271 research outputs found

    Precedence-constrained scheduling problems parameterized by partial order width

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    Negatively answering a question posed by Mnich and Wiese (Math. Program. 154(1-2):533-562), we show that P2|prec,pj∈{1,2}p_j{\in}\{1,2\}|Cmax⁥C_{\max}, the problem of finding a non-preemptive minimum-makespan schedule for precedence-constrained jobs of lengths 1 and 2 on two parallel identical machines, is W[2]-hard parameterized by the width of the partial order giving the precedence constraints. To this end, we show that Shuffle Product, the problem of deciding whether a given word can be obtained by interleaving the letters of kk other given words, is W[2]-hard parameterized by kk, thus additionally answering a question posed by Rizzi and Vialette (CSR 2013). Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled. Oper. 7(1):75-82), we show that the more general Resource-Constrained Project Scheduling problem is fixed-parameter tractable parameterized by the partial order width combined with the maximum allowed difference between the earliest possible and factual starting time of a job.Comment: 14 pages plus appendi

    Decision Problems on Copying and Shuffling

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    We study decision problems of the form: given a regular or linear context-free language LL, is there a word of a given fixed form in LL, where given fixed forms are based on word operations copy, marked copy, shuffle and their combinations

    Topological Sorting with Regular Constraints

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    We introduce the constrained topological sorting problem (CTS): given a regular language K and a directed acyclic graph G with labeled vertices, determine if G has a topological sort that forms a word in K. This natural problem applies to several settings, e.g., scheduling with costs or verifying concurrent programs. We consider the problem CTS[K] where the target language K is fixed, and study its complexity depending on K. We show that CTS[K] is tractable when K falls in several language families, e.g., unions of monomials, which can be used for pattern matching. However, we show that CTS[K] is NP-hard for K = (ab)^* and introduce a shuffle reduction technique to show hardness for more languages. We also study the special case of the constrained shuffle problem (CSh), where the input graph is a disjoint union of strings, and show that CSh[K] is additionally tractable when K is a group language or a union of district group monomials. We conjecture that a dichotomy should hold on the complexity of CTS[K] or CSh[K] depending on K, and substantiate this by proving a coarser dichotomy under a different problem phrasing which ensures that tractable languages are closed under common operators

    GENROUTE: A genetic algorithm printed wire board (printed wire board (PWB) Router)

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    The major effort of this thesis was to develop an electronic circuit routing system that utilizes genetic algorithms to perform Printed Wire Board (PWB) routing rather than brute force exhaustive searching methods. This problem can be classified as an NP-Hard optimization problem searching a large solution space. Some desirable characteristics of an electronic routing system are that it: Minimize the number of potential solutions Minimize the number of board layers and tap holes Minimize trace lengths and the number of jogs Minimize trace cross-talk and the board capacitance

    Algorithms, Bounds, and Strategies for Entangled XOR Games

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    We study the complexity of computing the commuting-operator value ω∗\omega^* of entangled XOR games with any number of players. We introduce necessary and sufficient criteria for an XOR game to have ω∗=1\omega^* = 1, and use these criteria to derive the following results: 1. An algorithm for symmetric games that decides in polynomial time whether ω∗=1\omega^* = 1 or ω∗<1\omega^* < 1, a task that was not previously known to be decidable, together with a simple tensor-product strategy that achieves value 1 in the former case. The only previous candidate algorithm for this problem was the Navascu\'{e}s-Pironio-Ac\'{i}n (also known as noncommutative Sum of Squares or ncSoS) hierarchy, but no convergence bounds were known. 2. A family of games with three players and with ω∗<1\omega^* < 1, where it takes doubly exponential time for the ncSoS algorithm to witness this (in contrast with our algorithm which runs in polynomial time). 3. A family of games achieving a bias difference 2(ω∗−ω)2(\omega^* - \omega) arbitrarily close to the maximum possible value of 11 (and as a consequence, achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and Vidick. 4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games: that is, we show that there exists a constant CkunsatC_k^{\text{unsat}} depending only on the number kk of players, such that a random kk-XOR game over an alphabet of size nn has ω∗<1\omega^* < 1 with high probability when the number of clauses is above CkunsatnC_k^{\text{unsat}} n. 5. A lower bound of Ω(nlog⁥(n)/log⁥log⁥(n))\Omega(n \log(n)/\log\log(n)) on the number of levels in the ncSoS hierarchy required to detect unsatisfiability for most random 3-XOR games. This is in contrast with the classical case where the nn-th level of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all possible solutions.Comment: 55 page

    Recent results and open problems on CIS Graphs

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