Let A be a m n{matrix. Linear and hereditary discrepancy are deened as lindisc(A) := maxfminfkA(p ")k 1 j " 2 ff1; 1g n g jp 2 1; 1] n g herdisc(A) := maxfminfk(a ij) i2m];j2J "k 1 j " 2 ff1; 1g n g jJ n]g: We're investigating connections between the two concepts. Best results known are lindisc(A) 2(1 2 2 n) herdisc(A) for any matrix A and an example of an A satisfying lindisc(A) = 2(1 1 n+1) h...

Many problems of practical interest can be solved using tree search methods because carefully tuned successor ordering heuristics guide the search toward regions of the space that are likely to contain solutions. For some problems, the heuristics often lead directly to a solution| but not always. Limited discrepancy search addresses the problem of what to do when the heuristics fail. Our intuit...

The linear discrepancy of a poset P , denoted ld(P ), is the minimum, over all linear extensions L, of the maximum distance in L between two elements incomparable in P . We prove that ld(P ) ≤ ⌊(3r − 1)/2⌋ when P has width 2. Tanenbaum, Trenk, and Fishburn asked whether this upper bound holds for all posets. We answer this in the negative by giving a randomized construction of bipartite posets ...

We present an improvement to Harvey and Ginsberg’s limited discrepancy search algorithm, which eliminates much of the redundancy in the original, by generating each path from the root to the maximum search depth only once. For a complete binary tree of depth d, this reduces the asymptotic complexity from O(y2”) to O(2”). Th e savings is much less in a partial tree search, or in a heavily pruned...

Many search trees are impractical ly large to explore exhaustively. Recently, techniques like l imi ted discrepancy search have been proposed for improving the chance of f inding a goal in a l imi ted amount of search. Depth-bounded discrepancy search offers such a hope. The motivat ion behind depth-bounded discrepancy search is that branching heuristics are more likely to be wrong at the top o...

Many search trees are impractically large to explore exhaustively. Recently, techniques like limited discrepancy search have been proposed for improving the chance of nding a goal in a limited amount of search. Depth-bounded discrepancy search ooers such a hope. The motivation behind depth-bounded discrepancy search is that branching heuristics are more likely to be wrong at the top of the tree...

It is proven that there is a two-coloring of the first n integers forwhich all arithmetic progressions have discrepancy less than const.n1/4. Thisshows that a 1964 result of K. F. Roth is, up to constants, best possible. Department of Applied Mathematics, Charles University, Malostranské nám. 25,118 00 Praha 1, Czech RepublicE-mail address: [email protected] Courant Insti...