper has been withdrawn because Theorem 21 and Corollary 22 are in error; The modeling idea is OK, but it needs 9-dimensional variables instead of the 8-dimensional variables defined in notations 6.9. Examples of the correct model (with 9-index variables) are: (1) Diaby, M., "Linear Programming Formulation of the Set Partitioning Problem," International Journal of Operational Research 8:4 (August 2010) pp. 399-427; (2) Diaby, M., "Linear Programming Formulation of the Vertex Coloring Problem," International Journal of Mathematics in Operational Research 2:3 (May 2010) pp. 259-289; (3) Diaby, M., "The Traveling Salesman Problem: A Linear Programming Formulation," WSEAS Transactions on Mathematics, 6:6 (June 2007) pp. 745-754.

cs.DM↗
arXiv2013

On Limits to the Scope of the Extended Formulations "Barriers"

In this paper, we introduce the notion of augmentation for polytopes and use it to show the error in two presumptions that have been key in arriving at over-reaching/over-scoped claims of "impossibility" in recent extended formulations (EF) developments. One of these presumptions is that: "If Polytopes P and Q are described in the spaces of variables x and y respectively, and there exists a linear map x=Ay between the feasible sets of P and Q, then Q is an EF of P". The other is: "(An augmentation of Polytope A projects to Polytope B) ==> (The external descriptions of A and B are related)". We provide counter-examples to these presumptions, and show that in general: (1) If polytopes can always be arbitrarily augmented for the purpose of establishing EF relations, then the notion of EF becomes degenerate/meaningless in some cases, and that: (2) The statement: "(Polytope B is the projection of an augmentation of Polytope A) ==> (Polytope B is the projection of Polytope A)" is not true in general (although, as we show, the converse statement, "(B is the projection of A) ==> (B is the projection of every augmentation of A)", is true in general). We illustrate some of the ideas using the minimum spanning tree problem, as well as the "lower bounds" developments in Fiorini et al. (2011; 2012), in particular.

cs.DM↗
arXiv2007

A Reply to Hofman On: "Why LP cannot solve large instances of NP-complete problems in polynomial time"

Using an approach that seems to be patterned after that of Yannakakis, Hofman argues that an NP-complete problem cannot be formulated as a polynomial bounded-sized linear programming problem. He then goes on to propose a "construct" that he claims to be a counter-example to recently published linear programming formulations of the Traveling Salesman Problem (TSP) and the Quadratic Assignment Problems (QAP), respectively. In this paper, we show that Hofman's construct is flawed, and provide further proof that his "counter-example" is invalid.

cs.CC↗